tomleslie

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9 years, 301 days

MaplePrimes Activity


These are answers submitted by tomleslie

is shown in the attached - but I can think of a few others!!

  restart;
  p:=piecewise( t-2*Pi*trunc(t/(2*Pi))<=Pi,
                Pi,
                (2*Pi-t+2*Pi*trunc(t/(2*Pi)))
              );
  plot(p, t=0..20*Pi, size=[1000,200]);

p := piecewise(t-2*Pi*trunc(t/(2*Pi)) <= Pi, Pi, 2*Pi-t+2*Pi*trunc(t/(2*Pi)))

 

 

 

Download perPW.mw

 

Relevant comments are highlighted in the attached

The only thing that bothers me about the explanation in the attached is what is reported when an even doesn't fire! I would have expected some kind of "warning" or NULL answer, rather than returning "t=0, x(t)=0"

If you think about this explanation for long enough, then it should be obvious why reversing the order of events always works! First time through the "offending" loop the 'sol' procedure will been evaluated from t=0 to t=1, so the event at t=0.5 is available. Second time throught the loop the 'sol' procedure has been evaluated from t=0 to t=0.5, so the event at t=0.3 is available - and so on

  restart;

  interface(version);

`Standard Worksheet Interface, Maple 2019.2, Windows 7, November 26 2019 Build ID 1435526`

(1)

  sys := { diff(x(t), t) = 1, x(0) = 0 }:
  evs := [ [x(t)-0.1, none],  [x(t)-0.3, none], [x(t)-0.5, none] ]:
  sol := dsolve(sys, numeric, events=evs);
  plots:-odeplot(sol, [t, x(t)], t=0..0.5, gridlines=true);

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 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([Array(1..4, 1..21, {(1, 1) = 1.0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = 1.0, (1, 8) = undefined, (1, 9) = -.1, (1, 10) = 1.0, (1, 11) = undefined, (1, 12) = undefined, (1, 13) = undefined, (1, 14) = undefined, (1, 15) = undefined, (1, 16) = undefined, (1, 17) = undefined, (1, 18) = undefined, (1, 19) = undefined, (1, 20) = undefined, (1, 21) = undefined, (2, 1) = 2.0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = 1.0, (2, 8) = undefined, (2, 9) = -.3, (2, 10) = 1.0, (2, 11) = undefined, (2, 12) = undefined, (2, 13) = undefined, (2, 14) = undefined, (2, 15) = undefined, (2, 16) = undefined, (2, 17) = undefined, (2, 18) = undefined, (2, 19) = undefined, (2, 20) = undefined, (2, 21) = undefined, (3, 1) = 3.0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = 1.0, (3, 8) = undefined, (3, 9) = -.5, (3, 10) = 1.0, (3, 11) = undefined, (3, 12) = undefined, (3, 13) = undefined, (3, 14) = undefined, (3, 15) = undefined, (3, 16) = undefined, (3, 17) = undefined, (3, 18) = undefined, (3, 19) = undefined, (3, 20) = undefined, (3, 21) = undefined, (4, 1) = 3.0, (4, 2) = .0, (4, 3) = 100.0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (4, 8) = undefined, (4, 9) = undefined, (4, 10) = 0.10e-6, (4, 11) = undefined, (4, 12) = .0, (4, 13) = undefined, (4, 14) = .0, (4, 15) = .0, (4, 16) = undefined, (4, 17) = undefined, (4, 18) = undefined, (4, 19) = undefined, (4, 20) = undefined, (4, 21) = undefined}, datatype = float[8], order = C_order), proc (t, Y, Ypre, n, EA) EA[1, 7+2*n] := Y[1]-.1; EA[1, 8+2*n] := 1; EA[2, 7+2*n] := Y[1]-.3; EA[2, 8+2*n] := 1; EA[3, 7+2*n] := Y[1]-.5; EA[3, 8+2*n] := 1; 0 end proc, proc (e, t, Y, Ypre) return 0 end proc, Array(1..3, 1..2, {(1, 1) = undefined, (1, 2) = undefined, (2, 1) = undefined, (2, 2) = undefined, (3, 1) = undefined, (3, 2) = undefined}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 1, (2) = 1, (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) = 3, (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, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.5047658755841546e-2, (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..1, {(1) = .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..1, {(1) = .1}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8]), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 1.0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..1, {(1, 1) = .0, (2, 0) = .0, (2, 1) = .0, (3, 0) = .0, (3, 1) = .0, (4, 0) = .0, (4, 1) = .0, (5, 0) = .0, (5, 1) = .0, (6, 0) = .0, (6, 1) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = x(t)]`; YP[1] := 1; 0 end proc, -1, 0, 0, 0, 0, proc (t, Y, Ypre, n, EA) EA[1, 7+2*n] := Y[1]-.1; EA[1, 8+2*n] := 1; EA[2, 7+2*n] := Y[1]-.3; EA[2, 8+2*n] := 1; EA[3, 7+2*n] := Y[1]-.5; EA[3, 8+2*n] := 1; 0 end proc, proc (e, t, Y, Ypre) return 0 end proc, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = x(t)]`; YP[1] := 1; 0 end proc, -1, 0, 0, 0, 0, proc (t, Y, Ypre, n, EA) EA[1, 7+2*n] := Y[1]-.1; EA[1, 8+2*n] := 1; EA[2, 7+2*n] := Y[1]-.3; EA[2, 8+2*n] := 1; EA[3, 7+2*n] := Y[1]-.5; EA[3, 8+2*n] := 1; 0 end proc, proc (e, t, Y, Ypre) return 0 end proc, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..1, {(1) = 0.}); _vmap := array( 1 .. 1, [( 1 ) = (1)  ] ); _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); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `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 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 _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; 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]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _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, x(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 := 1; _ndsol := _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

