tomleslie

7135 Reputation

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10 years, 292 days

MaplePrimes Activity


These are answers submitted by tomleslie

The attached shows the results for your specific problem. It also duplicates the results for the "sample" problem on the Wikipedia page https://en.wikipedia.org/wiki/Euler_method - which illustrates just how bad the Euler method can be!

restart

a := 0; b := 4; alpha := 1.0; Npts := 20; f := proc (t, y) options operator, arrow; exp(3.0-y) end proc; ode := diff(y(t), t) = f(t, y(t)); ic := y(a) = alpha; exact := simplify(dsolve([ode, ic])); odeMethods := proc (func, start, finish, initVal, N) local k1, k2, k3, k4, i, res, h; res := Matrix(N+1, 3); h := evalf((finish-start)/N); res[1, 1] := start; res[1, 2] := initVal; res[1, 3] := initVal; for i from 2 to N+1 do res[i, 2] := res[i-1, 2]+h*func(res[i-1, 1], res[i-1, 2]); k1 := func(res[i-1, 1], res[i-1, 3]); k2 := func(res[i-1, 1]+(1/2)*h, res[i-1, 3]+(1/2)*h*k1); k3 := func(res[i-1, 1]+(1/2)*h, res[i-1, 3]+(1/2)*h*k2); k4 := func(res[i-1, 1]+h, res[i-1, 3]+h*k3); res[i, 3] := res[i-1, 3]+h*((1/6)*k1+(1/3)*k2+(1/3)*k3+(1/6)*k4); res[i, 1] := res[i-1, 1]+h end do; return eval(res) end proc; ans := rk4(f, a, b, alpha, Npts); plots:-display([plot(rhs(exact), t = a .. b, color = red, thickness = 8), plot(ans[() .. (), 1], ans[() .. (), 2], color = blue, style = point, symbol = solidcircle), plot(ans[() .. (), 1], ans[() .. (), 3], color = green, style = point, symbol = solidcircle)], legend = ["Exact", "Euler", "Runge-Kutta4"])

diff(y(t), t) = exp(3.0-y(t))

 

y(t) = 3+ln(t+exp(-2))

 

 

a := 0; b := 4; alpha := 1.0; Npts := 4; f := proc (t, y) options operator, arrow; y end proc; ode := diff(y(t), t) = f(t, y(t)); ic := y(a) = alpha; exact := simplify(dsolve([ode, ic])); ans := rk4(f, a, b, alpha, Npts); plots:-display([plot(rhs(exact), t = a .. b, color = red, thickness = 8), plot(ans[() .. (), 1], ans[() .. (), 2], color = blue, style = point, symbol = solidcircle), plot(ans[() .. (), 1], ans[() .. (), 3], color = green, style = point, symbol = solidcircle)], legend = ["Exact", "Euler", "Runge-Kutta4"])

ode := diff(y(t), t) = y(t)

 

exact := y(t) = exp(t)

 

Matrix(%id = 36893488148175941492)

 

 

 

Download odeMethods.mw

in the attached

restart

      # S(t) = # susceptibles at time t
      # I(t) = # infected at time t

      # R(t) = # recovered/dead/vaccinated at time t

      # S+I+R=N constant (assuming population size is constant over a short time interval)``

deS := diff(S(t), t) = -beta*S(t)*I0(t); deI := diff(I0(t), t) = beta*S(t)*I0(t)-alpha*I0(t); deR := diff(R(t), t) = alpha*I0(t); beta := .2; alpha := .1; F := dsolve([deS, deI, deR, S(0) = .99, I0(0) = 0.1e-1, R(0) = 0], [S(t), I0(t), R(t)], numeric); with(plots); odeplot(F, [[t, S(t)], [t, I0(t)], [t, R(t)]], t = 0 .. 100, numpoints = 100, colour = [blue, red, green], legend = ["S(t)", "I0(t)", "R(t)"]); eval(S(t), F(100))

diff(S(t), t) = -beta*S(t)*I0(t)

 

diff(I0(t), t) = beta*S(t)*I0(t)-alpha*I0(t)

 

diff(R(t), t) = alpha*I0(t)

 

