Rouben Rostamian

MaplePrimes Activity


These are answers submitted by Rouben Rostamian

Finite differences work more naturally in Maple if we use function form rather than indexed name for the variables. Here is an illustration.

Finite differences in Maple

Finite difference calculations work more naturally in Maple

if `u__i j` is expressed as a function u(i, j).

restart;

"(∂)/(∂ x)" and "(∂)/(∂ y)"

dx := (i,j) -> (u(i+1,j) - u(i,j))/h;
dy := (i,j) -> (u(i,j+1) - u(i,j))/h;

proc (i, j) options operator, arrow; (u(i+1, j)-u(i, j))/h end proc

proc (i, j) options operator, arrow; (u(i, j+1)-u(i, j))/h end proc

"((∂)^2)/((∂)^( )x^2)" and "((∂)^2)/((∂)^( )y^2)"

(dx(i,j) - dx(i-1,j))/h:
simplify(%):
d2x := unapply(%, i, j);

(dy(i,j) - dy(i,j-1))/h:
simplify(%):
d2y := unapply(%, i, j);

proc (i, j) options operator, arrow; (u(i+1, j)-2*u(i, j)+u(i-1, j))/h^2 end proc

proc (i, j) options operator, arrow; (u(i, j+1)-2*u(i, j)+u(i, j-1))/h^2 end proc

"(∂)/(∂ x)(((∂)^2)/((∂)^( )x^2))"

(d2x(i+1,j) - d2x(i,j))/h:
simplify(%):
d3x := unapply(%, i, j);

proc (i, j) options operator, arrow; (u(i+2, j)-3*u(i+1, j)+3*u(i, j)-u(i-1, j))/h^3 end proc

The Laplacian

d2x(i,j) + d2y(i,j):
simplify(%):
Lap := unapply(%, i, j);

proc (i, j) options operator, arrow; (u(i+1, j)-4*u(i, j)+u(i-1, j)+u(i, j+1)+u(i, j-1))/h^2 end proc

The x derivative of the Laplacian

(Lap(i+1,j) - Lap(i,j))/h:
simplify(%);

(u(i+2, j)-5*u(i+1, j)+5*u(i, j)+u(i+1, j+1)+u(i+1, j-1)-u(i-1, j)-u(i, j+1)-u(i, j-1))/h^3

 

Download finite-difference-expressions.mw

 

I can't tell whether you have picked the coefficients of the DEs based on some specific need or just randomly.  The initial conditions and the plotting range are certainly picked randomly.  A phase portrait is interesting near the equilibria.  You should indentify the locations of the equilibria and then set up the plotting scene near that location.

Have a look at this modified version of your worksheet.

restart;

with(plots):

with(DEtools):

The vector field:

rhs1 := (A,l,S) -> `αAP`*A - `μA`*A;
rhs2 := (A,l,S) -> `αlS`*l*S - `μl`*l - `βlA`*l*A;
rhs3 := (A,l,S) -> `αS`*S - `βSl`*l*S - 0.25*S*A;

proc (A, l, S) options operator, arrow; `αAP`*A-`μA`*A end proc

proc (A, l, S) options operator, arrow; `αlS`*l*S-`μl`*l-`βlA`*l*A end proc

proc (A, l, S) options operator, arrow; `αS`*S-`βSl`*l*S-.25*S*A end proc

Equilibria:

equilibria := solve([rhs1(A,l,S), rhs2(A,l,S), rhs3(A,l,S)], [A,l,S]);

[[A = 0., l = 0., S = 0.], [A = 0., l = `αS`/`βSl`, S = `μl`/`αlS`]]

The differential equations:

de1 := diff(A(t), t) = rhs1(A(t),l(t),S(t));
de2 := diff(l(t), t) = rhs2(A(t),l(t),S(t));
de3 := diff(S(t), t) = rhs3(A(t),l(t),S(t));

diff(A(t), t) = `αAP`*A(t)-`μA`*A(t)

diff(l(t), t) = `αlS`*l(t)*S(t)-`μl`*l(t)-`βlA`*l(t)*A(t)

diff(S(t), t) = `αS`*S(t)-`βSl`*l(t)*S(t)-.25*S(t)*A(t)

Parameters:

`αAP` := 2;
`μA` := 1;
`αlS` := 2;
`μl` := 0.2;
`βlA` := 0.5;
`αS` := 100;
`βSl` := 1;
`βSA` = 0.25;

2

1

2

.2

.5

100

1

`βSA` = .25

Where are the equilibria?

equilibria;

[[A = 0., l = 0., S = 0.], [A = 0., l = 100, S = .1000000000]]

Pick initial conditions near the second equilibrium:

ICs := seq([A(0)=0,l(0)=z1,S(0)=0.3], z1=90..110, 10),
                seq([A(0)=0.1,l(0)=z1,S(0)=0.15], z1=90..110, 10);

[A(0) = 0, l(0) = 90, S(0) = .3], [A(0) = 0, l(0) = 100, S(0) = .3], [A(0) = 0, l(0) = 110, S(0) = .3], [A(0) = .1, l(0) = 90, S(0) = .15], [A(0) = .1, l(0) = 100, S(0) = .15], [A(0) = .1, l(0) = 110, S(0) = .15]

How many initial conditions?

nops([ICs]);

6

Specify colors for plotting orbits, one color per initial condition:

DEplot3d([de1, de2, de3], [A,l,S], t=0..3, [ICs],
        linecolor=["Gold", "Cyan", "Magenta", "Red", "Green", "Blue"],
        thickness=5, stepsize=0.01, arrows=hybrid, dirgrid=[5,5,5],
        animatecurves=true);

Warning, numpoints adjusted from 301 to 313 for animation

>

 

Download deplot3d-animate-orbits.mw

 

restart;

Rounds the entries of the matrix A to n decimal places.

