mmcdara

6274 Reputation

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7 years, 324 days

MaplePrimes Activity


These are answers submitted by mmcdara

dsolve does not return a formal solution.
I don't lnow if it is possible to get one but I suggest you to dsolve numerically.

At the end of the attached fie you will find an attempt to solve formally using the option type='series'.
Nevertheless this seems a dead end unless you use a very large number of terms in the series expansion.

 

restart

with(plots):

NULL

xi := .5;

.5

 

.5

 

0.5e-1

 

diff(U(t), t) = .5*U(t)-U(t)*V(t)

 

diff(w(t), t) = U(t)*V(t)-w(t)

 

diff(V(t), t) = 0.5e-1-.5*V(t)+w(t)

 

U(0) = 1

 

w(0) = 0

 

V(0) = 0

(1)

# Solve numerically:

soln := dsolve({eq1, eq2, eq3, ic1, ic2, ic3}, numeric)

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 24, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..54, {(1) = 3, (2) = 3, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = .10095317511683091, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..3, {(1) = 1.0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..3, {(1) = .1, (2) = .1, (3) = .1}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = 0, (2) = 0, (3) = 0}, datatype = integer[8]), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..3, {(1) = 1.0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .5, (2) = 0.5e-1, (3) = .0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = U(t), Y[2] = V(t), Y[3] = w(t)]`; YP[1] := .5*Y[1]-Y[1]*Y[2]; YP[2] := 0.5e-1-.5*Y[2]+Y[3]; YP[3] := Y[1]*Y[2]-Y[3]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = U(t), Y[2] = V(t), Y[3] = w(t)]`; YP[1] := .5*Y[1]-Y[1]*Y[2]; YP[2] := 0.5e-1-.5*Y[2]+Y[3]; YP[3] := Y[1]*Y[2]-Y[3]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..3, {(1) = 0., (2) = 1., (3) = 0.}); _vmap := array( 1 .. 3, [( 1 ) = (1), ( 2 ) = (2), ( 3 ) = (3)  ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) end if; `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch:  end try else try procname(procname("right")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch:  end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error  end try; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; _dat[4][26] := _EnvDSNumericSaveDigits; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [t, U(t), V(t), w(t)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

(2)

pn := plots:-display(
  plots:-odeplot(soln, [t, U(t)], t=0..10, color=red  , legend=typeset(U(t))),
  plots:-odeplot(soln, [t, V(t)], t=0..10, color=blue , legend=typeset(V(t))),
  plots:-odeplot(soln, [t, w(t)], t=0..10, color=green, legend=typeset(w(t)))
):

display(pn)

 

Order := 20:

sols:= dsolve({eq1, eq2, eq3, ic1, ic2, ic3}, {U(t), V(t), w(t)}, type='series'):
sols := map(convert, sols, polynom):

(3)

ps := plots:-display(
  plot(eval(U(t), sols), t=0..10, color=red  , style=point, symbol=circle),
  plot(eval(V(t), sols), t=0..10, color=blue , style=point, symbol=circle),
  plot(eval(w(t), sols), t=0..10, color=green, style=point, symbol=circle)
):

display(pn, ps, view=[default, 0..4])

 
   

Do_It_Numerically.mw

There is no need to build the Rough (Rough-Hurwitz) matrix for a polynomial of order 3.
The conditions for its 3 roots have a strictly negative real parts are very simple.

I propose you do the things this way

restart

with(LinearAlgebra):

J := Matrix(3, 3, [[0, 0, -m+Rstar/m], [xi, -1, m-Rstar/m], [0, 1, -m]])

J := Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = -m+Rstar/m, (2, 1) = xi, (2, 2) = -1, (2, 3) = m-Rstar/m, (3, 1) = 0, (3, 2) = 1, (3, 3) = -m})

(1)

char_poly := CharacteristicPolynomial(J, lambda)

lambda^3-(-1-m)*lambda^2+Rstar*lambda/m-xi*(-m^2+Rstar)/m

(2)

# Coefficients of lambda in char_poly sorted by decreasing powers of lambda

c := [coeffs(char_poly, lambda, 'p')];
sort([p]);
c := c[sort([p], output=permutation)]

[-xi*(-m^2+Rstar)/m, Rstar/m, 1, 1+m]

 

[1, lambda, lambda^2, lambda^3]

 

[-xi*(-m^2+Rstar)/m, Rstar/m, 1+m, 1]

(3)

# Condiitons for char_poly to be a Hurwitz polynomial

HurwitzConditions := { (c >~ 0)[], c[2]*c[3] - c[1]*c[4] > 0};

{0 < 1, 0 < Rstar/m, 0 < -xi*(-m^2+Rstar)/m, 0 < 1+m, 0 < xi*(-m^2+Rstar)/m+Rstar*(1+m)/m}

(4)

# char_poly is a Hurwitz polynomial iif

sol := {solve(HurwitzConditions)}:
print~(%):
 

{xi = -m-1, 0 < m, m^2 < Rstar}

 

{0 < m, 0 < xi, Rstar < m^2, xi*m^2/(xi+m+1) < Rstar}

 

{0 < m, Rstar < xi*m^2/(xi+m+1), xi < -m-1, m^2 < Rstar}

 

{0 < m, xi < 0, m^2 < Rstar, -m-1 < xi}

 

{-1 < m, Rstar < xi*m^2/(xi+m+1), m < 0, xi < 0, -m-1 < xi}

(5)

# Check that random choices of (m, xi, Rstar) which verifie sol[2] lead
# to characteristic polynomials whose roots have a negative real part.

ind := indets(sol):

K    := 1:
sol[K];
V    := {m = rand(0.0 .. 1.)()}:
W    := select(has, eval(sol[K], V), ind):
V    := V union select(type, W, `=`):
W    := select(has, eval(sol[K], V), ind):
V    := V union {Rstar = rand(lhs(W[1])..2*lhs(W[1]))()};

eval(char_poly, V):
evalf([solve(%)]):
Re~(%);

{xi = -m-1, 0 < m, m^2 < Rstar}

 

{Rstar = 0.8681332282e-1, m = .2322382978, xi = -1.232238298}

 

[-0.9917798169e-1, -1.033882335, -0.9917798169e-1]

(6)

# Check that random choices of (m, xi, Rstar) which verifie sol[2] lead
# to characteristic polynomials whose roots have a negative real part.

ind := indets(sol):

K    := 2:
sol[K];
V    := {m = rand(0.0 .. 1.)(), xi = rand(0.0 .. 1.)()}:
W    := select(has, eval(sol[K], %), ind):
V    := V union {Rstar = rand(lhs(W[1])..rhs(W[2]))()};

eval(char_poly, V):
evalf([solve(%)]):
Re~(%);

{0 < m, 0 < xi, Rstar < m^2, xi*m^2/(xi+m+1) < Rstar}

 

{Rstar = 0.3847120558e-2, m = 0.9447763000e-1, xi = 0.6722224402e-1}

 

[-0.1761063369e-1, -1.059256363, -0.1761063369e-1]

(7)

# Check that random choices of (m, xi, Rstar) which verifie sol[3] lead
# to characteristic polynomials whose roots have a negative real part.

ind := indets(sol):

K    := 3:
sol[K];
V    := {m = rand(0.0 .. 1.)()}:
W    := select(has, eval(sol[K], V), ind):
V    := V union {xi = rand(2*rhs(W[3])..rhs(W[3]))()}:
W    := select(has, eval(sol[K], V), ind):
V    := V union {Rstar = rand(lhs(W[1])..rhs(W[2]))()};

eval(char_poly, V):
evalf([solve(%)]):
Re~(%);

{0 < m, Rstar < xi*m^2/(xi+m+1), xi < -m-1, m^2 < Rstar}

 

{Rstar = 0.3401466279e-4, m = 0.5199090505e-2, xi = -1.910165343}

 

[-0.1987352406e-2, -1.001224386, -0.1987352406e-2]

(8)

# Check that random choices of (m, xi, Rstar) which verifie sol[4] lead
# to characteristic polynomials whose roots have a negative real part.

ind := indets(sol):

K    := 4:
sol[K];
V    := {m = rand(0.0 .. 1.)()}:
W    := select(has, eval(sol[K], V), ind):
V    := V union {xi = rand(lhs(W[1])..rhs(W[3]))()}:
W    := select(has, eval(sol[K], V), ind):
V    := V union {Rstar = rand(lhs(W[1])..2*lhs(W[1]))()};

eval(char_poly, V):
evalf([solve(%)]):
Re~(%);

{0 < m, xi < 0, m^2 < Rstar, -m-1 < xi}

 

{Rstar = .8538659689, m = .8412320805, xi = -1.427625121}

 

[-.3490193892, -1.143193302, -.3490193892]

(9)

# Check that random choices of (m, xi, Rstar) which verifie sol[5] lead
# to characteristic polynomials whose roots have a negative real part.

ind := indets(sol):

K    := 5:
sol[K];
V    := {m = rand(-1.0 .. 0.)()};
W    := select(has, eval(sol[K], V), ind);
V    := V union {xi = rand(lhs(W[1])..rhs(W[3]))()};
W    := select(has, eval(sol[K], V), ind);
V    := V union {Rstar = rand(2*rhs(W[1])..rhs(W[1]))()};

eval(char_poly, V):
evalf([solve(%)]):
Re~(%);

{-1 < m, Rstar < xi*m^2/(xi+m+1), m < 0, xi < 0, -m-1 < xi}

 

{m = -0.3278705062e-1}

 

{-.9672129494 < xi, Rstar < 0.1074990688e-2*xi/(xi+.9672129494), xi < 0}

 

{m = -0.3278705062e-1, xi = -.7612353590}

 

{Rstar < -0.3972863848e-2}

 

{Rstar = -0.5790344183e-2, m = -0.3278705062e-1, xi = -.7612353590}

 

[-0.5228045574e-2, -.9567568583, -0.5228045574e-2]

(10)