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 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(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) = 37, (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) = 2.159194204929377, (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) = .99, (2) = 0.1e-1, (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) = .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..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), Array(1..6, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = 0, (2) = 0, (3) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..3, {(1) = .99, (2) = 0.1e-1, (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) = -0.19800000000000004e-2, (2) = 0.9800000000000004e-3, (3) = 0.10e-2}, 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] = S(t), Y[2] = I0(t), Y[3] = R(t)]`; YP[1] := -.2*Y[1]*Y[2]; YP[2] := .2*Y[1]*Y[2]-.1*Y[2]; YP[3] := .1*Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 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] = S(t), Y[2] = I0(t), Y[3] = R(t)]`; YP[1] := -.2*Y[1]*Y[2]; YP[2] := .2*Y[1]*Y[2]-.1*Y[2]; YP[3] := .1*Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..3, {(1) = 0., (2) = .99, (3) = 0.1e-1}); _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); _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, S(t), I0(t), R(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

 

 

HFloat(0.20983463862927937)

(1)

F := dsolve([deS, deI, deR, S(0) = .74, I0(0) = 0.1e-1, R(0) = .25], [S(t), I0(t), R(t)], numeric); odeplot(F, [[t, S(t)], [t, I0(t)], [t, R(t)]], t = 0 .. 100, numpoints = 100, colour = [blue, red, green], legend = ["S(t)", "I0(t)", "R(t)"]); eval(S(t), F(100))

 

HFloat(0.35161989045942577)

(2)

ds := proc (p) options operator, arrow; dsolve({deI, deR, deS, I0(0) = 0.1e-1, R(0) = p, S(0) = .99-p, Y(t) = diff(I0(t), t)}, {I0(t), R(t), S(t), Y(t)}, numeric) end proc; G := proc (par) options operator, arrow; `if`(type(par, numeric), eval(Y(t), (ds(par))(0)), 'G(par)') end proc; ans := fsolve(G(p), p); odeplot(ds(ans), [[t, S(t)], [t, I0(t)], [t, R(t)]], t = 0 .. 100, numpoints = 100, colour = [blue, red, green], legend = ["S(t)", "I0(t)", "R(t)"])

.4900000000

 

 

display([seq(odeplot(ds(j), [t, I0(t)], t = 0 .. 100, numpoints = 100, colour = red, legend = "I0(t)"), j = ans-0.2e-1 .. ans+0.2e-1, 0.1e-1)])

 

 

Download SIR.mw

is in the attached

  restart;
#
# The ODE system
#
  odeSys:=[ diff(C(t),t) = F - k5*(C(t)-Am)   - k3*(C(t)-A(t)) - k1*(C(t)-B(t)),
            diff(B(t),t) =   - k1*(B(t)-C(t)) - k3*(B(t)-A(t)) - k2*(B(t)-Am),
            diff(A(t),t) =   - k3*(A(t)-C(t)) - k3*(A(t)-B(t)) - k4*(A(t)-Am)
          ];

[diff(C(t), t) = F-k5*(C(t)-Am)-k3*(C(t)-A(t))-k1*(C(t)-B(t)), diff(B(t), t) = -k1*(B(t)-C(t))-k3*(B(t)-A(t))-k2*(B(t)-Am), diff(A(t), t) = -k3*(A(t)-C(t))-k3*(A(t)-B(t))-k4*(A(t)-Am)]

(1)

#
# As t -> infinity,
#
#  diff(A(t),t) = diff(B(t),t) = diff(C(t),t) = 0
#
# so set the rhs of odeSys = 0 and solve
#
  sol:=solve( rhs~(odeSys), [A(t), B(t), C(t)])[]:
#
# Tidy up the solutions
#
  collect(sol[1], [Am, F]);
  collect(sol[2], [Am, F]);
  collect(sol[3], [Am, F]);

A(t) = Am+(2*k1*k3+k2*k3+k3^2)*F/(2*k1*k2*k3+k1*k2*k4+2*k1*k3*k4+2*k1*k3*k5+k1*k4*k5+k2*k3^2+k2*k3*k4+2*k2*k3*k5+k2*k4*k5+k3^2*k4+k3^2*k5+k3*k4*k5)

 

B(t) = Am+(2*k1*k3+k1*k4+k3^2)*F/(2*k1*k2*k3+k1*k2*k4+2*k1*k3*k4+2*k1*k3*k5+k1*k4*k5+k2*k3^2+k2*k3*k4+2*k2*k3*k5+k2*k4*k5+k3^2*k4+k3^2*k5+k3*k4*k5)