Removes trailing zeros.

doit := proc(A::Matrix, n::posint)
        map[3](sprintf, "%.*g", n, A);
        parse~(%);
end proc:

Give it a try:

A := Matrix(3,3, (i,j) -> 1.0/(i+j));

Matrix(3, 3, {(1, 1) = .5000000000, (1, 2) = .3333333333, (1, 3) = .2500000000, (2, 1) = .3333333333, (2, 2) = .2500000000, (2, 3) = .2000000000, (3, 1) = .2500000000, (3, 2) = .2000000000, (3, 3) = .1666666667})

doit(A, 4);

Matrix(3, 3, {(1, 1) = .5, (1, 2) = .3333, (1, 3) = .25, (2, 1) = .3333, (2, 2) = .25, (2, 3) = .2, (3, 1) = .25, (3, 2) = .2, (3, 3) = .1667})

Download mw.mw

 

restart;

de := (diff(H(K),K))^3+4*K^4*H(K)^4*diff(H(K),K)+8*K^4*H(K)^5=0;

(diff(H(K), K))^3+4*K^4*H(K)^4*(diff(H(K), K))+8*K^4*H(K)^5 = 0

The differential equation above is not well-defined.  A properly defined first order

differential equation should be of the form dH/dK = something.  Therefore we

begin with solving your equation for the derivative:

convert(de, D);
sol := [ solve(%, D(H)(K)) ];

(D(H))(K)^3+4*K^4*H(K)^4*(D(H))(K)+8*K^4*H(K)^5 = 0

[2*((1/6)*(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3)-2*K^2*H(K)^2/(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3))*H(K)*K, 2*(-(1/12)*(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3)+K^2*H(K)^2/(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/6)*(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3)+2*K^2*H(K)^2/(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3)))*H(K)*K, 2*(-(1/12)*(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3)+K^2*H(K)^2/(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/6)*(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3)+2*K^2*H(K)^2/(-108*K*H(K)^2+12*(12*K^6*H(K)^6+81*H(K)^4*K^2)^(1/2))^(1/3)))*H(K)*K]

nops(sol);

3

We see that there are three possible solutions.  Two of them are complex-valued.

We discard those and keep what is left:

DE := diff(H(K),K) = simplify(remove(has, sol, I))[] assuming K > 0;

diff(H(K), K) = -(1/3)*12^(1/3)*(K^2*H(K)^2*12^(1/3)-K^(2/3)*((4*K^4*H(K)^6+27*H(K)^4)^(1/2)*3^(1/2)-9*H(K)^2)^(2/3))*H(K)*K^(2/3)/((4*K^4*H(K)^6+27*H(K)^4)^(1/2)*3^(1/2)-9*H(K)^2)^(1/3)

This is the differential equation that you should be working with.
Let's solve it:

dsol := dsolve({DE, H(0)=1/2}, numeric);

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 28, [( 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..65, {(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) = 0, (17) = 0, (18) = 15, (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, (64) = -1, (65) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.10e-9, (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) = .5}, 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) = .5}, 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), 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] = H(K)]`; if 1.73205080756888*evalf((4*X^4*Y[1]^6+27*Y[1]^4)^(1/2))-9*Y[1]^2 < 0 then YP[1] := undefined; return 0 end if; if 1.73205080756888*evalf((4*X^4*Y[1]^6+27*Y[1]^4)^(1/2))-9*Y[1]^2 < 0 then YP[1] := undefined; return 0 end if; if X < 0 then YP[1] := undefined; return 0 end if; YP[1] := -.763142828368887*(2.28942848510666*X^2*Y[1]^2-evalf(X^(2/3))*evalf((1.73205080756888*evalf((4*X^4*Y[1]^6+27*Y[1]^4)^(1/2))-9*Y[1]^2)^(2/3)))*Y[1]*evalf(X^(2/3))*evalf(1/(1.73205080756888*evalf((4*X^4*Y[1]^6+27*Y[1]^4)^(1/2))-9*Y[1]^2)^(1/3)); 0 end proc, -1, 0, 0, 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] = H(K)]`; if 1.73205080756888*evalf((4*X^4*Y[1]^6+27*Y[1]^4)^(1/2))-9*Y[1]^2 < 0 then YP[1] := undefined; return 0 end if; if 1.73205080756888*evalf((4*X^4*Y[1]^6+27*Y[1]^4)^(1/2))-9*Y[1]^2 < 0 then YP[1] := undefined; return 0 end if; if X < 0 then YP[1] := undefined; return 0 end if; YP[1] := -.763142828368887*(2.28942848510666*X^2*Y[1]^2-evalf(X^(2/3))*evalf((1.73205080756888*evalf((4*X^4*Y[1]^6+27*Y[1]^4)^(1/2))-9*Y[1]^2)^(2/3)))*Y[1]*evalf(X^(2/3))*evalf(1/(1.73205080756888*evalf((4*X^4*Y[1]^6+27*Y[1]^4)^(1/2))-9*Y[1]^2)^(1/3)); 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 27 ) = (""), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0), ( 28 ) = (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 elif type(_xin, `=`) and lhs(_xin) = "setdatacallback" then if not type(rhs(_xin), 'nonegint') then error "data callback must be a nonnegative integer (address)" end if; _dtbl[1][28] := rhs(_xin) 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) = [K, H(K)], (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

plots:-odeplot(dsol, K=0..7);