Download Hurwitz_mmcdara.mw

restart:
lst := [24300, 18907875, 151200, 147000]:

ilst := map(ifactor, lst);
     [   2    5    2     2    3    5     5    3    2         3       3   2]
     [(2)  (3)  (5) , (3)  (5)  (7) , (2)  (3)  (5)  (7), (2) (3) (5) (7) ]

xlst := eval(ilst, ``(2)=``(x))
     [   2    5    2     2    3    5     5    3    2         3       3   2]
     [(x)  (3)  (5) , (3)  (5)  (7) , (x)  (3)  (5)  (7), (x) (3) (5) (7) ]

     

One_way.mw

To understand why this works use lprint:

lprint(ilst[1])
``(2)^2*``(3)^5*``(5)^2

As (2)^p is ``(2)^p `` prevents the evaluation), just replace ``(2) by ``(x) ( ``(x) will preserve the representation (x)^p)

To retrieve the numbers:

expand(xlst)
             [      2                  5         3]
             [6075 x , 18907875, 4725 x , 18375 x ]

Three errors:

  1. You define delta(k) at the top of your worksheet but use delta[k] further on.
  2. You use sum instead of add.
  3. You did some mistake in the indices of Theta in the loop.

restart; _local(gamma, delta, Zeta)

with(PDEtools):

Theta[0] := 1:

for k from 0 to 7 do Theta[k+2] := -(beta*add((k-r+1)*Theta[k-r+1]*(r+1)*Theta[r+1], r = 0 .. k)+beta*add(Theta[k-r]*(r+1)*(r+2)*Theta[r+2], r = 0 .. k)+beta*add(add(Theta[r-m]*(k-r+1)*Theta[k-r+1]/(1+delta[m-1]), m = 0 .. r), r = 0 .. k)+add((k-r+1)*Theta[k-r+1]/(1+delta[r-1]), r = 0 .. k)-2*lambda*Theta[k])/((k+1)*(k+2)) end do

-.1500000000*A^2-.3000000000*Theta[2]-.6500000000*A+.1000000000

 

Error, too many levels of recursion

 

# For k=0 the computation of T2 is responsible of the error.
# When r=0 you invoke Theta[r+2]= Theta[2]=Theta[k+2] (being processed) for k=2.
#
# Check your indices.

for k  from 0 to 7 do
  Theta_Indices_in_T1 := seq([k-r+1, r+1], r = 0 .. k);
  T1 := beta*add((k-r+1)*Theta[k-r+1]*(r+1)*Theta[r+1], r = 0 .. k);
  Theta_Indices_in_T2 := seq([k-r, r+2], r = 0 .. k);
  T2 := beta*add(Theta[k-r]*(r+1)*(r+2)*Theta[r+2], r = 0 .. k);

  T3 := beta*add(add(Theta[r-m]*(k-r+1)*Theta[k-r+1]/(1+delta[m-1]), m = 0 .. r), r = 0 .. k);

  T4 :=add((k-r+1)*Theta[k-r+1]/(1+delta[r-1]), r = 0 .. k)-2*lambda*Theta[k];

  Theta[k+2] := -1/((k+1)*(k+2))*(T1+T2+T3+T4)
end do;

[1, 1]

 

.3*A^2

 

[0, 2]

 

Error, too many levels of recursion

 

 

Download AF_mmcdara.mw

My advice: Verify the expressions of Theta[k+2]

The radius of convergence of function tanh being Pi/2, there necessarily exists a finite range for xx and yy outside of which the Taylor expansion doesn't converge whatever the nuimber of terms you use.

This is reason why your plot (and @dharr's) exhibit sush large values.

More of this, given that the range for yy where h (@dharr's g) has significant variations (obviously has values in (0, 1); along xx the variations of h  are smaller) is 0.5512527188e-1 .. 0.5569069537e-1  (definition given in the file [BL, BR]), you can simplify f by taking only a 6 terms, not the one hundred you manipulate.

All the details are in the attached file.

So, to answeer your question

         "Why Taylor series can not estimate my function in desired interval [-1<x,y<1]?"

         " Because your are not in the range where this expansion converges"

The convergence range is

yy = 0.55291125973999346567e-1 .. 0.55524841270134109157e-1

The xx convergence range is likely larger for the reason I gave below (slow variations in the xx direction). A rough estimation is 

fsolve(g(eval(f0, yy=AT))=0.001) .. fsolve(g(eval(f0, yy=AT))=0.999);
        -3.3962547 .. 0.1126052

 

 

restart

with(Student[MultivariateCalculus]):

f := 35122173917479363738100862234581108137514304171*xx^2*(1/22300745198530623141535718272648361505980416)-2255097230860381206152749351617455809672044745*xx*yy^9*(1/11150372599265311570767859136324180752990208)+2168816628024980374461014350770096009019357665*xx*yy^8*(1/5575186299632655785383929568162090376495104)-17449701902039745490242163912540688306429882361*xx^2*yy*(1/696898287454081973172991196020261297061888)-11540959773500599403794316292492996114189538863*xx^2*yy^2*(1/5575186299632655785383929568162090376495104)+27287439738914744607616926917914225474665410565*xx^2*yy^3*(1/174224571863520493293247799005065324265472)+929769947314964740179937673332890647768037984465*xx^2*yy^4*(1/11150372599265311570767859136324180752990208)-100809382380090436397261413740272360141145204891*xx^2*yy^5*(1/348449143727040986586495598010130648530944)-930314746723434588666177195703059675161177190255*xx^2*yy^6*(1/5575186299632655785383929568162090376495104)+36390552938954376406834468187448925576623439893*xx^2*yy^7*(1/174224571863520493293247799005065324265472)+1872760743346397986120124413411813119412045269675*xx^2*yy^8*(1/22300745198530623141535718272648361505980416)-35643509355104072817665294345590475660747146425*xx^2*yy^9*(1/696898287454081973172991196020261297061888)-125283292999146417157156696376640452081866835*xx^3*(1/1393796574908163946345982392040522594123776)+5011420945327438626354964312196465908094234685*xx^3*yy*(1/11150372599265311570767859136324180752990208)+29341459645317546529685572705520876577051855*xx^3*yy^2*(1/87112285931760246646623899502532662132736)-15637727799880882327290754576104647826715168925*xx^3*yy^3*(1/11150372599265311570767859136324180752990208)-62755544772437504320590342390381422715234113715/89202980794122492566142873090593446023921664-851688199122087410134053760306093104684621525*xx^3*yy^4*(1/696898287454081973172991196020261297061888)+23458516464006675395891679247259419002768896835*xx^3*yy^5*(1/11150372599265311570767859136324180752990208)+39584968580329795728950940517214770307434335*xx^3*yy^6*(1/21778071482940061661655974875633165533184)-20361225581568567923686744589522827658576624955*xx^3*yy^7*(1/11150372599265311570767859136324180752990208)-1174244552874873223035231031480900497934023075*xx^3*yy^8*(1/1393796574908163946345982392040522594123776)+35696532930567486560276536615522532283474689213*yy*(1/2787593149816327892691964784081045188247552)+43423414494451507811145033075147441881593811799*yy^2*(1/22300745198530623141535718272648361505980416)+941109349474535911451616661821106567867537125*xx^3*yy^9*(1/1393796574908163946345982392040522594123776)-48412290717709997717153300332089796247538326265*xx^4*(1/44601490397061246283071436545296723011960832)-56566850002827011453690682806041619180254985625*yy^3*(1/696898287454081973172991196020261297061888)+17196469545705046799299985950707233685621881055*xx^4*yy*(1/1393796574908163946345982392040522594123776)-9551461763890264957289963973620923748598225435*xx^4*yy^2*(1/11150372599265311570767859136324180752990208)-26051472095770585704126329008135447818638784275*xx^4*yy^3*(1/348449143727040986586495598010130648530944)-765302392604646459013613426858243443467023490875*xx^4*yy^4*(1/22300745198530623141535718272648361505980416)+94251624724512021502035994822030873708141367565*xx^4*yy^5*(1/696898287454081973172991196020261297061888)+843981485493394825713526892530506348990296828805*xx^4*yy^6*(1/11150372599265311570767859136324180752990208)-2038600361316622246653155899145012259420048867785*yy^4*(1/44601490397061246283071436545296723011960832)-33218490572036542393092937176469859040906121155*xx^4*yy^7*(1/348449143727040986586495598010130648530944)-1758702445038817232726176779731884586549332868025*xx^4*yy^8*(1/44601490397061246283071436545296723011960832)+211134394987302797546644924545169826774270265159*yy^5*(1/1393796574908163946345982392040522594123776)+31380186488931551370058361496245928395816772575*xx^4*yy^9*(1/1393796574908163946345982392040522594123776)+184838927094446995029201369223921105703104647*xx^5*(1/2787593149816327892691964784081045188247552)+1970986683407627074325019523003479974617451789943*yy^6*(1/22300745198530623141535718272648361505980416)-6817973449093402642853212701104432585928821163*xx^5*yy*(1/22300745198530623141535718272648361505980416)-113510140727511300460098712979462156361337425*xx^5*yy^2*(1/348449143727040986586495598010130648530944)+23570688854853763073042723518782612790921757535*xx^5*yy^3*(1/22300745198530623141535718272648361505980416)+1613038118657167505912389296857854524947676825*xx^5*yy^4*(1/1393796574908163946345982392040522594123776)-44608078263668464626393951292252447406629869273*xx^5*yy^5*(1/22300745198530623141535718272648361505980416)-588774433706353379897742534304221654039246663*xx^5*yy^6*(1/348449143727040986586495598010130648530944)+47950825635610780986659544491454706340397108297*xx^5*yy^7*(1/22300745198530623141535718272648361505980416)+2207379816207475241162406248223006569040862935*xx^5*yy^8*(1/2787593149816327892691964784081045188247552)-77131555128675321096947207038878222843991869993*yy^7*(1/696898287454081973172991196020261297061888)-5023626067733175609651265492842895195168362165*xx^5*yy^9*(1/5575186299632655785383929568162090376495104)+5795161625895678368156852916105373987594511979*xx^6*(1/22300745198530623141535718272648361505980416)-3917684154726736823398471536296978037714283086195*yy^8*(1/89202980794122492566142873090593446023921664)-539977758872163289054492124375185771143918033*xx^6*yy*(1/696898287454081973172991196020261297061888)+782685832362921584689673760969891945953777553*xx^6*yy^2*(1/5575186299632655785383929568162090376495104)+749877940244270735637721966049124917356845885*xx^6*yy^3*(1/174224571863520493293247799005065324265472)+14159347676475748959036290080103848146860867025*xx^6*yy^4*(1/11150372599265311570767859136324180752990208)-2937701213452088192123555543440803264914467299*xx^6*yy^5*(1/348449143727040986586495598010130648530944)-23673134207774883972271882396704370580007933039*xx^6*yy^6*(1/5575186299632655785383929568162090376495104)+1173296429365947392287371443632107462978009165*xx^6*yy^7*(1/174224571863520493293247799005065324265472)+57447439083834576362467553225131370438848237035*xx^6*yy^8*(1/22300745198530623141535718272648361505980416)-1277356081222180962342283013232991241852904465*xx^6*yy^9*(1/696898287454081973172991196020261297061888)-29946355461657315300256240552185966952551471*xx^7*(1/1393796574908163946345982392040522594123776)+998213736763384913910074759047227544847506773*xx^7*yy*(1/11150372599265311570767859136324180752990208)+10578825782023300845453772557509072093336001*xx^7*yy^2*(1/43556142965880123323311949751266331066368)-4303517165264733669855129139552505045324631645*xx^7*yy^3*(1/11150372599265311570767859136324180752990208)+76828297887427851822683521168415270943435162685*yy^9*(1/2787593149816327892691964784081045188247552)-652299342907430898149182084981866414949696905*xx^7*yy^4*(1/696898287454081973172991196020261297061888)+11170081785792631086653879206603595320491089331*xx^7*yy^5*(1/11150372599265311570767859136324180752990208)+116540829629507365267125159526451609264014215*xx^7*yy^6*(1/87112285931760246646623899502532662132736)-14785537121406447202257499440081382142298519099*xx^7*yy^7*(1/11150372599265311570767859136324180752990208)+220816865194317615868568855814620996552449073*xx*(1/5575186299632655785383929568162090376495104)-868641325364973493898126340263842300348545855*xx^7*yy^8*(1/1393796574908163946345982392040522594123776)+216255546256559295251079313253452049445763455*xx^7*yy^9*(1/348449143727040986586495598010130648530944)-4089215965643055747590786827106386135115380275*xx^8*(1/89202980794122492566142873090593446023921664)+1869246621670048362557342074310025153518449965*xx^8*yy*(1/2787593149816327892691964784081045188247552)+18712604797880071317805036942199122521197359575*xx^8*yy^2*(1/22300745198530623141535718272648361505980416)-3479476522267890993628796487849129439635143625*xx^8*yy^3*(1/696898287454081973172991196020261297061888)-206512033439850904054937113093163624192322042825*xx^8*yy^4*(1/44601490397061246283071436545296723011960832)+15350689937843699961175740256400109996121380375*xx^8*yy^5*(1/1393796574908163946345982392040522594123776)+157001869330425518481531763580902779395436599415*xx^8*yy^6*(1/22300745198530623141535718272648361505980416)-6686861200533386632065997818427854246215113305*xx^8*yy^7*(1/696898287454081973172991196020261297061888)-9205355621994819342146712860571987786619361601*xx*yy*(1/44601490397061246283071436545296723011960832)-285743684916570536194588196441080828723328178675*xx^8*yy^8*(1/89202980794122492566142873090593446023921664)+8094790880015327525694605814920739418439287725*xx^8*yy^9*(1/2787593149816327892691964784081045188247552)+30423874459994412977383604476886160940746185*xx^9*(1/5575186299632655785383929568162090376495104)-1197236208181378637639504269592639035279087665*xx^9*yy*(1/44601490397061246283071436545296723011960832)-72716798311978341010558827315982986191821905*xx^9*yy^2*(1/696898287454081973172991196020261297061888)+5138909461003175489938484170634052266819688725*xx^9*yy^3*(1/44601490397061246283071436545296723011960832)+1206817075246069632318716986669541278160772775*xx^9*yy^4*(1/2787593149816327892691964784081045188247552)-12993287722661922638788467553649639108437064835*xx^9*yy^5*(1/44601490397061246283071436545296723011960832)-431284328058774504067793959976795724976545555*xx^9*yy^6*(1/696898287454081973172991196020261297061888)+17639360745426635511855086638766468926126459875*xx^9*yy^7*(1/44601490397061246283071436545296723011960832)-104255809907916433055923335622932126645726549*xx*yy^2*(1/696898287454081973172991196020261297061888)-2146702909675882809503682033933399905335826325*xx^9*yy^9*(1/11150372599265311570767859136324180752990208)+1587967252519403636411870604735180043125989625*xx^9*yy^8*(1/5575186299632655785383929568162090376495104)+27484692689867334306687311759874973819976026005*xx*yy^3*(1/44601490397061246283071436545296723011960832)+1583056855557692418384969876461998197073089695*xx*yy^4*(1/2787593149816327892691964784081045188247552)-36304948749180317956941914133403396762716230691*xx*yy^5*(1/44601490397061246283071436545296723011960832)-590212436135125327923049635849260481403670583*xx*yy^6*(1/696898287454081973172991196020261297061888)+27046038795224386955728969793334632924015008227*xx*yy^7*(1/44601490397061246283071436545296723011960832):