 

C(t) = Am+(2*k1*k3+k1*k4+2*k2*k3+k2*k4+k3^2+k3*k4)*F/(2*k1*k2*k3+k1*k2*k4+2*k1*k3*k4+2*k1*k3*k5+k1*k4*k5+k2*k3^2+k2*k3*k4+2*k2*k3*k5+k2*k4*k5+k3^2*k4+k3^2*k5+k3*k4*k5)

(2)

 

Download heating.mw

Consider the attached, where I have defined the function to be integrated in terms of coordinates a, b, c rather than the (implied cartesian, x, y, z) - together with the excerpt from the help page for SurfaceInt() below (emphasis added)

Finally, an optional fourth argument can be coords=name or coordinates=name. It is the coordinate system in which v is interpreted. The variables in the integrand are then also interpreted in this same coordinate system.

restart;

VectorCalculus:-SurfaceInt( (a^2 + b^2)*c,
                            [a, b, c] = Surface( <1, s, t>,
                                                 s = 0 .. Pi/2,
                                                 t = 0 .. 2*Pi,
                                                 coords = spherical
                                               )
                          );

2*Pi^2*(-1+Pi)

(1)

 

Download SFInt.mw

comments in the attached


 

NULL

restart

f := proc (x, y, z) options operator, arrow; [x^2, x*sin(y)] end proc

proc (x, y, z) options operator, arrow; [x^2, x*sin(y)] end proc

(1)

phi[0] := [1, 2]; print(phi[0](t))

[1, 2]

 

[1, 2]

(2)

N = 5

N = 5

(3)

for k to 4 do phi[k+1] := phi[k]+unapply(int(f(phi[k](u)), u = 0 .. t), t); print(phi[k+1](t)) end do

Error, invalid input: f uses a 2nd argument, y, which is missing

 

``


 

Download Picard.mw

You cannot "force" the values of the independent variable (aka abscissa) which Maple will use in a plot() command - so in your case the value t=0 might never occur!!

Obviously there are various ways to "workaround" this problem - the attached shows one of these.

  delta := t -> piecewise(t = 0, 1, 0);
  plot( [ seq
          ( [j, delta(j)],
            j=-1..1, 0.1
          )
        ],
        axes=boxed,
        style=point,
        symbol=solidcircle,
        symbolsize=20
      );

delta := proc (t) options operator, arrow; piecewise(t = 0, 1, 0) end proc

 

 

 

Download plt.mw

the attached

restart

NULL

lambda1 := -(3*(-6.003*(1+(1/2)*x^2)^3+15.400672*(1+(1/2)*x^2)^2-8.745236204-7.393580208*x^2))/(20*(1+(1/2)*x^2)^4)

-(3/20)*(-6.003*(1+(1/2)*x^2)^3+15.400672*(1+(1/2)*x^2)^2-8.745236204-7.393580208*x^2)/(1+(1/2)*x^2)^4

(1)

lambda := x -> -1/20*(3*(-1)*6.003*(1 + 1/2*x^2)^3 + 3*15.400672*(1 + 1/2*x^2)^2 + 3*(-8.745236204) + 3*(-1)*7.393580208*x^2)/(1 + 1/2*x^2)^4

proc (x) options operator, arrow; -(1/20)*(-18.009*(1+(1/2)*x^2)^3+46.202016*(1+(1/2)*x^2)^2-26.23570861-22.18074062*x^2)/(1+(1/2)*x^2)^4 end proc

(2)

data := Matrix([seq([lambda(j), j], j = 0 .. 1, .1)])

_rtable[18446744074401277942]

(3)

#
# Generate a polynomial fit and convert this fit
# to a function
#
  c:= unapply
      ( CurveFitting:-PolynomialInterpolation
                      ( data,
                        lambda
                      ),
        lambda
      );

proc (lambda) options operator, arrow; -0.6298938178e12*lambda^10-0.1443095781e12*lambda^9-7314620066.*lambda^8+489000147.0*lambda^7+45718341.60*lambda^6-73451.8926*lambda^5-68414.68490*lambda^4-508.640042*lambda^3+39.75513129*lambda^2+5.699824301*lambda+.6702161828 end proc

(4)

#
# Plot the data used for the fit and the fit itself
#
  plots:-display
         ( [ plot
             ( data,
               style=point,
               symbol=solidcircle,
               symbolsize=16,
               color=black
             ),
             plot
             ( c(lambda),
               lambda=min(data[..,1])..max(data[..,1]),
               color=red
             )
           ],
           labels=[lambda, x]
         );

 

 

Download cFit.mw

 

need to supply the quantities being plotted as a list - ie enclosed in '[]'. See the attached.