 

Download mw.mw

 

 

restart;

with(plots):

We wish to determine the equation of the surface obtained by rotating
 the line C: x = 0, y = z about the line L; y = 0, z = x-1, and plot that surface.

 

We observe that the rotation axis, "L," may be parameterized as
`<,>`(x, 0, x-1) or equivalently
L(t) = `<,>`(0, 0, -1)+u*t, where u = `<,>`(1, 0, 1):

u := < 1, 0, 1 >;
L := t -> <0,0,-1> + u*t;

Vector(3, {(1) = 1, (2) = 0, (3) = 1})

proc (t) options operator, arrow; `<,>`(0, 0, -1)+u*t end proc

Introduce the mutually orthogonal unit vectors v and w, both of which
are also orthogonal to
u:

v := <0,1,0>;

Vector(3, {(1) = 0, (2) = 1, (3) = 0})

LinearAlgebra:-CrossProduct(u,v):
w := %/sqrt(% . %);

Vector(3, {(1) = -(1/2)*sqrt(2), (2) = 0, (3) = (1/2)*sqrt(2)})

Also observe that the line C may be parametrized as

C := s -> <0,s,s>;

proc (s) options operator, arrow; `<,>`(0, s, s) end proc

For any point P on the line C there is a point Q on the line L so that the line segment

PQ is orthogonal to L,  that is, Q is the nearest point on the line L to the point P.

Let P = `<,>`(0, s, s),  and Q = `<,>`(t, 0, t-1)". "We determine Q by asserting that

Q-P is perpendicular to u:

P := <0, s, s>;
Q := <t, 0, t-1>;
(Q - P)^+ . u = 0;
isolate(%, t);
Q := eval(Q, %);

Vector(3, {(1) = 0, (2) = s, (3) = s})

Vector(3, {(1) = t, (2) = 0, (3) = t-1})

2*t-1-s = 0

t = 1/2+(1/2)*s

Vector[column](%id = 36893628651414056100)

As the line C rotates about L, the point P traces a circle about the point Q.

The circle's radius "r, "is the length of PQ:

r := sqrt(simplify((Q - P)^+ . (Q-P)));

(1/2)*(6*s^2+4*s+2)^(1/2)

The entire surface of revolution consists of a family of such circles.  We conclude that

the parametric equation of the surface of revolution is

S := Q + r*(v*cos(q) + w*sin(q));

Vector(3, {(1) = 1/2+(1/2)*s-(1/4)*sqrt(6*s^2+4*s+2)*sin(q)*sqrt(2), (2) = (1/2)*sqrt(6*s^2+4*s+2)*cos(q), (3) = -1/2+(1/2)*s+(1/4)*sqrt(6*s^2+4*s+2)*sin(q)*sqrt(2)})

Plotting

display(
        tubeplot([seq](L(t)), t=-1..2, radius=0.06, color="Red"),   # the line L
        tubeplot([seq](C(s)), s=-2..2, radius=0.06, color="Green"), # the line C
        plot3d(S, s=-2..2, q=-Pi..Pi, color="Gold"),
        scaling=constrained, style=surface, lightmodel=light2);

 

Download surface-of-revolution.mw

 

You are right, the equation has no (real) solutions.

If you were solving that equation by hand, you would square both sides and get

x^2 - 10*x + 1 = -8*x^2 + 9*x - 1

and then solve the quadratic and obtain x = 1/9, 2.  Then you would substitute these into the original equation to check whether they solve the original equation.  If it happens that they don't, they are referred to as spurious solutions and should be thrown out.

In the present case, x = 1/9 and 2 are both spurious.  Ideally Maple should check for that but it doesn't.  Just be aware of the issue and check it yourself.

Your equ4 is
Z1T = ZL*(I1T + I2)/I1T

which is equivalent to 

Z1T*I1T = ZL*(I1T + I2)

The left-hand side of that equation is the product of two of your unknowns:

vars_unknown := [V1, I1T, V1T, Z1, Z1T]

and therefore your system of equations is nonlinear.  It's no wonder that 
LinearAlgebra:-GenerateMatrix produces nonsense.

It seems that you believe that the system of equations should be linear.  If so, then you must have entered some of the equations incorrectly.  Check!

After you calculate your w, do:

latex(w);

That produces a latex code for your expression.  Copy and paste that code into your latex document.

If that does not answer your question, be sure to upload your worksheet so that we can see what you see.