# Preliminaries

f0 := evalf(f):
g  := phi -> .5*(1+tanh(phi)):  # note that 0 <= g(..) <= 1

Tf := (n, p, q) -> mtaylor(f0, [xx=p, yy=q], n);
Tg := (m, n, p, q) -> mtaylor(g(Tf(n, p, q)), [xx=p, yy=q], m);

# examples

Tf(5, 0, 0):
Tg(4, 5, 0, 0):

 

proc (n, p, q) options operator, arrow; mtaylor(f0, [xx = p, yy = q], n) end proc

 

proc (m, n, p, q) options operator, arrow; mtaylor(g(Tf(n, p, q)), [xx = p, yy = q], m) end proc

(1)

# Observe expression g(f0)):

# Observation 1:
#
# As the expression below is numerical equal to 0, and because the min value
# of g(any_function) is 0 too, xx=yy=0 is already extremely far from the
# region where g(..) takes values significantly larger than 0 and lower then 1.


eval(f0, [xx=0, yy=0]);
g(%);
 

-703.5139881

 

0.

(2)

# Observation 2:
#
# Can we get an approximation of where the previous region takes place?
# To do this we search(arbitrarily on the direction <xx, yy=xx> and get

BL := fsolve(eval(g(f0), yy=xx)=0.001);
BR := fsolve(eval(g(f0), yy=xx)=0.999);
plot3d(
  g(f0)
  , xx=BL..BR, yy=BL..BR
  , color = blue
  , style = surface
);

0.5512527188e-1

 

0.5569069537e-1

 

 

# Consider now the taylor expansion of g(f0) around xx = yy = AT
# where AT stands for (BL+BR)/2).
#
# Comparing the values of g(g0) and TG(5, AT, AT) shows a good agreement.


AT := (BL+BR)/2;
eval(g(f0), [xx=AT, yy=AT]);


Tg(5, (BL+BR)/2, (BL+BR)/2):
eval(%, [xx=AT, yy=AT]);

0.5540798362e-1

 

.5005873193

 

.500587183

(3)

# Before computing the Taylor expansion of g(f0), let us
# begin observing how close could be g(f0) and g(Tf(n, AT, AT)).
#
# Taking arbitrarily n=2 already gives a good agreement.
# For more security I will nevertheless use the value n = 3.

plot3d(
  [g(f0),  g(Tf(3, AT, AT))]
  , xx=BL..BR, yy=BL..BR
  , color = [blue, red]
  , style = [surface$2]
);

 

# Finally ask yourself theis simple question:
#  "Can the Taylor expansion of tanh(x) around x=0 be reasonably acute
#   in the range, let's say -4..4, where on considers than tanh(x)
#   takes values in its almost complete rnge of variation 1..1?@
#
# Let us see:

plot(
  [tanh(x), seq(mtaylor(tanh(x), x, k), k=4..16, 4)]
  , x=-4..4
  , color=[blue, red, orange, cyan, green]
  , legend=[typeset(tanh(x)), seq(cat("k=", k), k=4..16, 4)]
  , view=[default, -2..2]
  , gridlines=true
)

 

# The conclusion is obvious: as no Taylor expansion of tanh(x) gives
# an acceptable representation of tanh(x) outside of a limited range
# (here around -1..1), there is absolutely no chance at all that
# a Taylor expansion og g(f0) will give you a reliable representation
# in a quite limited range around yy=AT (as you see the variation of
# g(f0) is far more rapid along yy than along xx).


# Defining the range where tanh(x) has significant variations this way

SignificantRange := fsolve(tanh(x)=-0.999)..fsolve(tanh(x)=0.999);

# one observes this range extends far from 0 for about 2.4 times the radius of
# convergence of its Taylor expansion.
# Converting this radius of convergence in terms of BR and BL then gives:

xi := (Pi/2) / op(2, SignificantRange);

# and then

ConvergenceRange := AT-(AT-BL)*xi .. AT+(BR-AT)*xi;

g35f3 := Tg(35, 3, AT, AT):

plot3d(
  [g(f0), g35f3]
  , xx=ConvergenceRange, yy=ConvergenceRange
  , color = [blue, red]
  , style = [surface$2]
);

-3.8002011672502000318 .. 3.8002011672502000318

 

.41334557242176583540

 

0.55291125973999346567e-1 .. 0.55524841270134109157e-1

 

 

# But if push further out from this range

PossibleRange := AT-(AT-BL)*(xi*1.1) .. AT+(BR-AT)*(xi*1.1);


plot3d(
  [g(f0), g35f3]
  , xx=PossibleRange, yy=PossibleRange
  , color = [blue, red]
  , style = [surface$2]
);

0.55279440209399281223e-1 .. 0.55536527035147520073e-1

 

 

``

Download taylorProblem_mmcdara.mw

Generally speaking things are done in the other way: starting from an ODE/PDE one derive a numerical scheme which is convergent and consistent with this equation (a fex other properties are required).