By the way the second plot is not very interesting becuase the expressions being potted (ie 'eig') do not depend on the plotting variable beta2.

restart

with(LinearAlgebra); A := Matrix(6, 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) = `&epsilon;`*`&beta;hh`*Mh/`&mu;h`, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (3, 5) = 0, (3, 6) = `&epsilon;`*omega*`&beta;hh`*Mh/`&mu;h`, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 0, (4, 5) = 0, (4, 6) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = 0, (5, 6) = 0, (6, 1) = 0, (6, 2) = `&epsilon;`*beta1*Mb/`&mu;b`, (6, 3) = `&epsilon;`*beta1*Mb/`&mu;b`, (6, 4) = 0, (6, 5) = 0, (6, 6) = 0}); B := Matrix(6, 6, {(1, 1) = `&mu;h`, (1, 2) = 0, (1, 3) = 0, (1, 4) = `&sigma;h`, (1, 5) = 0, (1, 6) = 0, (2, 1) = 0, (2, 2) = `&mu;h`, (2, 3) = 0, (2, 4) = 0, (2, 5) = 0, (2, 6) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = `&mu;h`-delta1, (3, 4) = 0, (3, 5) = 0, (3, 6) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = `&mu;h`-`&sigma;h`, (4, 5) = 0, (4, 6) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = `&mu;b`, (5, 6) = 0, (6, 1) = 0, (6, 2) = 0, (6, 3) = 0, (6, 4) = 0, (6, 5) = 0, (6, 6) = `&mu;b`}); C := A.(1/B); Rank(C); evs := Eigenvalues(C); eig := op(`minus`({entries(evs, nolist)}, {0}))

Matrix(6, 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) = `&epsilon;`*`&beta;hh`*Mh/`&mu;h`, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (3, 5) = 0, (3, 6) = `&epsilon;`*omega*`&beta;hh`*Mh/`&mu;h`, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 0, (4, 5) = 0, (4, 6) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = 0, (5, 6) = 0, (6, 1) = 0, (6, 2) = `&epsilon;`*beta1*Mb/`&mu;b`, (6, 3) = `&epsilon;`*beta1*Mb/`&mu;b`, (6, 4) = 0, (6, 5) = 0, (6, 6) = 0})

 

Matrix(6, 6, {(1, 1) = `&mu;h`, (1, 2) = 0, (1, 3) = 0, (1, 4) = `&sigma;h`, (1, 5) = 0, (1, 6) = 0, (2, 1) = 0, (2, 2) = `&mu;h`, (2, 3) = 0, (2, 4) = 0, (2, 5) = 0, (2, 6) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = `&mu;h`-delta1, (3, 4) = 0, (3, 5) = 0, (3, 6) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = `&mu;h`-`&sigma;h`, (4, 5) = 0, (4, 6) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = `&mu;b`, (5, 6) = 0, (6, 1) = 0, (6, 2) = 0, (6, 3) = 0, (6, 4) = 0, (6, 5) = 0, (6, 6) = `&mu;b`})

 

Matrix(6, 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) = `&epsilon;`*`&beta;hh`*Mh/(`&mu;h`*`&mu;b`), (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (3, 5) = 0, (3, 6) = `&epsilon;`*omega*`&beta;hh`*Mh/(`&mu;h`*`&mu;b`), (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 0, (4, 5) = 0, (4, 6) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = 0, (5, 6) = 0, (6, 1) = 0, (6, 2) = `&epsilon;`*beta1*Mb/(`&mu;h`*`&mu;b`), (6, 3) = `&epsilon;`*beta1*Mb/(`&mu;b`*(`&mu;h`-delta1)), (6, 4) = 0, (6, 5) = 0, (6, 6) = 0})