 

 

# times that fired the events
#
# Technically this is not an "initialization".
# In order to obtain the result at t=1, Maple
# has to "run" the solution module "sol" from
# t=0 to t=1. During this process any events
# which are "fired" will be recorded and can
# be accessed
#
  sol(1); # initialization
  sol(eventfired=[1]);
  sol(eventfired=[2]);
  sol(eventfired=[3]);
 

[t = 1., x(t) = HFloat(1.0)]

 

[HFloat(0.1)]

 

[HFloat(0.3)]

 

[HFloat(0.49999999999999994)]

(2)

# Same times computed  within a loop
#
# Does the same as above - the solution has
# still be evaluaetd between t=0 and t=1 and
# any events which are "fired" are recorded
# and can be accessed
#

  for i from 1 to 3 do
      te := op(sol(eventfired=[i]));
  end do;

HFloat(0.1)

 

HFloat(0.3)

 

HFloat(0.49999999999999994)

(3)

# Values of x(t) computed  within a loop
#
# Why are calues for events 2 and 3 wrong ?
  for i from 1 to 3 do
    #
    # First time through this loop the 'sol'
    # procedure has been evaluated above
    # between t=0 and t=1 so all "fired"
    # events can be accessed
    #
      te := op(sol(eventfired=[i]));
    #
    # Now the 'sol' procedure will be
    # re-initialized and re-evaluated
    # between t=0 and t=te, ie t=0 and
    # t=0.1. Since the stopping point for
    # this evaluation is t=0.1, then any
    # evaents which occur after this time
    # will not be accessible in the next
    # iteration of the loop. After all, if
    # you have run the procedure from t=0.0
    # to t=0.1, then you would not expect
    # to fire an eveent at t=0.3 - would you?
    #
      xe := sol(te);
  end do;

HFloat(0.1)

 

[t = HFloat(0.1), x(t) = HFloat(0.1)]

 

HFloat(0.0)

 

[t = HFloat(0.0), x(t) = HFloat(0.0)]

 

HFloat(0.0)

 

[t = HFloat(0.0), x(t) = HFloat(0.0)]

(4)

 


 