I will show you how to solve the problem if we are looking for a quadratic (2nd degree) polynomial that goes through three prescribed points.  I leave it to you to adapt it to the case of a cubic (3rd degree) polynomial that goes through four prescribed points.

restart;

Want to fund a quadratic function f as in:

f := x -> a[2]*x^2 + a[1]*x + a[0];

proc (x) options operator, arrow; a[2]*x^2+a[1]*x+a[0] end proc

so that it fits the following data:

eqns := { f(-2) = -5, f(1) = 1, f(4) = 61 };

{a[2]+a[1]+a[0] = 1, 4*a[2]-2*a[1]+a[0] = -5, 16*a[2]+4*a[1]+a[0] = 61}

Solve that system of equations to get the quadratic's coefficients:

the_coeffs := solve(eqns);

{a[0] = -7, a[1] = 5, a[2] = 3}

Substitute the calculated coefficients into the quadratic's formula

to see what it looks like:

y = eval(f(x), the_coeffs);

y = 3*x^2+5*x-7

Download quadratic-fit.mw

You will find that in the Common Symbols palette, fourth row, fourth column.

In your failed version you attempt to solve three equations for two unknowns.  That's not good.  What you want is to solve your three equations for three unknowns, as in:

solve({I__cm = m*r^2, r*T = I__cm*a/r, g*m - T = m*a}, {T, a, I__cm});

 

This ought to do it:

p := v -> R*T/(v-b) - a/v^2;
numer(p(v)) = 0;

The result is -R*T*v^2 - a*b + a*v = 0, which quadratic in v.  There is no v^3 term.

A couple of errors:

  1. In your equations you have variables named mu and mu[1].  That's bad.  One equation declares mu as a scalar, the other makes it an array.  Hence the confusion.  If you want a subscripted mu1, write it as mu__1 (note the two underscore characters!) not mu[1].  That will not conflict with the variable named mu.
  2. Your equations involve a variable T.  It needs to be assigned a numerical value.

A general adivse:  You have:

dsn := dsolve(eval({eqn1, eqn2, eqn3, eqn4, i(0) = 11437, r(0) = 1077, s(0) = 1770000, t(0) = 1087}), {i(t), r(t), s(t), t(t)}, numeric)

Aren't you interested in seeing the system of equations that you are passing to dsolve() to solve?  In general, you should be.  So define:

sys := eval({eqn1, eqn2, eqn3, eqn4, i(0) = 11437, r(0) = 1077, s(0) = 1770000, t(0) = 1087}), {i(t), r(t), s(t), t(t)};

and then examine sys to make sure that what you are passing to dsolve() is what you really mean, and only then do:

dsolve(sys, numeric);

 

Let's say you have y = f(x) for some function f.  Then dy / dx = f '(x) is a (trivial) differential equation which has y = f(x) as a solution.  You don't need Maple to tell you that.

For instance, if y = cos (x),  then dy/dx = − sin (x) meets your requirements.

I had noted earlier that it would be possible to solve your system of equations through the method of lines.  Now I have worked out the details.  See if this worksheet makes sense.

restart;

with(plots):

The PDEs

 

I have changed the notation somewhat in order to get consistency

across the board.  Here are the changes:
 
"xi -> x  tau -> t  phi -> eta  Theta -> theta   Phi -> phi  "
I solve the system in the spatial range 0 < x and x < L.  Furthermore, I have changed
the forcing term v(0, L) = cos(omega*t) to v(0, L) = sin(omega*t) so that at t = 0 it

matches the initial condition v(x, 0) = 0.  

The PDEs are given as equ1, equ2, equ3:

equ1 := diff(v(x,t),t) = phi__7*diff(v(x,t),x$2) + phi__7*diff(v(x,t),t,x$2) - phi__8*M*v(x,t) + phi__9*Gr*theta(x,t) + phi__10*Gm*phi(x,t);
equ2 := diff(theta(x,t),t) = lambda__f*diff(v(x,t),x$2)/(Pe*phi__11);
equ3 := diff(phi(x,t),t) = (1-eta)*diff(phi(x,t),x$2)/Sc;

diff(v(x, t), t) = phi__7*(diff(diff(v(x, t), x), x))+phi__7*(diff(diff(diff(v(x, t), t), x), x))-phi__8*M*v(x, t)+phi__9*Gr*theta(x, t)+phi__10*Gm*phi(x, t)

diff(theta(x, t), t) = lambda__f*(diff(diff(v(x, t), x), x))/(Pe*phi__11)

diff(phi(x, t), t) = (1-eta)*(diff(diff(phi(x, t), x), x))/Sc

We are going to solve this system on the interval 0 < x and x < L for a prescribed L 

subject to prescribed initial and boundary conditions

ic_v := v(x,0) = v__0(x);
ic_theta := theta(x,0) = theta__0(x);
ic_phi := phi(x,0) = phi__0(x);
bc_v := v(0,t) = v__left(t), v(L,t) = v__right(t);
bc_theta := theta(0,t) = theta__left(t), theta(L,t) = theta__right(t);
bc_phi := phi(0,t) = phi__left(t), phi(L,t) = phi__right(t);

v(x, 0) = v__0(x)

theta(x, 0) = theta__0(x)

phi(x, 0) = phi__0(x)

v(0, t) = v__left(t), v(L, t) = v__right(t)

theta(0, t) = theta__left(t), theta(L, t) = theta__right(t)

phi(0, t) = phi__left(t), phi(L, t) = phi__right(t)