 

2

 

Vector[column](%id = 18446744074396590198)

 

((`&mu;h`-delta1)*Mb*Mh*beta1*`&beta;hh`*(`&mu;h`*omega+`&mu;h`-delta1))^(1/2)*epsilon/((`&mu;h`-delta1)*`&mu;h`*`&mu;b`), -((`&mu;h`-delta1)*Mb*Mh*beta1*`&beta;hh`*(`&mu;h`*omega+`&mu;h`-delta1))^(1/2)*epsilon/((`&mu;h`-delta1)*`&mu;h`*`&mu;b`)

(1)

params := {`&mu;b` = 1, `&mu;h` = .99, Mb = .5000, Mh = .5000, beta1 = .4972432119, beta2 = 0.2756788171e-2, delta1 = 0.1957235865e-2, omega = 0.2756788171e-2, `&beta;hh` = 0.2756788171e-2, `&beta;hl` = .4972432120, `&epsilon;` = .5000}

{`&mu;b` = 1, `&mu;h` = .99, Mb = .5000, Mh = .5000, beta1 = .4972432119, beta2 = 0.2756788171e-2, delta1 = 0.1957235865e-2, epsilon = .5000, omega = 0.2756788171e-2, `&beta;hh` = 0.2756788171e-2, `&beta;hl` = .4972432120}

(2)

plot([eval(eig, remove(has, params, beta1))], beta1 = 0 .. 0.2756788171e-2, filled = false, thickness = 4, color = blue, transparency = .4)

 

plot([eval(eig, remove(has, params, beta2))], beta2 = 0 .. 0.2756788171e-2, filled = false, thickness = 4, color = blue, transparency = .4)

 

NULL


 

Download pltMat.mw

You mean you want something like the attached?????

  restart;
  x:= p->2*p:
  y:= p->p:
  z:= p->(2-2*p);
  F:= p->x(p)*y(p)+y(p)+z(p):
#
# Plot the the quantity 'F' as function of the parameter 't'
#
  plot( F(t), t=0..1, color=red);
#
# Plot the curve along which the line integral is computed
#
  plots:-spacecurve( [x(t), y(t), z(t)], t=0..1, color=red);
#
# Compute the line integral
#
  int(F(t)*sqrt( diff(x(t),t)^2+diff(y(t),t)^2+diff(z(t),t)^2), t=0..1);

proc (p) options operator, arrow; 2-2*p end proc

 

 

 

13/2

(1)

 


 

Download lint.mw

of 2-D input, document mode and "Units" probably gives you the three "flakiest" options you can come uo with in Maple.

For your specific problem I can "fix" it as shown in the attached - but for me the biggest problem is the 'Units' package. It really *ought* to be so much easier to use!

(1) The software chosen to generate the data was Maple. This software was chose due to the fact that it was reccomennded by many, and due to its wide popularity, due to its ease of use.

 

a) The acceleration will be found by taking a time derivative of the velocity function using the differentiate command. Once the velocity is found the data table of the velocity in 4 seconds intervals wil be generated using the eval command in a for loop.

 

b) The Velocity is given by the following function:

v := u*ln(m__o/(-q*t+m__o))-g*t

12000*Units:-Unit(ft/s)*ln(2400*Units:-Unit(lb)/(2400*Units:-Unit(lb)-25*Units:-Unit(lb/s)*t))-32.2*Units:-Unit(ft/s^2)*t

(1)

Just like for the acceleration, the eval command will be used in a for loop to generate the data table of the velocity every 4 seconds.

c) The velocity function will be integrated to find the elevation function. The starting parameters of the rocket when its elevation is 0 will be plugged into the function to find the constant of integration (in this code, the variable c). The same command will be used to generate the data table as before.

with(Units)

Automatically loading the Units[Simple] subpackage
 

 

UseSystem('FPS')

q := 25*Unit('lb'/'s'); m__o := 2400*Unit('lb'); g := 32.2*Unit('ft'/'s'^2); u := 12000*Unit('ft'/'s')

12000*Units:-Unit(ft/s)

(2)

a := diff(u*ln(m__o/(-q*t+m__o))-g*t, t)

300000*Units:-Unit(ft/s)*Units:-Unit(lb/s)/(2400*Units:-Unit(lb)-25*Units:-Unit(lb/s)*t)-32.2*Units:-Unit(ft/s^2)