The only situation where we start from a discrete equaation (or a set of)  and want to buid a continuous ODE/PDE is when you want to answer this question

"I have a continuous ODE/PDE C  and its discrete version D, I know that D doesn't represents exactly C and because of some truncation error I would like to tknow what is the continuous ODE/PDE Ctrue that D represents withe a null truncation error

See here for a Maple example and the references within.
If you are in this framework please let me know.

Otherwise here is a step by step explanation  on how to find a continuous PDE that z11 would discretize:
(do not use "h" and "1" to denote the same space step, firstly it's confusing, and secondly one can get nothing wifh that: use "h" only)

restart

with(LinearAlgebra):

with(PDEtools):

with(Physics):

with(plots):

Physics:-Setup(mathematicalnotation = true);

[mathematicalnotation = true]

(1)

NULL

U := proc (i, t) options operator, arrow; Matrix([[1+I*(q(i+h, t)-q(i, t))/lambda, I*(r(i+h, t)-r(i, t))/lambda], [I*(r(i+h, t)-r(i, t))/lambda, 1-I*(q(i+h, t)-q(i, t))/lambda]]) end proc:

NULL

V := proc (i, t) options operator, arrow; Matrix([[-((1/2)*I)*lambda, -r(i, t)], [r(i, t), ((1/2)*I)*lambda]]) end proc:

 

z := diff(U(i, t), t)+Typesetting:-delayDotProduct(U(i, t), V(i, t))-Typesetting:-delayDotProduct(V(i+h, t), U(i, t)):

z11 := simplify(Physics:-`*`(Physics:-`*`(Physics:-`^`(h, -1), lambda), z[1, 1]), size) = 0;

I*(r(i+h, t)^2-r(i, t)^2+(D[2](q))(i+h, t)-(diff(q(i, t), t)))/h = 0

(2)

R0, NoR0 := selectremove(has, [op(expand(lhs(z11)))], r);
R0, NoR0 := map(add, [R0, NoR0])[]

[I*r(i+h, t)^2/h, -I*r(i, t)^2/h], [I*(D[2](q))(i+h, t)/h, -I*(diff(q(i, t), t))/h]

 

I*r(i+h, t)^2/h-I*r(i, t)^2/h, -I*(diff(q(i, t), t))/h+I*(D[2](q))(i+h, t)/h

(3)

R0 := factor(R0)

I*(r(i+h, t)-r(i, t))*(r(i+h, t)+r(i, t))/h

(4)

# Remark:
#
# The notation f(i+(1/2)*h, t) is commonly ised to represent symbolically the
# mean value of fome function f(x, t) at time t within the interval [i, i+h].
#
# More precisely, in discretization scheme f(i+(1/2)*h, t) is th approximation
# of int(r(x, t), x=i..i+h) got by the mid-point rule.
#
# This strategy, when it comes to a function representing a mass is named
# "mass lumping". This name is used whatever the function f represents
#
# Thus here

MassLumping := r(i+(1/2)*h, t) = ``(r(i+h, t)+r(i, t))/2

r(i+(1/2)*h, t) = (1/2)*``(r(i+h, t)+r(i, t))

(5)

# Generally one starts from the continuous equation and then derive a discretization
# scheme from it.
# For instance, assuming is mass-liming is used, a term such as r(x, t)*diff(r(x, t), x)
# could be discretized as:
r(x, t)*diff(r(x, t), x) = r(i+(1/2)*h, t) . ``((r(i+h, t)-r(i, t)) / h);

r(x, t)*(diff(r(x, t), x)) = r(i+(1/2)*h, t)*``((r(i+h, t)-r(i, t))/h)

(6)

# Now let's try to buid the continuous PDE from its discretzation
# ASSUMING THE MASS-LUMPING STRATEGY IS USED
#
# Step 1 , for convenience set: I/h = C

R0 := algsubs(I/h = C, R0);

(r(i+h, t)-r(i, t))*(r(i+h, t)+r(i, t))*C

(7)

# Set 2:


FromMassLumping := 2*~(rhs=lhs)(value(MassLumping));

R := subsop(2=rhs(FromMassLumping), R0)

``(r(i+h, t)+r(i, t)) = 2*r(i+(1/2)*h, t)

 

2*(r(i+h, t)-r(i, t))*r(i+(1/2)*h, t)*C

(8)

# Set 3: identify the second term in R using a 2nd order Taylor expansion

MyRule := op(2, R) = convert(convert(taylor(op(2, R), h, 2), polynom), diff)

r(i+h, t)-r(i, t) = (diff(r(i, t), i))*h

(9)

# Set 4: use MyRule to rewrite R

R := eval(eval(R, C=I/h), MyRule)

(2*I)*(diff(r(i, t), i))*r(i+(1/2)*h, t)

(10)

# Set 5: use a 1st order expansion of r(i+(1/2)*h, t) as h --> 0

UnlumpedMass := op(-1, R) = convert(taylor(op(-1, R), h, 1), polynom)

r(i+(1/2)*h, t) = r(i, t)

(11)

# Set 6: use a 1st order expansion of r(i+(1/2)*h, t) as h --> 0

R := eval(R, UnlumpedMass)

(2*I)*(diff(r(i, t), i))*r(i, t)

(12)

# Set 7: finally replace "i" by "x"

R := eval(R, i=x)

(2*I)*(diff(r(x, t), x))*r(x, t)

(13)

# Step 8: focus on NoR:
#
# Basically I apply here some of the same steps as above

NoR := algsubs(I/h = C, NoR0);
NoR := eval(NoR, i=x);
NoR := convert(convert(taylor(NoR, h, 2), polynom), diff);
NoR := eval(NoR, C=I/h)

-(diff(q(i, t), t))*C+(D[2](q))(i+h, t)*C

 

-(diff(q(x, t), t))*C+(D[2](q))(x+h, t)*C

 

(diff(q(x, t), x, t))*C*h

 

I*(diff(diff(q(x, t), t), x))

(14)

# Step 9: assembling

ContinuousPDE := R + NoR

(2*I)*(diff(r(x, t), x))*r(x, t)+I*(diff(diff(q(x, t), t), x))

(15)

 

Download ContinuousEquation.mw

I've been stuck several hours until I saw that the expression of eq I copid-pasted from your file 123.mw was WRONG.
You wrote indeed

eq := a^3*b*T^2*(diff(V(xi), `$`(xi, 2)))+a^3*b*T*(diff(V(xi), xi))+3*a^2*b*V(xi)^2-(3*a*c+2*b*omega)*V(xi)

instead of

eq := a^3*b*xi^2*(diff(V(xi), `$`(xi, 2)))+a^3*b*xi*(diff(V(xi), xi))+3*a^2*b*V(xi)^2-(3*a*c+2*b*omega)*V(xi)

!

Once the corrections are done one can get several solutions among them the two solutions given at formula (3.4) in the image you provide:

restart

 

From equation (3.4)
 

`eq (3.4)` := a^3*b*xi^2*(diff(V(xi), `$`(xi, 2)))+a^3*b*xi*(diff(V(xi), xi))+3*a^2*b*V(xi)^2-(3*a*c+2*b*omega)*V(xi)

a^3*b*xi^2*(diff(diff(V(xi), xi), xi))+a^3*b*xi*(diff(V(xi), xi))+3*a^2*b*V(xi)^2-(3*a*c+2*b*omega)*V(xi)

(1)


The "constant V" case:

# The "constant V" case:
# Let K a non null constant

Cst := factor(eval(`eq (3.4)`, V(xi)=K)):

# K being non nul by definition one keeps onlu

Cst := Cst/K:

# Then Cst is null if

isolate(Cst, omega):

# The special case displayed in the excerpt you reproduce corresponds to K=a/3
# (why this choice?).
map(factor, eval(%, K=a/3))

omega = (1/2)*a*(a^2*b-3*c)/b

(2)


The "non constant V" case:

verbose := false:

Guess := xi -> (p[0]+p[1]*xi+p[2]*xi^(2)+p[3]*xi^(3))/(q[0]+q[1]*xi+q[2]*xi^(2)+q[3]*xi^(3))

proc (xi) options operator, arrow; (p[0]+p[1]*xi+p[2]*xi^2+p[3]*xi^3)/(q[0]+q[1]*xi+q[2]*xi^2+q[3]*xi^3) end proc

(3)

# Plug this guess into `eq (3.4)`

f := eval(`eq (3.4)`, V=(xi -> Guess(xi))):

if verbose then print(f): end if:

# Assuming that the denominator doesn't vanish one can concentrate on the numerator of f.
# This numerator is to be seen seen like a polynom with indeterminate xi (9th degree)?
# For it to be identically null whatever xi, it is necessary that all the coefficients
# of this polynom are equal to 0

collect(numer(f), xi):
coefficients := [coeffs(%, xi)]:
if verbose then print(%): end if:

SOL := solve(coefficients):
if verbose then print~({SOL}): end if:

SOL := allvalues([SOL]):

AllPossibleSolutions := NULL:

for s in map(op, {SOL}) do
  try
   eval(Guess(xi), s):
   AllPossibleSolutions := AllPossibleSolutions, simplify(%);
  catch:
  end try
end do:

AllNonConstantSolutions := {AllPossibleSolutions}:                    # eliminate potential multiple occurrences
AllNonConstantSolutions := select(has, AllNonConstantSolutions, xi):  # select non constant guesses
AllNonConstantSolutions := remove(has, AllNonConstantSolutions, I):   # remove possibly complex solutions

print~(AllNonConstantSolutions):

(xi^3*p[3]+xi^2*p[2]+xi*p[1]+p[0])/(xi^3*q[3]+xi^2*q[2]+xi*q[1]+q[0])

 

(1/3)*(-xi^2*p[3]+4*(p[1]*p[3])^(1/2)*xi-p[1])*a/(xi^2*p[3]+2*(p[1]*p[3])^(1/2)*xi+p[1])

 

(1/3)*(xi^2*p[3]+4*(p[1]*p[3])^(1/2)*xi+p[1])*a/(-xi^2*p[3]+2*(p[1]*p[3])^(1/2)*xi-p[1])

 