Download events.mw

See the attached - Note that you will have to change the file path in the ExcelTools:-Import() to something appropriate for where you have the data stored

restart:
with(DiscreteTransforms):
data := ExcelTools:-Import("C:/Users/TomLeslie/Desktop/rr.xlsx"):
plots:-listplot(abs~(FourierTransform(data[..,2])), axis[1]=[mode=log], size=[1200, 400]);
plots:-listplot(abs~(FourierTransform(data[..,2]))^~2, axis[1]=[mode=log], size=[1200, 400]);

 

 

 


 

Download dfts.mw

which I found too painful to debug using 2-D math input. Using 1D input in worksheet mode (IMO the easiest way to understand/debug) the attached shows the calculation I think you are trying to

  restart;

  pde := ((diff(C(x, t), t) = k*diff(C(x, t), x, x)) assuming (0 < x and x < h__1, 0 < t));
  bc1 := ((C(h__1, t) = C1) assuming (0 < t));
  bc2 := C(x, 0) = C2;
  bc3 := ((D[1](C)(0, t) = 0) assuming (0 <= t));
  ans:=pdsolve([pde, bc1, bc2, bc3]);

diff(C(x, t), t) = k*(diff(diff(C(x, t), x), x))

 

C(h__1, t) = C1

 

C(x, 0) = C2

 

(D[1](C))(0, t) = 0

 

C(x, t) = Sum(-4*cos((1/2)*Pi*csgn(1/h__1)*x*(1+2*n)/h__1)*exp(-(1/4)*k*Pi^2*(1+2*n)^2*t/h__1^2)*(-1)^n*(C1-C2)/((1+2*n)*Pi), n = 0 .. infinity)+C1

(1)

  h__1, h__2 := 7*10^(-6), 250*10^(-6);
  E__1, E__2, nu__1, nu__2 := 1.13*10^9, 130*10^9, 0.32, 0.28;
  `E__1`, `E__2` := E__1/(1 - nu__1), E__2/(1 - nu__2);
  C1 := 8.23*10^(-3);
  C2 := 20.58*10^(-3);
  alpha := beta*(rhs(ans) - C2)/C2:
  kappa__T0 := 0:
  kappa__T:=kappa__T0+6*alpha* (E__1*h__1^2 - E__2*h__2^2)^2/(E__1*E__2*h__1*h__2(h__1 + h__2)) + 4*(h__1 + h__2);
 

7/1000000, 1/4000

 

1130000000., 130000000000, .32, .28

 

1661764706., 0.1805555556e12

 

0.8230000000e-2

 

0.2058000000e-1

 

257/250000+0.7070708340e-1*beta*(Sum(0.1572450838e-1*cos((500000/7)*Pi*x*(1+2*n))*exp(-(250000000000/49)*k*Pi^2*(1+2*n)^2*t)*(-1)^n/(1+2*n), n = 0 .. infinity)-0.1235000000e-1)

(2)

#
# 1D plot of kappa_T with fixed 'x'
#
  plot(eval(kappa__T, [x=5.0, beta=0.078, k=3*10^(-12)]), t = 0 .. 20);
#
# 2D plot of kappa_T  as function of (x,t)
#
  plot3d(eval(kappa__T, [beta=0.078, k=3*10^(-12)]), t = 0 .. 20, x=0..10)

 

 

 


 

Download pdeProb.mw

as in the attached

  restart;
  with(GraphTheory):

  G := SpecialGraphs[DodecahedronGraph]():
  DrawGraph(G);
  A := AdjacencyMatrix(G):
  plots:-sparsematrixplot(A, size=[900,900]);

 

 

  S := [$1..20] =~ StringTools:-Char~(96 +~  [$1..20]):
  plots:-sparsematrixplot(A, axis[1]=[tickmarks=[S]],axis[2]=[tickmarks=[S]], size=[900,900])

 

 

 

 

 