(3)

``

v := u*ln(m__o/(-q*t+m__o))-g*t

12000*Units:-Unit(ft/s)*ln(2400*Units:-Unit(lb)/(2400*Units:-Unit(lb)-25*Units:-Unit(lb/s)*t))-32.2*Units:-Unit(ft/s^2)*t

(4)

h := int(u*ln(m__o/(-q*t+m__o))-g*t, t)

-1152000.*Units:-Unit(ft)-16.10000000*t^2*Units:-Unit(ft/s^2)-1152000.*Units:-Unit(ft)*ln(-96./(-96.+t/Units:-Unit(s)))+12000.*t*ln(-96./(-96.+t/Units:-Unit(s)))*Units:-Unit(ft/s)+12000.*t*Units:-Unit(ft/s)

(5)

c := 0-(eval(h, t = 0*Unit('s')))

1152000.*Units:-Unit(ft)

(6)

h := c+h

-16.10000000*t^2*Units:-Unit(ft/s^2)-1152000.*Units:-Unit(ft)*ln(-96./(-96.+t/Units:-Unit(s)))+12000.*t*ln(-96./(-96.+t/Units:-Unit(s)))*Units:-Unit(ft/s)+12000.*t*Units:-Unit(ft/s)

(7)

h := (1/5280)*h

-0.3049242424e-2*t^2*Units:-Unit(ft/s^2)-218.1818182*Units:-Unit(ft)*ln(-96./(-96.+t/Units:-Unit(s)))+2.272727273*t*ln(-96./(-96.+t/Units:-Unit(s)))*Units:-Unit(ft/s)+2.272727273*t*Units:-Unit(ft/s)

(8)

for i from -4 by 4 to 80 do if i = -4 then printf("%11s %18s %14s %14s \n", "time", "acceleration", "velocity", "altitude") else printf("%10.2f %s %10.2f %s %10.2f %s %10.2f %s\n", i, "s", op(1, Simple:-eval(a, t = i*Unit('seconds'))), "ft/s^2", op(1, Simple:-eval(v, t = i*Unit('seconds'))), "ft/s", op(1, Simple:-eval(h, t = i*Unit('seconds'))), "ft") end if end do

       time       acceleration       velocity       altitude
      0.00 s      92.80 ft/s^2       0.00 ft/s       0.00 ft
      4.00 s      98.23 ft/s^2     381.92 ft/s       0.14 ft
      8.00 s     104.16 ft/s^2     786.54 ft/s       0.58 ft
     12.00 s     110.66 ft/s^2    1215.98 ft/s       1.34 ft
     16.00 s     117.80 ft/s^2    1672.66 ft/s       2.43 ft
     20.00 s     125.69 ft/s^2    2159.38 ft/s       3.88 ft
     24.00 s     134.47 ft/s^2    2679.38 ft/s       5.71 ft
     28.00 s     144.27 ft/s^2    3236.49 ft/s       7.95 ft
     32.00 s     155.30 ft/s^2    3835.18 ft/s      10.63 ft
     36.00 s     167.80 ft/s^2    4480.84 ft/s      13.77 ft
     40.00 s     182.09 ft/s^2    5179.96 ft/s      17.43 ft
     44.00 s     198.57 ft/s^2    5940.45 ft/s      21.64 ft
     48.00 s     217.80 ft/s^2    6772.17 ft/s      26.45 ft
     52.00 s     240.53 ft/s^2    7687.50 ft/s      31.92 ft
     56.00 s     267.80 ft/s^2    8702.42 ft/s      38.12 ft
     60.00 s     301.13 ft/s^2    9837.95 ft/s      45.14 ft
     64.00 s     342.80 ft/s^2   11122.55 ft/s      53.07 ft
     68.00 s     396.37 ft/s^2   12596.12 ft/s      62.04 ft
     72.00 s     467.80 ft/s^2   14317.13 ft/s      72.21 ft
     76.00 s     567.80 ft/s^2   16376.19 ft/s      83.81 ft
     80.00 s     717.80 ft/s^2   18925.11 ft/s      97.15 ft

 

plot(a, t = 0*Unit(seconds) .. 80*Unit(seconds))