The system is not a "standard" PDE and it requires a bit of manipulation

to put it in a manageable form.  Toward that end,

we introduce a new variable u through

pde0 := u(x,t) = v(x,t) - phi__7*diff(v(x,t),x$2);

u(x, t) = v(x, t)-phi__7*(diff(diff(v(x, t), x), x))

or equivalently:

eq := isolate(pde0, diff(v(x,t),x$2));

diff(diff(v(x, t), x), x) = (-u(x, t)+v(x, t))/phi__7

Using the expression above to eliminate the second derivative of v from equ1.  Call the result pde1:

diff(eq, t):
subs(%, equ1):
subs(eq, %):
pde1 := isolate(%, diff(u(x,t),t));

diff(u(x, t), t) = -u(x, t)+v(x, t)-phi__8*M*v(x, t)+phi__9*Gr*theta(x, t)+phi__10*Gm*phi(x, t)

Similarly, eliminate the second derivative of v from equ2.  Call the result pde2:

pde2 := subs(eq, equ2);

diff(theta(x, t), t) = lambda__f*(-u(x, t)+v(x, t))/(phi__7*Pe*phi__11)

Keep equ3 as is.  Call it pde3:

pde3 := equ3;

diff(phi(x, t), t) = (1-eta)*(diff(diff(phi(x, t), x), x))/Sc

Equations pde0, pde1, pde2, pde3 form a system of four PDEs in the
four unknowns u, v, theta, phi.  It's not a standard PDE because pde0 has
no time derivative.

 

Let's observe that the initial condition
on u may be determined from its definition:

ic_u := subs(t=0, diff(ic_v, x$2), ic_v, pde0);

u(x, 0) = v__0(x)-phi__7*(diff(diff(v__0(x), x), x))

 

Discretization through the method of lines

 

We apply the method of lines to solve the PDEs.  Toward that end, we partition the

interval [0, L] into n+1 subintervals of equal lengths h = L/(n+1) through the n+2 
equally-spaced points x__0 = 0, x__1 = h, x__2 = 2*h, () .. (), x__n = nh, `x__n+1` = n+1 and n+1 = Lb.
We discretize the system's unknowns in space, through

"`U__j`(t)=u(`x__j`,t),  `V__j`(t)=v(`x__j`,t),  `Theta__j`(t)=theta(`x__j`,t), "

`&Phi;__j`(t) = phi(x__j, t), j = 0, 1, 2, () .. (), n+1.
Furthermore, we discretize the 2nd order derivatives through centered differences,

as in
"((&PartialD;)^2phi)/((&PartialD;)^( )x^2)(`x__j`,t)&approx;(`Phi__j-1`(t)-2 `Phi__j`(t)+`Phi__j+1`(t))/(h^(2)) ",
and similarly for "((&PartialD;)^2v)/((&PartialD;)^( )x^2)".
Here we fix the value of n and express the PDEs and the initial
and boundary conditions in terms of the discretized variables.
We choose a small n for this demonstration but n will be larger

for practical applications.

n := 8;   # change as needed

8

The initial condition on U:

convert(ic_u, D):
subs(x=h*j, %):
IC_U := seq(U[j](0) = rhs(%), j=1..n);

U[1](0) = v__0(h)-phi__7*((D@@2)(v__0))(h), U[2](0) = v__0(2*h)-phi__7*((D@@2)(v__0))(2*h), U[3](0) = v__0(3*h)-phi__7*((D@@2)(v__0))(3*h), U[4](0) = v__0(4*h)-phi__7*((D@@2)(v__0))(4*h), U[5](0) = v__0(5*h)-phi__7*((D@@2)(v__0))(5*h), U[6](0) = v__0(6*h)-phi__7*((D@@2)(v__0))(6*h), U[7](0) = v__0(7*h)-phi__7*((D@@2)(v__0))(7*h), U[8](0) = v__0(8*h)-phi__7*((D@@2)(v__0))(8*h)

 

These equations involve V__0(t) and `V__n+1`(t) which are known from the boundary conditions,

V[0](t) = subs(bc_v, v__left(t)):
V[n+1](t) = subs(bc_v, v__right(t)):
BC_V := %%, %;

V[0](t) = v__left(t), V[9](t) = v__right(t)

Shortly we will need the boundary conditions on Phi so let's calculate them now:

Phi[0](t) = subs(bc_phi, phi__left(t)):
Phi[n+1](t) = subs(bc_phi, phi__right(t)):
BC_Phi := %%, %;

Phi[0](t) = phi__left(t), Phi[9](t) = phi__right(t)

Now we turn to pde0, pde1, pde2, pde3, and express them in discretized forms:

DE0 := U[j](t) = V[j](t) - phi__7*(V[j-1](t) - 2*V[j](t) + V[j+1](t))/h^2;