(1/3)*(6*2^(2/3)*(p[0]^2*p[3])^(2/3)*xi^2+2*p[3]*xi^3*p[0]+9*2^(1/3)*(p[0]^2*p[3])^(1/3)*xi*p[0]+2*p[0]^2)*a/(p[0]*(-2*p[3]*xi^3+3*2^(1/3)*(p[0]^2*p[3])^(1/3)*xi-2*p[0]))

 

-4*p[2]*a*p[1]*xi/(4*xi^2*p[2]^2-4*xi*p[1]*p[2]+p[1]^2)

 

4*p[2]*xi*a^2*q[1]/(4*a^2*q[1]^2+4*a*xi*p[2]*q[1]+xi^2*p[2]^2)

(4)

# Check the last solution
#
# Note the result is obviously not 0
eval(`eq (3.4)`, diff=Diff);
eval(%, V(xi) = AllNonConstantSolutions[-1]);
simplify(value(%));

a^3*b*xi^2*(Diff(Diff(V(xi), xi), xi))+a^3*b*xi*(Diff(V(xi), xi))+3*a^2*b*V(xi)^2-(2*b*omega+3*c*a)*V(xi)

 

a^3*b*xi^2*(Diff(Diff(4*p[2]*xi*a^2*q[1]/(4*a^2*q[1]^2+4*a*xi*p[2]*q[1]+xi^2*p[2]^2), xi), xi))+a^3*b*xi*(Diff(4*p[2]*xi*a^2*q[1]/(4*a^2*q[1]^2+4*a*xi*p[2]*q[1]+xi^2*p[2]^2), xi))+48*a^6*b*p[2]^2*xi^2*q[1]^2/(4*a^2*q[1]^2+4*a*xi*p[2]*q[1]+xi^2*p[2]^2)^2-4*(2*b*omega+3*c*a)*p[2]*xi*a^2*q[1]/(4*a^2*q[1]^2+4*a*xi*p[2]*q[1]+xi^2*p[2]^2)

 

4*a^2*xi*p[2]*q[1]*(b*a^3-2*b*omega-3*c*a)/(4*a^2*q[1]^2+4*a*xi*p[2]*q[1]+xi^2*p[2]^2)

(5)

# A few words
#
# The first solution is generic and not interesting:
#
# The last solution corresponds to the solution given in equation (3.5)
# once done the transformation {a*q[1] -> q[1], p[2] -> q[2]} (remember a <> 0).
# The previous one is also the solution given in equation (3.5)
# under the transformation p[1] --> q[1] and p[2] -->  -q[2]).
#
# I didn't dig into these solutions to understand the meaning of
# AllPossibleSolutions[2..4].
 

# Give a look now to constant solutions

AllConstantSolutions := {AllPossibleSolutions}:                 # eliminate potential multiple occurrences
AllConstantSolutions := remove(has, AllConstantSolutions, xi):  # select non constant guesses
AllConstantSolutions := remove(has, AllConstantSolutions, I):   # remove possibly complex solutions

print~(AllConstantSolutions):

0

 

-(4/3)*a

 

-(1/3)*a

 

(1/3)*(2*b*omega+3*c*a)/(b*a^2)

(6)

# A few words
#
# The first solution is obvious
#
# The last solution corresponds to the constant olution given in equation (3.5)
 

 

Download 123_mmcdara_2.mw

It remains a point which puzzles me look to the text in red in the attached file)

It would be interesting to have more information on the paper you cite.
It looks like the non constant solution is the eigen function of the operator which describes lhs(`eq (3.4)`) and the constant solution is the associated eigen value.

@Aung 

 

I don't understand your "can you check please...look like need to determine.constraints..more effectvely?"  sentence.

The attached file presents a quick analysis of your problem.
I'm convinced that either your model, or your data, the units you use, are not correct.
By the way, what is this model? It looks like a kind of viscoelasticity model.

restart

with(LinearAlgebra):

with(plots):

with(plottools):

with(Statistics):

with(Optimization):

with(CurveFitting):

E__1 := 126*10^9:

varepsilon := [0., 0.1500000000e-2, 0.3000000000e-2, 0.4500000000e-2, 0.6000000000e-2, 0.7500000000e-2, 0.9000000000e-2, 0.1050000000e-1, 0.1200000000e-1, 0.1350000000e-1, 0.1500000000e-1]:

true_strain := [0, .310734, .720339, 1.15819, 1.68079, 2.18927, 2.75424, 3.26271, 3.81356, 4.32203]:

for strain in varepsilon do sigma[strain] := strain/(1.811315924*10^(-11)+.2500000000/(1.010000000*10^10+sum(G[a]*exp(-strain/(strain_rate*tau[a])), a = 1 .. 9))-3.968253968*10^(-12)*nu[12]) end do:

stress := [sigma[0.], sigma[0.1500000000e-2], sigma[0.3000000000e-2], sigma[0.4500000000e-2], sigma[0.6000000000e-2], sigma[0.7500000000e-2], sigma[0.9000000000e-2], sigma[0.1050000000e-1], sigma[0.1200000000e-1], sigma[0.1350000000e-1], sigma[0.1500000000e-1]]:

NULL

``

obj := log[10](add((stress[i]-true_stress[i])^2, i = 1 .. 10)):

indets(obj, name);

{G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8], G[9], tau[1], tau[2], tau[3], tau[4], tau[5], tau[6], tau[7], tau[8], tau[9]}

(1)

constraints := [G[1] <= 0, G[2] <= 0, G[3] <= 0, G[4] <= 0, G[5] <= 0, G[6] <= 0, G[7] <= 0, G[8] <= 0, 1540 <= tau[1], 155260 <= tau[2], 10546880 <= tau[3], 6.000000000*10^8 <= tau[4], 1.080000000*10^10 <= tau[5], 4.040000000*10^11 <= tau[6], 1.100000000*10^13 <= tau[7], 1.080000000*10^14 <= tau[8], 2.040000000*10^16 <= tau[9]];

[G[1] <= 0, G[2] <= 0, G[3] <= 0, G[4] <= 0, G[5] <= 0, G[6] <= 0, G[7] <= 0, G[8] <= 0, 1540 <= tau[1], 155260 <= tau[2], 10546880 <= tau[3], 600000000.0 <= tau[4], 0.1080000000e11 <= tau[5], 0.4040000000e12 <= tau[6], 0.1100000000e14 <= tau[7], 0.1080000000e15 <= tau[8], 0.2040000000e17 <= tau[9]]

(2)

 

A SIMPLE ANALYSIS

 

# Let's tke for instance stress[5]:

s5 := stress[5]

0.6000000000e-2/(0.1700204813e-10+.2500000000/(0.1010000000e11+G[1]*exp(-60./tau[1])+G[2]*exp(-60./tau[2])+G[3]*exp(-60./tau[3])+G[4]*exp(-60./tau[4])+G[5]*exp(-60./tau[5])+G[6]*exp(-60./tau[6])+G[7]*exp(-60./tau[7])+G[8]*exp(-60./tau[8])+G[9]*exp(-60./tau[9])))

(3)

# You impose quite large values for the owar bounds of tau.
# So s5 is almost equal to

s5_1 := eval(s5, {seq(tau[i]=+infinity, i=1..9)})

0.6000000000e-2/(0.1700204813e-10+.2500000000/(0.1010000000e11+1.*G[1]+1.*G[2]+1.*G[3]+1.*G[4]+1.*G[5]+1.*G[6]+1.*G[7]+1.*G[8]+1.*G[9]))

(4)

# Assuming that the Gshave almost the same value g one gets

s5_2 := normal(eval(s5_1, {seq(G[i]=g, i=1..9)}))

0.6000000000e-2*(0.1010000000e11+9.*g)/(.4217206861+0.1530184332e-9*g)

(5)

# s5_2 is singular at

singular(s5_2);
plot(s5_2, g=-1e10..-1e8)

{g = -2756012314.}

 

 

# What is the range of positive values for s5_2?
# As you impose the Gs are negative:
solve(s5_2 > 0) assuming g < 0;
domains := solve({s5_2 > 0, g < 0})

RealRange(-infinity, Open(-2756012314.)), RealRange(Open(-1122222222.), infinity)

 

{g < -2756012314.}, {-1122222222. < g, g < 0.}

(6)

# In the two domains 5_2 is contiously increasing:

normal(diff(s5_2, g))

0.1350000000e-1/(.4217206861+0.1530184332e-9*g)^2

(7)

# The minium value of s5_2 in domains[1] is

s5_2_min_1 := limit(s5_2, g=-infinity);

# a value 10^6 times larger than true_stress[5]

352898659.8

(8)

# The minium value of s5_2 in domains[1] is

s5_2_min_2 := 0;

# So if you expect than s5 canbe close to true_stress[5]=191,
# it seems reasonable to impose that g verifies

domains[2]

0

 

{-1122222222. < g, g < 0.}

(9)

# Note that the value of s5_2 are rapidly much higher than 191 in the
# range of interest.

plot(s5_2, g=lhs(op(1, domains[2]))..0, axis[2]=[mode=log])

 

# When is s5_2 equal to true_stress[5]?

solve({s5_2 = true_stress[5], domains[2][]})

{g = -1122221336.}

(10)

# Now, assuming each tau is large enough for the limit tau -> +oo
# is reasonable, lets define this simplified problem

ObjLimit := eval(obj, {seq(tau[i]=+infinity, i=1..9)}):

# And, all the G[1],.., G[9] only intervene through their sum SG, set:

ObjSimple := algsubs(add(1.*G[i], i=1..9) = SG, ObjLimit):

ObjSimple := simplify(ObjSimple, size);

(0.2262907678e39+0.4481007768e29*SG+0.2218321900e19*SG^2)/(SG+0.2480411083e11)^2

(11)

# Try to minimize ObjSimple without taking any precautions at all:

minimize(ObjSimple, location=true)

1160584793., {[{SG = -0.1009999443e11}, 1160584793.]}

(12)

# Even for its minimizer ObjSimple gets the amazing high value 10^9.
# Might one reason be the crude approximation tau = +oo?
#
# Let's look to the values of exp(-strain/(strain_rate*tau[a])) for the
# constraints you impose to the taus.

exp(-strain/(strain_rate*tau[a]));

TauMin := {1540 = tau[1], 155260 = tau[2], 10546880 = tau[3], 6.000000000*10^8 = tau[4], 1.080000000*10^10 = tau[5], 4.040000000*10^11 = tau[6], 1.100000000*10^13 = tau[7], 1.080000000*10^14 = tau[8], 2.040000000*10^16 = tau[9]};


eval( [seq([seq(exp(-x/(strain_rate*tau[i])), i=1..9)], x in varepsilon)], (rhs=lhs)~(TauMin)):
print~(%):

exp(-150.0000000/tau[a])

 

{1540 = tau[1], 155260 = tau[2], 10546880 = tau[3], 600000000.0 = tau[4], 0.1080000000e11 = tau[5], 0.4040000000e12 = tau[6], 0.1100000000e14 = tau[7], 0.1080000000e15 = tau[8], 0.2040000000e17 = tau[9]}

 

[1., 1., 1., 1., 1., 1., 1., 1., 1.]