 

plot(v, t = 0*Unit(seconds) .. 80*Unit(seconds))

 

plot(h, t = 0*Unit(seconds) .. 80*Unit(seconds))

 

NULL

Download unitProb.mw

it is difficult to be precise - but the overall approach "sounds" wrong

  1. You do not ever want to "integrate" the numerical solution of an ODE. The best approach is to add an additional equation to the ODE system, whihc computes the integral you want
  2. If you then use the output=listprocedure option which will return a list of "procedures" to compute any of the dependent variables - including the integral which you specified in (1) above - you just have to select it
  3. This particular procedure, which will return a value of the dependent variable for any supplied value of the independent variable, can then be used in a PDE system which is to be solved numerically
  4. Ther may well be other (better?) ways to address your problem - but since you do not supply details- who knows?

why you put the equations in two differetn arrays in the first place!. Anyhow see the attached

restart

T1 := array(1 .. 2); T1[1] := x = 1; T1[2] := y = 2

array( 1 .. 2, [ ] )

 

x = 1

 

y = 2

(1)

T2 := array(1 .. 2); T2[1] := z = 3; T2[2] := r = x+y+z

array( 1 .. 2, [ ] )

 

z = 3

 

r = x+y+z

(2)

solve({entries(T1, 'nolist'), entries(T2, 'nolist')}, {r, x, y, z})

{r = 6, x = 1, y = 2, z = 3}

(3)

``


 

Download arrSol.mw

You can either 'Remove()' columns which you don't want, or select columns you do want, as shown in the attached 'toy' example.

Might be a version issue I suppose - Which Maple version are you using?

  restart;
  df:= DataFrame( Matrix
                  ( 3, 5,
                    (i, j) -> 2*i-j
                  ),
                  rows = [a, b, c],
                  columns = [A, B, C, D, E]
                );
#
# Remove columns labelled A, C, E
#
  df2:=Remove(df,[A, C, E]);
#
# Select coumns labelled B and D
#
  df3:=df[ [B, D] ];

DataFrame(Matrix(3, 5, {(1, 1) = 1, (1, 2) = 0, (1, 3) = -1, (1, 4) = -2, (1, 5) = -3, (2, 1) = 3, (2, 2) = 2, (2, 3) = 1, (2, 4) = 0, (2, 5) = -1, (3, 1) = 5, (3, 2) = 4, (3, 3) = 3, (3, 4) = 2, (3, 5) = 1}), rows = [a, b, c], columns = [A, B, C, D, E])

 

Vector[row](3, {(1) = 1, (2) = 3, (3) = 5})

 

DataFrame(Matrix(3, 2, {(1, 1) = 0, (1, 2) = -2, (2, 1) = 2, (2, 2) = 0, (3, 1) = 4, (3, 2) = 2}), rows = [a, b, c], columns = [B, D])

 

_m755491744

(1)

 

Download frame.mw

which will 'trap' *most* errors. See the help at ?try, or consider the 'toy' examples in the attached which show how to trap a simple divide-by-zero error

  restart;
  div1:= proc(a, b)
              local d:
              d:=a/b:
              return d
         end proc:
#
# Not a divide by zero error
# so just returns the "answer"
#
  try ans:=div1(6,3)
  catch "numeric exception: division by zero":
         ans:=infinity;
  end try:
  ans;
#
# Produce a divide by zero error, but "catch"
# the error and return infinity
#
  try   ans:=div1(6,0)
  catch "numeric exception: division by zero":
        ans:=infinity;
  end try:
  ans;

2

 

infinity

(1)

#
# More or less the same functionality as above, but
# with tthe try/catch statement within the procedure
#
  div2:=proc(a, b)
             local d;
             try d:=a/b
             catch "numeric exception: division by zero":
                   d:=infinity;
             finally return d;
             end try;
        end proc:
  ans:=div2(6,3);
  ans:=div2(6,0);

2

 

infinity

(2)

 


 

Download trycat3.mw

see the attached

restart;
Explore( plot( x^2*sin(a*x),
               x=-2*Pi..2*Pi
             ),
         animate=true,
         autorun=true,
         loop=true,
         parameters=[ a=[ seq(j, j=-15..-7),
                          seq(j, j=7..15)
                        ]
                    ]
      );

 

Download animExplore.mw

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