Download ticks.mw

Plotting the roots is simple. However Maple's plot routines use floating point arithmetic: you should read the help at ?plot, computation ) to see how Maple's plot routines handle floating point caculations which (may?) result in complex numbers and "decides" whether or not the imaginary parts arise from "round-off" error (or not)

  restart;
#
# Increase Digits because evaluating roots
# at default setting leads to "small"
# imaginary parts (rounding errors in
# floating point calculation?), which may(?)
# or may not(?) be valid
#
  Digits:=15:
  Sol:=[solve(x^3+(a-3)^3*x^2-a^2*x+a^3=0,x)]:
#
# OP should read the help in the Complex Values
# section at ?plot, computation
#
# Plot all roots on one graph.
#
  plot(Sol, a=0..1, numpoints=1000, axes=boxed, color=[red, green, blue]);
#
# Or individually
#
  plot(Sol[1], a=0..1, numpoints=1000, color=[red]);
  plot(Sol[2], a=0..1, numpoints=1000, color=[green]);
  plot(Sol[3], a=0..1, numpoints=1000, color=[blue]);
 

 

 

 

 

#
# Check the "exact" roots at supplied value of 'a'
# (which should be given as an "exact" (ie rational)
# number
#
  radnormal~(eval(Sol, a=5/100));
  evalf~(%);

[(1/24000)*((8663142827534939+24000*(-26002084130697117)^(1/2))^(2/3)+205379*(8663142827534939+24000*(-26002084130697117)^(1/2))^(1/3)+42181013641)/(8663142827534939+24000*(-26002084130697117)^(1/2))^(1/3), (1/48000)*(I*(8663142827534939+(24000*I)*8667361376899039^(1/2)*3^(1/2))^(2/3)*3^(1/2)-(8663142827534939+(24000*I)*8667361376899039^(1/2)*3^(1/2))^(2/3)-(42181013641*I)*3^(1/2)+410758*(8663142827534939+(24000*I)*8667361376899039^(1/2)*3^(1/2))^(1/3)-42181013641)/(8663142827534939+(24000*I)*8667361376899039^(1/2)*3^(1/2))^(1/3), -(1/48000)*(I*(8663142827534939+(24000*I)*8667361376899039^(1/2)*3^(1/2))^(2/3)*3^(1/2)-(42181013641*I)*3^(1/2)+(8663142827534939+(24000*I)*8667361376899039^(1/2)*3^(1/2))^(2/3)-410758*(8663142827534939+(24000*I)*8667361376899039^(1/2)*3^(1/2))^(1/3)+42181013641)/(8663142827534939+(24000*I)*8667361376899039^(1/2)*3^(1/2))^(1/3)]

 

[25.6724721909119+0.988934040154763e-17*I, -0.225571803210691e-2+0.871120565894926e-13*I, 0.215852712035943e-2-0.829776832684670e-13*I]

(1)

 

 


 

Download plotRoot.mw

 

 

You could just provide all the condtions where a '1' is required as part of an 'or' statement - otherwise '0'

As in the attached

  restart;
  with(LinearAlgebra[Modular]):
  A := Mod( 2,
            Matrix( 3,
                    6,
                    (i,j)-> if   `or`( i<2 and j<3,
                                       i=2 and j>2 and j<5,
                                       i=3
                                     )
                            then 1
                            else 0
                            end if
                  ),
            float[8]
          );

Matrix(3, 6, {(1, 1) = 1.0, (1, 2) = 1.0, (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (1, 6) = 0., (2, 1) = 0., (2, 2) = 0., (2, 3) = 1.0, (2, 4) = 1.0, (2, 5) = 0., (2, 6) = 0., (3, 1) = 1.0, (3, 2) = 1.0, (3, 3) = 1.0, (3, 4) = 1.0, (3, 5) = 1.0, (3, 6) = 1.0})

(1)

 

Download initMat.mw

 

  1. If I download and re-execute your worksheet (using !!!) I see the same effect
  2. It persists on multiple re-executions (using !!!)
  3. However, if I remove output using Evaluate->Remove Output from Worksheet, then re-execute, plot labels become consistent and correct, and stay this way on multiple re-executions (without ever using Evaluate->Remove Output from Worksheet again)
  4. So having used Evaluate->Remove Output from Worksheet once, the problem "goes away" and stays gone
  5. That's pretty weird

For Digits=15, every version of Maple I can check gives the same answer, ie 0.167084168057542.