U[j](t) = V[j](t)-phi__7*(V[j-1](t)-2*V[j](t)+V[j+1](t))/h^2

DE1 := diff(U[j](t),t) = -U[j](t) + V[j](t) - phi__8*M*V[j](t) + phi__9*Gr*Theta[j](t) + phi__10*Gm*Phi[j](t);

diff(U[j](t), t) = -U[j](t)+V[j](t)-phi__8*M*V[j](t)+phi__9*Gr*Theta[j](t)+phi__10*Gm*Phi[j](t)

DE2 := diff(Theta[j](t),t) = lambda__f/(phi__7*Pe*phi__11)*(-U[j](t) + V[j](t));

diff(Theta[j](t), t) = lambda__f*(-U[j](t)+V[j](t))/(phi__7*Pe*phi__11)

DE3 := diff(Phi[j](t),t) = (1 - sigma)/Sc*(Phi[j-1](t) - 2*Phi[j](t) + Phi[j+1](t))/h^2;

diff(Phi[j](t), t) = (1-sigma)*(Phi[j-1](t)-2*Phi[j](t)+Phi[j+1](t))/(Sc*h^2)

We apply each of DE0 through DE3 at the n interior nodes x__j and thus obtain a system of

4*n equations in the 4*n unknownsU__j, V__j, `&Theta;__j`, `&Phi;__j`.   The system involves the values

of V__j and `&Phi;__j` at the domain's left and right boundaries but we eliminate them through

 substituting their values from the boundary conditions BC_V and BC_Phi.

seq(DE0, j=1..n), seq(DE1, j=1..n), seq(DE2, j=1..n), seq(DE3, j=1..n):
SYS := subs(BC_V, BC_Phi, {%});

{diff(Phi[1](t), t) = (1-sigma)*(phi__left(t)-2*Phi[1](t)+Phi[2](t))/(Sc*h^2), diff(Phi[2](t), t) = (1-sigma)*(Phi[1](t)-2*Phi[2](t)+Phi[3](t))/(Sc*h^2), diff(Phi[3](t), t) = (1-sigma)*(Phi[2](t)-2*Phi[3](t)+Phi[4](t))/(Sc*h^2), diff(Phi[4](t), t) = (1-sigma)*(Phi[3](t)-2*Phi[4](t)+Phi[5](t))/(Sc*h^2), diff(Phi[5](t), t) = (1-sigma)*(Phi[4](t)-2*Phi[5](t)+Phi[6](t))/(Sc*h^2), diff(Phi[6](t), t) = (1-sigma)*(Phi[5](t)-2*Phi[6](t)+Phi[7](t))/(Sc*h^2), diff(Phi[7](t), t) = (1-sigma)*(Phi[6](t)-2*Phi[7](t)+Phi[8](t))/(Sc*h^2), diff(Phi[8](t), t) = (1-sigma)*(Phi[7](t)-2*Phi[8](t)+phi__right(t))/(Sc*h^2), diff(Theta[1](t), t) = lambda__f*(-U[1](t)+V[1](t))/(phi__7*Pe*phi__11), diff(Theta[2](t), t) = lambda__f*(-U[2](t)+V[2](t))/(phi__7*Pe*phi__11), diff(Theta[3](t), t) = lambda__f*(-U[3](t)+V[3](t))/(phi__7*Pe*phi__11), diff(Theta[4](t), t) = lambda__f*(-U[4](t)+V[4](t))/(phi__7*Pe*phi__11), diff(Theta[5](t), t) = lambda__f*(-U[5](t)+V[5](t))/(phi__7*Pe*phi__11), diff(Theta[6](t), t) = lambda__f*(-U[6](t)+V[6](t))/(phi__7*Pe*phi__11), diff(Theta[7](t), t) = lambda__f*(-U[7](t)+V[7](t))/(phi__7*Pe*phi__11), diff(Theta[8](t), t) = lambda__f*(-U[8](t)+V[8](t))/(phi__7*Pe*phi__11), diff(U[1](t), t) = -U[1](t)+V[1](t)-phi__8*M*V[1](t)+phi__9*Gr*Theta[1](t)+phi__10*Gm*Phi[1](t), diff(U[2](t), t) = -U[2](t)+V[2](t)-phi__8*M*V[2](t)+phi__9*Gr*Theta[2](t)+phi__10*Gm*Phi[2](t), diff(U[3](t), t) = -U[3](t)+V[3](t)-phi__8*M*V[3](t)+phi__9*Gr*Theta[3](t)+phi__10*Gm*Phi[3](t), diff(U[4](t), t) = -U[4](t)+V[4](t)-phi__8*M*V[4](t)+phi__9*Gr*Theta[4](t)+phi__10*Gm*Phi[4](t), diff(U[5](t), t) = -U[5](t)+V[5](t)-phi__8*M*V[5](t)+phi__9*Gr*Theta[5](t)+phi__10*Gm*Phi[5](t), diff(U[6](t), t) = -U[6](t)+V[6](t)-phi__8*M*V[6](t)+phi__9*Gr*Theta[6](t)+phi__10*Gm*Phi[6](t), diff(U[7](t), t) = -U[7](t)+V[7](t)-phi__8*M*V[7](t)+phi__9*Gr*Theta[7](t)+phi__10*Gm*Phi[7](t), diff(U[8](t), t) = -U[8](t)+V[8](t)-phi__8*M*V[8](t)+phi__9*Gr*Theta[8](t)+phi__10*Gm*Phi[8](t), U[1](t) = V[1](t)-phi__7*(v__left(t)-2*V[1](t)+V[2](t))/h^2, U[2](t) = V[2](t)-phi__7*(V[1](t)-2*V[2](t)+V[3](t))/h^2, U[3](t) = V[3](t)-phi__7*(V[2](t)-2*V[3](t)+V[4](t))/h^2, U[4](t) = V[4](t)-phi__7*(V[3](t)-2*V[4](t)+V[5](t))/h^2, U[5](t) = V[5](t)-phi__7*(V[4](t)-2*V[5](t)+V[6](t))/h^2, U[6](t) = V[6](t)-phi__7*(V[5](t)-2*V[6](t)+V[7](t))/h^2, U[7](t) = V[7](t)-phi__7*(V[6](t)-2*V[7](t)+V[8](t))/h^2, U[8](t) = V[8](t)-phi__7*(V[7](t)-2*V[8](t)+v__right(t))/h^2}