 

[.9903070230, .9999033925, .9999985778, .9999999750, .9999999986, 1.000000000, 1.000000000, 1.000000000, 1.000000000]

 

[.9807079997, .9998067944, .9999971556, .9999999500, .9999999972, .9999999999, 1.000000000, 1.000000000, 1.000000000]

 

[.9712020196, .9997102056, .9999957333, .9999999250, .9999999958, .9999999999, 1.000000000, 1.000000000, 1.000000000]

 

[.9617881807, .9996136261, .9999943111, .9999999000, .9999999944, .9999999999, 1.000000000, 1.000000000, 1.000000000]

 

[.9524655899, .9995170560, .9999928889, .9999998750, .9999999931, .9999999998, 1.000000000, 1.000000000, 1.000000000]

 

[.9432333628, .9994204952, .9999914667, .9999998500, .9999999917, .9999999998, 1.000000000, 1.000000000, 1.000000000]

 

[.9340906235, .9993239437, .9999900445, .9999998250, .9999999903, .9999999997, 1.000000000, 1.000000000, 1.000000000]

 

[.9250365045, .9992274015, .9999886223, .9999998000, .9999999889, .9999999997, 1.000000000, 1.000000000, 1.000000000]

 

[.9160701469, .9991308687, .9999872001, .9999997750, .9999999875, .9999999997, 1.000000000, 1.000000000, 1.000000000]

 

[.9071907000, .9990343452, .9999857779, .9999997500, .9999999861, .9999999996, 1.000000000, 1.000000000, 1.000000000]

(13)

# In ObjSimple I used the value "1" instead of the values above.
# As you can see my approximation is not that crude and cannot be responsible
# of the extremely high value of the minimum of ObjSimple.
#
# So the problem is elsewhere, on the side of your model certainly,
# maybe you should also look on the side of the units you use (are they consistent?)

Download 9terms_mmcdara.mw


LAST COMMENT:
As your model is governed by the sum SG = G[1]+..+G[7] when parameters tau[1]..tau[7] are "large enough" (which is your case),this means your model is not identifiable in the sense that an infinity of
7-uples (G[1], .., G[7]) can give the same value of SG.
Note that tau values are not identifiable neither when the (positive) lower bounds they lust verify are "large enough".

Next arrange the result the way it suits you (an example is provided at the end of the attached file):

restart

V := exp(lambda*S) = S^4*a4 + S^3*a3 + S^2*a2 + S*a1 + a0;

V1 := subs(S = 2, V);

V2 := subs(S = 1, V);

V3 := subs(S = 0, V);

V4 := subs(S = -1, V);

V5 := subs(S = -2, V);

exp(lambda*S) = S^4*a4+S^3*a3+S^2*a2+S*a1+a0

 

exp(2*lambda) = a0+2*a1+4*a2+8*a3+16*a4

 

exp(lambda) = a0+a1+a2+a3+a4

 

exp(0) = a0

 

exp(-lambda) = a0-a1+a2-a3+a4

 

exp(-2*lambda) = a0-2*a1+4*a2-8*a3+16*a4

(1)

sol := solve(subs(a0 = 1, {V1, V2, V4, V5}), {a1, a2, a3, a4});

{a1 = (2/3)*exp(lambda)-(2/3)*exp(-lambda)-(1/12)*exp(2*lambda)+(1/12)*exp(-2*lambda), a2 = (2/3)*exp(-lambda)-5/4+(2/3)*exp(lambda)-(1/24)*exp(2*lambda)-(1/24)*exp(-2*lambda), a3 = -(1/12)*exp(-2*lambda)-(1/6)*exp(lambda)+(1/6)*exp(-lambda)+(1/12)*exp(2*lambda), a4 = (1/24)*exp(2*lambda)+1/4-(1/6)*exp(lambda)-(1/6)*exp(-lambda)+(1/24)*exp(-2*lambda)}

(2)

trigsol := convert~(sol, trigh)

{a1 = (4/3)*sinh(lambda)-(1/6)*sinh(2*lambda), a2 = (4/3)*cosh(lambda)-5/4-(1/12)*cosh(2*lambda), a3 = (1/6)*sinh(2*lambda)-(1/3)*sinh(lambda), a4 = (1/12)*cosh(2*lambda)+1/4-(1/3)*cosh(lambda)}

(3)

rel := {
         sinh(2*lambda)=2*sinh(lambda)*cosh(lambda)
         , cos(2*lambda)=cos(lambda)^2+sin(lambda)^2
       }:
map(factor, simplify(trigsol, rel))

{a1 = -(1/3)*sinh(lambda)*(cosh(lambda)-4), a2 = -(1/6)*(cosh(lambda)-1)*(cosh(lambda)-7), a3 = (1/3)*sinh(lambda)*(cosh(lambda)-1), a4 = (1/6)*(cosh(lambda)-1)^2}

(4)

 

Download trigh.mw

The key is to define the rewritting rules rel that will lead you to the desire result.

The first point is that your couple of equations (eq1, eq2) cannot be solved formally.

THIS SECTION IS OBSOLETE GIVEN @Christian Wolinski's remark

I kept it however tio justify what @Christian Wolinski wrote

Having said that the natural approach is to give x a numerical value and solve  (eq1, eq2) numerically, which will provide you a single solution (if any).

I adopted a different way based on the observation that  eq1-eq2 is a polynomial equation which doesn't contain x. Its solution is easily computable, even by hand, and iis of the form y=f(z) where z can be coosed arbitrarily.
Things are a little bit more delicate for eq1-eq2 can be seen as a polynomial of degree 2 and y: then you can get 2 real solutions, a double real solution or 2 conjugate solutions depending on the value of z.  

In the attached file I considered that you would be interested by real solutions only. This constraints implies z is to be choosen in some subdomain of the real line (denoted Omega in the worksheet).
Each of the two relations y=f(z) is aimed to be plugged into equation eq1 (the simplest one). Once done the resulting equation, which depends only on z, is formally solved.

The results are of small direct interest as they are represented by a RootOf of a complex expression.
Using the classical command allvalues to force the evaluations of the roots (if any) of this equation still gives extremely complex, and IMO useless, expressions... but evaluating the output of allvalues as floats give, for each relation  y=f(z), a collection of numeric z-solutions.
The last step is to compute the corresponding y-solutions by applying f to each z-solution.

An attempt to represent the (y, z)-solutions for different values of x in the range 0..1 is also proposed.

xyz.mw

UPDATED ANSWER

The attached file contains a procedure which is aimed to find (I write this way for it seems that a few solutions are missed, likely for numerical reasons) all the solutions (y(x), z(x)) in a square domain 
-L <= y <= L, -L <= z <= L with L strictly positive.

Note that this domain could probably be reduced by ignoring negative y values (I didn't look into this point, but it seems that all solutions verify y > 0... something that is maybe easy to prove).

I only focused here on your claim "I need to find the first 3 or 5 solutions" (in fact this 

restart

eq1 := z*(1/150-2*y)*exp(-2*Pi*x*y)+1/5*((z^2-y^2)*sin(2*Pi*x*z)-2*y*z*cos(2*Pi*x*z))=0

z*(1/150-2*y)*exp(-2*Pi*x*y)+(1/5)*(-y^2+z^2)*sin(2*Pi*x*z)-(2/5)*y*z*cos(2*Pi*x*z) = 0

(1)

eq2 := (z^2-y^2+1/150*y-1)*exp(-2*Pi*x*y)+1/5*((z^2-y^2)*sin(2*Pi*x*z)+2*y*z*cos(2*Pi*x*z))=0

(z^2-y^2+(1/150)*y-1)*exp(-2*Pi*x*y)+(1/5)*(-y^2+z^2)*sin(2*Pi*x*z)+(2/5)*y*z*cos(2*Pi*x*z) = 0

(2)


How do I proceed

(1) Principles

# As no formal solution (y(x), z(x)) can be found a good start is to
# do some graphics, to understand what happens.


Digits := 15:

X := 0.1;   # my arbitrary choice in [0, 1]

L  := 20:
NG := 1000:

pp :=
plots:-implicitplot(
  eval([eq1, eq2], x=X)
  , y=-L..L, z=-L..L
  , color=[red, blue]
  , legend=["eq1=0", "eq2=0"]
  , grid=[NG, NG]
):

.1

(3)

# 10 solutions are likely to be seen here

plots:-display(pp)

 

NC1 := nops~([op(1, pp)])[]-2;  # NC1 red curves
NC2 := nops~([op(2, pp)])[]-2;  # NC2 blue curves

# Matrix representation of each curve

AllCurves := map2(op, -1, [plottools:-getdata(pp)]):

6

 

5

(4)

# Let J the objective function which is equal to 0 iif eq1 and eq2 areboth
# verified (log[10] is an artifact to lessen the range of variation of J)

J := log[10](add(lhs~([eq1, eq2])^~2)):

# Solution search algorithm

Digits := 15:

SignChanges      := NULL:
minima           := NULL:
SolutionsChecked := NULL:

# For each curve eq1=0
for ac in AllCurves[1..NC1] do
  pts1 := convert(ac, listlist):

  # Find the points on this curve where lhs(eq2) changes its sign
  # (points where a red and a blue curve will tangent will be missed)
  StartSign   := signum(eval(eval(lhs(eq2), x=X), [y, z] =~ pts1[1]));
  for i from 2 to numelems(pts1) do
    EndSign := signum(eval(eval(lhs(eq2), x=X), [y, z] =~ pts1[i]));
    if EndSign <> StartSign then
      StartSign   := EndSign;
      SignChanges := SignChanges, [pts1[max(1, i-2)], pts1[i]]
    end if:
  end do:
end do:
# Each couple of points (y, z) within which a sign change occurs defines
# a domain where to search a solution where J is minimal (theoritically null).
for s in [SignChanges] do
  dom := y = (min..max)(map2(op, 1, s)), z = (min..max)(map2(op, 2, s));
  try
    minima := minima, Optimization:-Minimize(eval(J, x=X), dom, optimalitytolerance=1e-8, iterationlimit=1000):
  catch:
    printf("No solution found in domain %a\n", [dom])
  end try:
end do:

# Evaluate eq1 and eq2 at the minima found.
for sol in [minima] do
  SolutionsChecked := SolutionsChecked, eval(eval(lhs~([eq1, eq2]), x=X), sol[2])
end do:
#print~(map2(op, 2, [minima]) implies~ [SolutionsChecked]):

# Due to numerical precision some minima appear to give significantly non nul solution.
# I arbitrarily discard them.
AlmostSolutions := zip((u, v) -> if `and`(is~(abs~(u) <=~ 1e-5)[]) then v end if, [SolutionsChecked], map2(op, 2, [minima])):
print~(AlmostSolutions):

plots:-display(
  pp,
  seq(
    plot([eval([y, z], AlmostSolutions[i])], style=point, symbol=circle, symbolsize=20, color=black),
    i = 1..numelems(AlmostSolutions)
  )
)

[y = HFloat(0.06410558905809292), z = HFloat(-1.0866319889014164)]

 

[y = HFloat(2.4579825552905987), z = HFloat(-11.399638564156827)]

 

[y = HFloat(12.489376121146398), z = HFloat(-12.501557424476289)]

 

[y = HFloat(12.489321824984936), z = HFloat(12.501551512780235)]

 

[y = HFloat(0.04887140292881092), z = HFloat(0.9406897626336884)]

 

[y = HFloat(3.321549126882268), z = HFloat(15.267477229081974)]

 

[y = HFloat(17.500667964088915), z = HFloat(17.499933396854253)]

 

[y = HFloat(7.618566248619008), z = HFloat(-7.465886048874683)]

 

[y = HFloat(7.628942370094447), z = HFloat(7.468027640667959)]

 

[y = HFloat(2.337410641856079), z = HFloat(4.602508208960851)]

 

[y = HFloat(2.4506539384845296), z = HFloat(3.880955998324246)]

 

 


How do I proceed

(2) The solution procedure

SearchSolutions := proc(X, L, {see::boolean:=false, verbose::boolean:=false})
  local NG, pp, NC1, NC2, AllCurves, SignChanges, minima, SolutionsChecked,
        ac, pts1, StartSign, i, EndSign, s, dom, sol, AlmostSolutions:
  
  uses plots:

  NG := 1000:
  pp := implicitplot(
    eval([eq1, eq2], x=X)
    , y=-L..L, z=-L..L
    , color=[red, blue]
    , legend=["eq1=0", "eq2=0"]
    , grid=[NG, NG]
    , title=typeset('x'=X)
  ):

  NC1 := nops~([op(1, pp)])[]-2;
  NC2 := nops~([op(2, pp)])[]-2;

  AllCurves := map2(op, -1, [plottools:-getdata(pp)]):

  SignChanges      := NULL:
  minima           := NULL:
  SolutionsChecked := NULL:


  for ac in AllCurves[1..NC1] do
    pts1        := convert(ac, listlist):
    StartSign   := signum(eval(eval(lhs(eq2), x=X), [y, z] =~ pts1[1]));
    for i from 2 to numelems(pts1) do
      EndSign := signum(eval(eval(lhs(eq2), x=X), [y, z] =~ pts1[i]));
      if EndSign <> StartSign then
        StartSign   := EndSign;
        SignChanges := SignChanges, [pts1[max(1, i-2)], pts1[i]]
      end if:
    end do:
  end do:

  for s in [SignChanges] do
    dom := y = (min..max)(map2(op, 1, s)), z = (min..max)(map2(op, 2, s));
    try
      minima := minima, Optimization:-Minimize(eval(J, x=X), dom, optimalitytolerance=1e-8, iterationlimit=1000):
    catch:
      if verbose then
        printf("X = %3a  No solution found in domain %a\n", X, [dom])
      end if;
    end try:
  end do:


  for sol in [minima] do
    SolutionsChecked := SolutionsChecked, eval(eval(lhs~([eq1, eq2]), x=X), sol[2])
  end do:
  AlmostSolutions := zip((u, v) -> if `and`(is~(abs~(u) <=~ 1e-5)[]) then v end if, [SolutionsChecked], map2(op, 2, [minima])):


  if see then
    print(
      plots:-display(
        pp,
        seq(
          plot([eval([y, z], AlmostSolutions[i])], style=point, symbol=circle, symbolsize=20, color=black),
          i = 1..numelems(AlmostSolutions)
        )
      )
    );
    print():
  end if:

  return AlmostSolutions
end proc:

 

SearchSolutions(0.2, 20, see=true)

 

 

[[y = HFloat(0.13344494780692642), z = HFloat(-1.155368320482175)], [y = HFloat(6.244754521797168), z = HFloat(-6.250785840087044)], [y = HFloat(6.2445978719784225), z = HFloat(6.250768327537615)], [y = HFloat(0.08665645857207374), z = HFloat(0.9089857044776194)], [y = HFloat(18.749999593205256), z = HFloat(-18.74999997792953)], [y = HFloat(2.0522651550046627), z = HFloat(17.6152331403813)], [y = HFloat(18.749999322420216), z = HFloat(18.750000024084212)], [y = HFloat(1.7363259754064275), z = HFloat(-15.280864064058637)], [y = HFloat(16.2500003960081), z = HFloat(-16.250000028937677)], [y = HFloat(16.249999975436403), z = HFloat(16.2500000042193)], [y = HFloat(1.88975713628911), z = HFloat(12.627904702824187)], [y = HFloat(13.750001092082368), z = HFloat(13.749999950711388)], [y = HFloat(8.750330843878837), z = HFloat(-8.74996646521638)], [y = HFloat(11.249981778755416), z = HFloat(-11.25000149528761)], [y = HFloat(11.249980778902769), z = HFloat(11.250001455194527)], [y = HFloat(1.6548260311033265), z = HFloat(7.6319076473128895)], [y = HFloat(8.750335605811678), z = HFloat(8.749967022737897)], [y = HFloat(3.8069703528898535), z = HFloat(-3.732476828019622)], [y = HFloat(3.8167718605611434), z = HFloat(3.734499990404839)]]

(5)

# Sweep the range 0..1 for x and collect all the solutions found in this range

Xs := [seq](0.1..1, 0.1):

Sol_wrt_X := table([ seq(xs = SearchSolutions(xs, 20, verbose=true), xs in Xs) ]):

X =  .4  No solution found in domain [y = 15.6066929420051 .. 15.6457631113523, z = 15.6245382892261 .. 15.6255081599189]
X =  .4  No solution found in domain [y = 11.8689358202139 .. 11.9077186173179, z = 11.8748079235297 .. 11.8760651664658]

X =  .5  No solution found in domain [y = 12.4853581013289 .. 12.5244290234156, z = 12.499626883656 .. 12.5005960016093]

X =  .8  No solution found in domain [y = 16.5565565565565 .. 16.5965965965966, z = 16.5624419179027 .. 16.5629167275484]

X =  .8  No solution found in domain [y = 12.7927927927928 .. 12.8328328328328, z = -12.8127796536036 .. -12.8121695242539]
X =  .8  No solution found in domain [y = 5.3053053053053 .. 5.3453453453453, z = 5.31222489410908 .. 5.31366125682572]

X =  .8  No solution found in domain [y = .314327082446639 .. .356413032179948, z = -2.59865527442221 .. -2.59660936472894]

X = 1.0  No solution found in domain [y = .4604604604604 .. .5005005005005, z = -10.0192011299064 .. -10.0185580322442]
X = 1.0  No solution found in domain [y = 13.2332332332332 .. 13.2732732732732, z = 13.2497897695701 .. 13.2502623826246]

X = 1.0  No solution found in domain [y = 2.23548973487505 .. 2.27305935450115, z = 2.24899474960935 .. 2.25146517002328]

 

# Plot all the solutions found for each value of X in Xs

plots:-display(
  seq(
    plot(
      [seq](eval([y, z], Sol_wrt_X[k][i]), i = 1..numelems(Sol_wrt_X[k]))
      , style=point
      , symbol=solidcircle
      , symbolsize=10
      ,  color=ColorTools:-Color([rand()/10^12, rand()/10^12, rand()/10^12])
      , legend=typeset('x'=k)
      , legendstyle=[location=right]
      , labels=[y, z]
    )
    , k in Xs[2..-1]
  )
  , gridlines=true
  , view=[-0.1..20, -20..20]
  , size=[500, 400]
)

 

 

yz_sol_1.mw

@C_R 

If you don't mind having r and t dimensionless:

r := t -> max(t, 0):
plot(r(t), t=-1..2, useunits = [Unit('s'), Unit(('W')/'m'^2)])

Finding the domains where f(x)^2-g(y)^2  > 0 requires knowing explicitely the expressions of f and g, if not you can simply consider the expression a^2-b^2  > 0 instead.