This begs the obvious questions; which version of Maple were you using, and exactly what was the code? - Please upload the offending worksheet using the big green up-arrow in the Mapleprimes toolbar

Check the attachements below for identical code in

Maple 18
Maple 2015
Maple 2016
Maple 2017
Maple 2018
Maple 2019

restart;

kernelopts(version);
Digits:=15:
phi:=(j,x)->piecewise(j=0,exp(x),1/(j-1)!*Integrate(exp((1-theta)*x)*theta^(j-1),theta=0..1)):
evalf(phi(3,0.01));

`Maple 18.02, X86 64 WINDOWS, Oct 20 2014, Build ID 991181`

 

.167084168057542

(1)

 

Download num18.mw

restart:

kernelopts(version);
Digits:=15:
phi:=(j,x)->piecewise(j=0,exp(x),1/(j-1)!*Integrate(exp((1-theta)*x)*theta^(j-1),theta=0..1)):
evalf(phi(3,0.01));

`Maple 2015.2, X86 64 WINDOWS, Dec 20 2015, Build ID 1097895`

 

.167084168057542

(1)

 

Download num2015.mw

restart;

kernelopts(version);
Digits:=15:
phi:=(j,x)->piecewise(j=0,exp(x),1/(j-1)!*Integrate(exp((1-theta)*x)*theta^(j-1),theta=0..1)):
evalf(phi(3,0.01));

`Maple 2016.2, X86 64 WINDOWS, Jan 13 2017, Build ID 1194701`

 

.167084168057542

(1)

 

Download num2016.mw

restart;

kernelopts(version);
Digits:=15:
phi:=(j,x)->piecewise(j=0,exp(x),1/(j-1)!*Integrate(exp((1-theta)*x)*theta^(j-1),theta=0..1)):
evalf(phi(3,0.01));

`Maple 2017.3, X86 64 WINDOWS, Sep 13 2017, Build ID 1262472`

 

.167084168057542

(1)

 

Download num2017.mw

restart;

kernelopts(version);
Digits:=15:
phi:=(j,x)->piecewise(j=0,exp(x),1/(j-1)!*Integrate(exp((1-theta)*x)*theta^(j-1),theta=0..1)):
evalf(phi(3,0.01));

`Maple 2018.2, X86 64 WINDOWS, Nov 16 2018, Build ID 1362973`

 

.167084168057542

(1)

 

Download num2018.mw

restart;

kernelopts(version);
Digits:=15:
phi:=(j,x)->piecewise(j=0,exp(x),1/(j-1)!*Integrate(exp((1-theta)*x)*theta^(j-1),theta=0..1)):
evalf(phi(3,0.01));

`Maple 18.02, X86 64 WINDOWS, Oct 20 2014, Build ID 991181`

 

.167084168057542

(1)

 

Download num2019.mw

you were trying to achieve something like the attached

  restart;

  eulerexp:= proc( fin::procedure, condin::Vector, h::rational, tmax::integer )
                   local n, j,
                         N:= tmax/h,
                         tab:= Matrix(N, 5);
                   tab[1,..]:= condin;
                   for n from 2 to N do
                       tab[n,..]:=tab[n-1, ..]+h*fin( tab[n-1,..] );
                   od;
                   return tab
             end proc:
  fin:= proc( v::Vector )
            return Vector[row]
                   ( eval
                     ( [ 1, 2*t-4*w+5*x-6*y-z, x, z, t ],
                       [ t=v[1], w=v[2], x=v[3], y=v[4], z=v[5] ]
                     )
                   )
        end proc:
  condin:= <25, 1, 2, 3, 4>:
  h:= 1/10:
  tmax:= 20:
  ans:= eulerexp(fin, condin, h, tmax);

_rtable[18446744074327425142]

(1)

 

 

Download eexp.mw

After making a few syntactic corrections, I managed to persuade fsolve() to come up with nine solutions. (Had no luck with solve() at all - expressions to be solved have *ridiculous" powers in 'a, and 'b' - up to 50 - so I would suggest that symbolic solution is impossible

A warning - the attached runs for about 10minutes on my machine. The time-consuming part is actually settinging up the system, I think as a result of "expression length explosion". The loop containing fsolve() solutions is *reasonably* quick - and could probably be improved if there were any doman restriction on the quantitis 'a' and 'b'

If one looks for more than 10 solutions by changing the value of 'numSols' in the final execution group, then the fsolve() command generates an error. This *looks* like a limit on the number of entries in the set provided to the 'avoid' option, but I can't be sure.