The SYS computed above is a system of linear first order DAEs (Differential-Algebraic Equations)
 in the 4*n unknowns "`U__j`, `V__j`, `Theta__j`,`Phi__j`, j=1,2,...,n."  These are DAEs because the

unknowns V__j  appear algebraically only; there are no derivatives of V__j.  

indets(SYS, function);

{diff(Phi[1](t), t), diff(Phi[2](t), t), diff(Phi[3](t), t), diff(Phi[4](t), t), diff(Phi[5](t), t), diff(Phi[6](t), t), diff(Phi[7](t), t), diff(Phi[8](t), t), diff(Theta[1](t), t), diff(Theta[2](t), t), diff(Theta[3](t), t), diff(Theta[4](t), t), diff(Theta[5](t), t), diff(Theta[6](t), t), diff(Theta[7](t), t), diff(Theta[8](t), t), diff(U[1](t), t), diff(U[2](t), t), diff(U[3](t), t), diff(U[4](t), t), diff(U[5](t), t), diff(U[6](t), t), diff(U[7](t), t), diff(U[8](t), t), v__left(t), phi__left(t), phi__right(t), v__right(t), Phi[1](t), Phi[2](t), Phi[3](t), Phi[4](t), Phi[5](t), Phi[6](t), Phi[7](t), Phi[8](t), Theta[1](t), Theta[2](t), Theta[3](t), Theta[4](t), Theta[5](t), Theta[6](t), Theta[7](t), Theta[8](t), U[1](t), U[2](t), U[3](t), U[4](t), U[5](t), U[6](t), U[7](t), U[8](t), V[1](t), V[2](t), V[3](t), V[4](t), V[5](t), V[6](t), V[7](t), V[8](t)}

The system of DAEs may be solved by applying the initial conditions IC_U
calculated earlier, and the initial conditions IC_Theta and IC_Phi which we calculate now:

subs(x=h*j, ic_theta):
IC_Theta := seq(Theta[j](0) = rhs(%), j=1..n);

Theta[1](0) = theta__0(h), Theta[2](0) = theta__0(2*h), Theta[3](0) = theta__0(3*h), Theta[4](0) = theta__0(4*h), Theta[5](0) = theta__0(5*h), Theta[6](0) = theta__0(6*h), Theta[7](0) = theta__0(7*h), Theta[8](0) = theta__0(8*h)

subs(x=h*j, ic_phi):
IC_Phi := seq(Phi[j](0) = rhs(%), j=1..n);

Phi[1](0) = phi__0(h), Phi[2](0) = phi__0(2*h), Phi[3](0) = phi__0(3*h), Phi[4](0) = phi__0(4*h), Phi[5](0) = phi__0(5*h), Phi[6](0) = phi__0(6*h), Phi[7](0) = phi__0(7*h), Phi[8](0) = phi__0(8*h)

The overall initial conditions are:

IC := { IC_U, IC_Theta, IC_Phi };

{Phi[1](0) = phi__0(h), Phi[2](0) = phi__0(2*h), Phi[3](0) = phi__0(3*h), Phi[4](0) = phi__0(4*h), Phi[5](0) = phi__0(5*h), Phi[6](0) = phi__0(6*h), Phi[7](0) = phi__0(7*h), Phi[8](0) = phi__0(8*h), Theta[1](0) = theta__0(h), Theta[2](0) = theta__0(2*h), Theta[3](0) = theta__0(3*h), Theta[4](0) = theta__0(4*h), Theta[5](0) = theta__0(5*h), Theta[6](0) = theta__0(6*h), Theta[7](0) = theta__0(7*h), Theta[8](0) = theta__0(8*h), U[1](0) = v__0(h)-phi__7*((D@@2)(v__0))(h), U[2](0) = v__0(2*h)-phi__7*((D@@2)(v__0))(2*h), U[3](0) = v__0(3*h)-phi__7*((D@@2)(v__0))(3*h), U[4](0) = v__0(4*h)-phi__7*((D@@2)(v__0))(4*h), U[5](0) = v__0(5*h)-phi__7*((D@@2)(v__0))(5*h), U[6](0) = v__0(6*h)-phi__7*((D@@2)(v__0))(6*h), U[7](0) = v__0(7*h)-phi__7*((D@@2)(v__0))(7*h), U[8](0) = v__0(8*h)-phi__7*((D@@2)(v__0))(8*h)}