Examples:

restart

expr := a^2-b^2;

a^2-b^2

(1)

solve(expr >= 0)

{a < b, b < 0}, {a = b, b < 0}, {a = -b, b < 0}, {b < 0, -b < a}, {a = a, b = 0}, {0 < b, a < -b}, {a = -b, 0 < b}, {a = b, 0 < b}, {0 < b, b < a}

(2)

plots:-inequal(expr >= 0, a=-3..3, b=-3..3 );

 

f := u -> u^2:
g := u -> u/(1+u^2):

expr := f(x)^2 - g(y)^2;

solve(expr >= 0)

x^4-y^2/(y^2+1)^2

 

{x < -(-y*(y^2+1))^(1/2)/(y^2+1), y < 0}, {x = -(-y*(y^2+1))^(1/2)/(y^2+1), y < 0}, {x = (-y*(y^2+1))^(1/2)/(y^2+1), y < 0}, {y < 0, (-y*(y^2+1))^(1/2)/(y^2+1) < x}, {x = x, y = 0}, {0 < y, x < -(y*(y^2+1))^(1/2)/(y^2+1)}, {x = -(y*(y^2+1))^(1/2)/(y^2+1), 0 < y}, {x = (y*(y^2+1))^(1/2)/(y^2+1), 0 < y}, {0 < y, (y*(y^2+1))^(1/2)/(y^2+1) < x}

(3)

plots:-inequal(expr >= 0, x=-3..3, y=-3..3 );

 

 

Download inequality.mw

This is an answer to your previous question "why solve gives noolution?" or something like that.

The answer is simple: solve can solve systems of 3 polynomial equations in 3 unknowns if each equation contans relatively low degrees of unwkowns combinations.
Beforetryink to use solve, give a look to your equations: it should be clear to you that yoour system cannot be solved in a formal way, that's it.

I end my worksheet with an advice and an example of a numeric solution.
It's up to you to take inspiration of this, or not and keep asking while  solve cannot solve your equations.

141123_Problem_mmcdara.mw

About your "reformulated" question:

You write: "Instead of solving my original equations, which are convoluted and not in polynomial form, I try to solve for their numerators first (since their numerators are polynomials). Broadly speaking, such solutions should also solve the original non-polynomial system"
I "solved for the numerators" in the attached file and no formal solution (using solve) could be got. 
If you say that the system in your new file can be "solved for the numerators", isn't it  because you have simplified your initial problem in somesense (isn't it the meaning of "no correlation" in the name of your new file?).

You write "They need to be verified":
What does "verifying a formal [thus exact] solution" mean to you? Checking that the roots ofthe numerator do not nullify the denominators?
if it is so do that:

num := numer~ (Eqs);
sol := solve(Eqs, Vars,...):  # or SolveTools:-PolynomialSystem maybe
den := denom~(Eqs):
eval(den, sol); # or eval(den, Vars=~sol) 
                # or seq(eval(den, Vars=~s), s in [sol]
                # or somethingelse depending on the structure of sol 

 

Here is an example where the olution (named sol_solve) has been obtained by using simply solve (no need to use SolveTools:-PolynomialSystem).
You will see at the end of the file that the zeroes of the numerators are also those of the denominators.

141123_Problem_NoCorrelation_mmcdara.mw

T := table([1=1, 2=4, 3=9]):
max(indices(T));
                               3
max(entries(T));
                               9

Note that for more complex forms of the indices the above can give unexpected results

T := table([[1, 3]=4, [3, 2]=5])

max(indices(T))
                               3

# You can use sort to find the largest index corresponding to a given order
ind := indices(T, nolist);
                         [1, 3], [3, 2]
max_index = sort([ind], key=(x -> x[2]))[-1]
                        max_index = [3, 2]


@Carl Love pointed out the incompleteness ogf my reply,
here is a way to get the index of the maximal entrie

T := table([ 1=23, 2=36, 3=14 ]);:
max_index = sort([entries(T)], output=permutation)[-1]
                         max_index = 2



@Carl Love :  using the 'pairs' option seems to do lead to something more complex. Maybe this can be made simpler?

max_index = lhs(sort([indices(T,'pairs')], key=(x -> rhs(x)))[-1]);
                         max_index = 2

I don't see here any advantage in using 'pairs' instead of op(op(T)).

max_index = lhs(sort(op(op(T)), key=(x -> rhs(x)))[-1]);
                         max_index = 2

Your set of equations can be rewritting under the form of a system with 3 equations and 3 unknowns f, q3 and q5.
Details are here

restart

with(PDEtools):

inf := 10:

deltaB := .5:

NULL

OdeSys := {diff(f(xi, eta), eta)-q1(xi, eta) = 0, diff(q1(xi, eta), eta)-q2(xi, eta) = 0, diff(q3(xi, eta), eta)-q4(xi, eta) = 0, diff(q5(xi, eta), eta)-q6(xi, eta) = 0, f(xi, eta)*q6(xi, eta)+(diff(q6(xi, eta), eta)+Nt*(diff(q4(xi, eta), eta))/Nb)/Sc-Kr*q5(xi, eta)-xi*(q1(xi, eta)*(diff(q5(xi, eta), xi))-q6(xi, eta)*(diff(f(xi, eta), xi))) = 0, q2(xi, eta)*f(xi, eta)-xi*(q1(xi, eta)*(diff(q1(xi, eta), xi))-q2(xi, eta)*(diff(f(xi, eta), xi)))-q1(xi, eta)^2+sin(xi)*cos(xi)/xi+(diff(q2(xi, eta), eta))*(1+deltaA)-(deltaA*deltaB*xi*xi)*q2(xi, eta)^2*(diff(q2(xi, eta), eta))-Ma*(q1(xi, eta)-sin(xi)/xi)+Lam*sin(xi)*(q3(xi, eta)+Nm*q3(xi, eta)*q3(xi, eta)-Nr*q5(xi, eta))/xi = 0, (1+4*Rd*(1/3))*(diff(q4(xi, eta), eta))/Pr+Nb*q4(xi, eta)*q6(xi, eta)+f(xi, eta)*q4(xi, eta)+(Ma*Ec*xi*xi)*(q1(xi, eta)-sin(xi)/xi)^2+Nt*q4(xi, eta)^2+((1+deltaA)*Ec*xi*xi)*q2(xi, eta)^2-(1/3)*Ec*deltaA*deltaB*xi^4*q2(xi, eta)^4-xi*(q1(xi, eta)*(diff(q3(xi, eta), xi))-q4(xi, eta)*(diff(f(xi, eta), xi))) = 0}:

numelems(OdeSys)

7

(1)

# Subsystem made of equations 1 to 4


SubSystem_1 := OdeSys[1..4]:

# Rewrite SubSystem_1 as a list of rewritting rules

SubSystem_1bis := map(eq -> -op(2, lhs(eq)) = op(1, lhs(eq)), SubSystem_1):

print();
RewrittingRules := [
                     SubSystem_1bis[1],
                     eval(SubSystem_1bis[2], SubSystem_1bis[1]),
                     SubSystem_1bis[3..4][]
                   ]:
print~(%):

 

q1(xi, eta) = diff(f(xi, eta), eta)

 

q2(xi, eta) = diff(diff(f(xi, eta), eta), eta)

 

q4(xi, eta) = diff(q3(xi, eta), eta)

 

q6(xi, eta) = diff(q5(xi, eta), eta)

(2)

# The remaining subsystem

SubSystem_2 := OdeSys[5..7]:

# The unknowns SubSystem_2 contains

udu := select(has, indets(SubSystem_2, function), eta):
du  := map2(op, 1, select(has, udu, diff)):
u   := remove(has, udu, diff):

Unknowns_2 := du union u

{f(xi, eta), q1(xi, eta), q2(xi, eta), q3(xi, eta), q4(xi, eta), q5(xi, eta), q6(xi, eta)}

(3)

# Rewrite SubSystem_2 according to RewrittingRules

SubSystem_2bis := eval(SubSystem_2, RewrittingRules):

# The unknowns SubSystem_2bis contains

udu := select(has, indets(SubSystem_2bis, function), eta);

S, Unknowns_2bis := selectremove(has, udu, diff):

while S <> {} do
  S, R := selectremove(has, map2(op, 1, S), diff);
  Unknowns_2bis := Unknowns_2bis union R
end do:

Unknowns_2bis;

{diff(diff(diff(f(xi, eta), eta), eta), eta), diff(diff(f(xi, eta), eta), eta), diff(diff(f(xi, eta), eta), xi), diff(diff(q3(xi, eta), eta), eta), diff(diff(q5(xi, eta), eta), eta), diff(f(xi, eta), eta), diff(f(xi, eta), xi), diff(q3(xi, eta), eta), diff(q3(xi, eta), xi), diff(q5(xi, eta), eta), diff(q5(xi, eta), xi), f(xi, eta), q3(xi, eta), q5(xi, eta)}

 

{f(xi, eta), q3(xi, eta), q5(xi, eta)}

(4)

# The remaining subsystem has 3 equations in 3 unknowns Unknowns_2bis
#
# Here are the derivatives it contains this should guide you to define
# a suitable set of boundary conditions

[select(has, udu, diff)[]]:
print~([ seq(select(has, %, u)[], u in Unknowns_2bis) ]):
 

diff(diff(diff(f(xi, eta), eta), eta), eta)

 

diff(diff(f(xi, eta), eta), eta)

 

diff(diff(f(xi, eta), eta), xi)

 

diff(f(xi, eta), eta)

 

diff(f(xi, eta), xi)

 

diff(diff(q3(xi, eta), eta), eta)

 

diff(q3(xi, eta), eta)

 

diff(q3(xi, eta), xi)

 

diff(diff(q5(xi, eta), eta), eta)

 

diff(q5(xi, eta), eta)

 

diff(q5(xi, eta), xi)

(5)

# Provided a solution of SubSystem_2bis augmented with ad hoc BCs has been obtained,
# the expressions of the remaining unknowns

RemainingUnknowns := remove(has, select(has, indets(OdeSys, function), eta), diff)
                     minus
                     Unknowns_2bis;

# Can be buid from the RewrittingRules

print~(RewrittingRules):

{q1(xi, eta), q2(xi, eta), q4(xi, eta), q6(xi, eta)}

 

q1(xi, eta) = diff(f(xi, eta), eta)

 

q2(xi, eta) = diff(diff(f(xi, eta), eta), eta)

 

q4(xi, eta) = diff(q3(xi, eta), eta)

 

q6(xi, eta) = diff(q5(xi, eta), eta)

(6)

 

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