Anyhow for what it is worth check the attached

restart;
n:=11:
M := 2:
Le := 5:
Lb := 2:
L:= 1:
l := 1/2:
Pr := 1:
Pe := 2:
Nt := 1/2:
Nb := 4/5:
F[0]:=0:
F[1]:=l*F[2]:
F[2]:=a:
T[0]:=1:
T[1]:=b:
d:=k->piecewise(k<>0,0,k=0,1):

for k from 0 to n do
    F[k+3]:=-1/(d(k)+(k+1)*(k+2)*F[k+2])*((add(F[k-m]*(m+1)*(m+2)*F[m+2],m=0..k))-(add((k-m+1)*(m+1)*F[k-m+1]*F[m+1],m=0..k))-M*(k+1)*F[k+1])*(factorial(k)/factorial(k+3));
    T[k+2]:=-1*(Pr/(k+1)*(k+2))*((add(F[k-m]*(m+1)*T[m+1],m=0..k))+Nt*(add((k-m+1)*(m+1)*T[k-m+1]*T[m+1],m=0..k))+Nb*(add((k-m+1)*(m+1)*T[k+1]*F[k-m+1],m=0..k)));
 end do:

with(numapprox):
f:=add(F[k]*y^k,k=0..n):
t:=add(T[k]*y^k,k=0..n):
ans:= {{}}:
eqsys:= { limit(pade(diff(f,y),y,[4,4]),y=infinity)=0.,
          limit(pade(t,y,[4,4]),y=infinity)=0.
        }:
#
# NB if numSols is set to 10 or higher, then
# fsolve() will generate an error - seems as
# if one can only "avoid" 10 solutions.
#
# Not sure why!!???
#
numSols:=8:
for i from 1 to numSols do
    sol:= fsolve( eqsys,
                  {a,b},
                  avoid = ans
               );
    ans:= `union`(ans,{sol});
od:
ans;

{{}, {a = -7.075494500, b = .3913361429}, {a = -1.134605000, b = 0.6719248299e-1}, {a = -.3028650866, b = -5.211998208}, {a = -.1827451829, b = -2.346056057}, {a = 0., b = -.2439671224}, {a = .9034417844, b = -1.486927916}, {a = 1.561732683, b = -2.156823308}, {a = 4.604997091, b = -3.085073661}}

(1)

 

Download solveSys.mw

 

  restart;
  L:= [0,3,0,7]:
  x:=[seq(`if`(L[i]>0, L[i]/2, NULL), i =1..numelems(L))];

 

The correct syntax for the command you are trying to implement is

dsolve({dl2,bbet}, f(t), numeric, output=Array([0,100, 200, 300, 400, 500, 600]))

although I doubt

  1. whether this is your only problem
  2. or is even the correct set of options for your purpose

From the error messages you are getting, I suspect that you have an undefined parameter either in the ODEs or BCs - obviously since you refuse to use the big green up-arrow in the Mapleprimes toolbar to upload your worksheet , I cannot check/correct this

Using the 'output=Array()' option will return an array of values for all dependent variables, at the values of the independent variable which you supply - ie at the points 0,100, 200, 300, 400, 500, 600. You cannot use the odeplot() command to plot these points.

I suspect that you just want a solution which can be continuously plotted in the range t=0..600 - in which case you don't want the 'output=Array()'.

Once again, if you could be bothered to use the big green up-arrow in the Mapleprimes toolbar to upload your worksheet, I could demonstrate how this is done.

But if you are determined to make life difficult for yourself..........

See my annotations of your "code" below. NB this is probably a non-exhaustive list, just the  glaringly obvious stupidities

restart;
restart; Digits := 1;

 Why do you restart twice?

You really want to work to a precision of one Digit?