 

Numerics

 

Here are the parameters/coefficients/functions that define the problem:

params :=

h = L/(n+1),
L = 10,

v__0 = (x -> 0),
theta__0 = (x -> 0),
phi__0 = (x -> 0),

v__left = (t -> 0),
theta__left = (t -> 0),
phi__left = (t -> 0),

v__right = (t -> sin(omega*t)),
theta__right = (t -> 1),
phi__right = (t -> 1),

phi__11 = 1 - eta + eta*rho__s*c__ps/(rho__f*c__pf),
phi__10 = phi__4/phi__1,
phi__9 = phi__3/phi__1,
phi__8 = phi__2/phi__1,
phi__7 = phi__6/(1 + lambda__1),
phi__6 = phi__5/(phi__1*Ra),
phi__5 = 1/(1 - eta)^2.5,
phi__4 = 1 - eta + eta*rho__s*beta__Cs/(rho__f*beta__Cf),
phi__3 = 1 - eta + eta*rho__s*beta__Ts/(rho__f*beta__Tf),
phi__2 = 1 + 3*(sigma - 1)*eta/(sigma + 2 - (sigma - 1)*eta),
phi__1 = 1 - eta + eta*rho__s/rho__f,

lambda__f = k__nf/k__f,
k__nf = (k__s + 2*k__f + 2*eta*(k__s - k__f))/(k__s + 2*k__f - eta*(k__s - k__f)),
lambda__f = k__nf/k__f,
a__0 = Pe*phi__11/lambda__f,
a__1 = Sc/(1 - eta),

M = 0.3,
Gr = 5,
Gm = 0.3,
Pe = 0.2,
Sc = 0.2,
Ra = 1,
lambda__1 = 0.5,
omega = 1,

rho__f = 5610,
rho__s = 2300,
c__pf = 0.880,
c__ps = 41.086,
beta__Tf = 0.0000125,
beta__Ts = 0.0000157,
beta__Cf = 0.0000125,
beta__Cs = 0.0000157,
eta = 0.02,
k__f = 1.046,
k__s = 1.160,
sigma = 0.2,

NULL:

Maple's dsolve() function is capable of solving DAEs.  Here is our solution:

eval(subs(params, SYS union IC)):
dsol := dsolve(%, numeric, output=operator, maxfun=0):

Let's plot a few samples:

odeplot(dsol, [t,U[3](t)], t=0..2*Pi);

odeplot(dsol, [t,V[3](t)], t=0..2*Pi);

odeplot(dsol, [t,Theta[3](t)], t=0..2*Pi);

odeplot(dsol, [t,Phi[3](t)], t=0..2*Pi);

 

 

Plotting in 3D

 

We wish to plot the solution v(x, t) over the domain

0 < x and x < L, 0 < t and t < T.  For the plotting grid, we take

the x__i = i*h, i = 0, 1, () .. (), n+1,   as we have done already,

and t__j = k*h, j = 0, 1, () .. (), m, where m is specified as desired,

and k = T/m.  Thus, the solution is represented as a grid
of points "(`x__i` , `t__j` , v(`x__i` , `t__j`))," that is, "(`x__i` , `t__j` , `V__i` (`t__j`)), "in 3D.  
The solutions  Theta(x, t) and Phi(x, t) are plotted in the same way.

 

A detail that needs toNULL be taken care of is that the solution

of the DAEs accounts for the interior nodes

i = 1, 2, () .. (), nonly.  The values of the solution corresponding to

the boundary nodes, that is, V__0(t), `V__n+1`(t), etc., are not

included in the solution of the DAE and need to be evaluated

separately through the prescribed boundary conditions.  These are:

V[0] := subs(params, v__left);
V[n+1] := subs(params, v__right);
Theta[0] := subs(params, theta__left);
Theta[n+1] := subs(params, theta__right);
Phi[0] := subs(params, phi__left);
Phi[n+1] := subs(params, phi__right);

proc (t) options operator, arrow; 0 end proc

proc (t) options operator, arrow; sin(t) end proc

proc (t) options operator, arrow; 0 end proc

proc (t) options operator, arrow; 1 end proc

proc (t) options operator, arrow; 0 end proc

proc (t) options operator, arrow; 1 end proc

Here is a proc for plotting the solution Q of the DAEs where Q is one of V, Theta, Phi.
over the domain 0 < x and x < L, 0 < t and t < T, and where m is the number of

time steps.

my_plot3d := proc(Q, T, m)
  local i, j, A, k:= T/m, L := subs(params, :-L);
  A := Array(0..n+1, 0..m, (i,j)->eval(Q[i](j*k), dsol), datatype=float[8]):
  plots:-display(GRID(0..L, 0..T, A), labels=[x,t,Q(x,t)], _rest);
end proc:

The solution v(x, t)

my_plot3d(V, 2*Pi, 20);

The solution theta(x, t):

my_plot3d(Theta, 2*Pi, 20);

The solution phi(x, t):

my_plot3d(Phi, 2*Pi, 20);

 

 

Download method-of-lines-and-DAEs.mw

 

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