Pr:=0.01:E:=1:a:=0:N:=10:

`&Delta;t`:=0.01:`&Delta;y`:=0.01:

#Discritization scheme

for i from 1 by 1 while i<=N do;  
end:

Above for loop 'ends' without doing anything

for j from 0 by 1 while j<=N do;
end:

Above for loop 'ends' without doing anything

eq1[i, j] := (U[i, j+1]-U[i, j])/`&Delta;t` = (1/2)*Gr*(theta[i, j+1]+theta[i, j])+(1/2)*Gc*(C[i, j+1]+C[i, j])+(U[i-1, j+1]-2.*U[i, j+1]-2.*U[i, j]+U[i+1, j])/(2.*`&Delta;y`)^2-(1/2)*M*(U[i, j+1]+U[i, j]):

eq2[i, j] := (theta[i, j+1]-theta[i, j])/`&Delta;t` = (1/Pr)*(theta[i-1, j+1]-2*theta[i, j+1]+theta[i+1, j+1]+theta[i-1, j]-2*theta[i,j]+theta[i+1,j])/(2.*`&Delta;y`)^2-E*((1/`&Delta;y`)^2*(U[i+1, j]-U[i, j])^2):

eq3[i, j] := (C[i, j+1]-C[i, j])/`&Delta;t` = (1/Sc)*(C[i-1, j+1]-2*C[i, j+1]+C[i+1, j+1]+C[i-1, j]-2*C[i,j]+C[i+1,j])/(2*`&Delta;y`)^2-(K/2)*(C[i, j+1]-C[i, j]):

eq1[], eq2[] and eq3[] will be evaluated once using the exit values from the earlier 'for' loops, which did nothing. Luckily none of eq1[], eq[2] and eq[3] are used for anything subsequently, so I suppose their (incorrect?) definition is irrelevant

end do;

Nothing to end - for loops all ended earlier

Error, reserved word `end` unexpected
end do:

Nothing to end - for loops all ended earlier

Error, reserved word `end` unexpected

# initial conditions
U[i, 0] := 0:
theta[i, 0]:= 0:
C[i, 0] := 0:

 In the above, three statements what do think the value of 'i' actually is?

NULL;

Well there's a statement which really does nothing!

U[0,j]:=exp(a*j*`&Delta;t`):
theta[0,j]:=(j*`&Delta;t`):
C[0,j]:=j*`&Delta;t`:

 In the above, three statements what do think the value of 'j' actually is?

U[N,j]:=0:
theta[N,j]:=0:
C[N,j]:=0:

sys := ([seq])(seq(eq[i, j], j = 0 .. N), i = 1 .. N):
nops(sys);
vars:=indets(sys):
nn := Matrix(N+1, N+1,(i, j)-> U[i-1, j-1]):
##
p:=proc(kk) local U_res,A;
  U_res:=solve(eval(sys,k=kk),vars);
  A:=eval(nn,U_res);
  plots:-matrixplot(A)
end proc;

The procedure 'p()' is defined, but is never called - so why does it exist?

plots:-(U,[M],Y=0..4);

Were you planning on using any specific command from the plots() package?

 

Your PDe is seconfd order in 'x' and first order in 't' - therefore you need three boundary conditions in order to obtain a complete solution.

If I "manufacture" a third boundary condition and set infinity to be something "large" but finite, then I can obtain the rather boring solution in the attached numerically

restart;
Inf:=100;
pde1:=diff(v(x,t),x,x)-diff(v(x,t),t)-v(x,t)-v(x,t)^2/(2*v(x,t)+0.5)=0;
bc1:=v(x,0)=0, v(0,t)=1, v(Inf,t)=0;
sol:=pdsolve( pde1, [bc1], numeric);
sol:-plot3d(v(x,t), x=0..Inf, t=0..10);

100

 

diff(diff(v(x, t), x), x)-(diff(v(x, t), t))-v(x, t)-v(x, t)^2/(2*v(x, t)+.5) = 0

 

v(x, 0) = 0, v(0, t) = 1, v(100, t) = 0

 

_m710005568

 

 

 

Download pdeProb.mw

 

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