[?] Solve the following PDE?

Questions
,
1

Hi I want to solve the following PDE

heqn = D[u[x, t], t] == c*D[u[x, t], {x, 2}] -A* u[x, t]- B; // A, B and c are constants

ic = u[x, 0] == 2*A/b; // b is a constant
bc = u[0,t] ==0, u[500,t]==0;
sol = DSolveValue[{heqn, ic,bc}, u[x, t], {x, t}]

The solver is not working, can you please let me know where I am doing a mistake.

Thanks,
Vishal

2

Hello,

heqn = D[u[x, t], t] == c*D[u[x, t], {x, 2}] - A*u[x, t] - B;
ic = u[x, 0] == 2*A/b;
bc = {u[0, t] == 0, u[500, t] == 0};
sol = DSolve[{heqn, ic, bc}, u[x, t], {x, t}]
(* returns unevaluated *)

If a symbolic solver returns unevaluated, then Mathematica can’t solve the problem.
DSolve[]'s symbolic support for PDE’s is still somewhat limited, so don’t be surprised if some things don’t work yet.

Numeric solution:

NDSolve[] needs numeric values for constans.

  A = 1;
  B = 2;
  c = 1/10;
  b = 3;
  heqn = D[u[x, t], t] == c*D[u[x, t], {x, 2}] - A*u[x, t] - B;
  ic = u[x, 0] == 2*A/b;
  bc = {u[0, t] == 0, u[500, t] == 0};
  sol = NDSolve[{heqn, ic, bc}, u, {x, 0, 1}, {t, 0, 1}]
  Plot3D[u[x, t] /. sol, {x, 0, 1}, {t, 0, 1}, AxesLabel -> Automatic]

121.png
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Comparison with Maple 2017.3 symbolic pdsolve solver:

321.png
321.png1174×313 13.4 KB

Regards,Mariusz.

3

Thanks, Mariusz for solving the equation. Can you tell me what is the solution when I have initial conditions which are as follows,

u(x,0) = exp(-Ax) + a*exp(Ax) -d

I tried using NDSolve, but it did not fetch me any results again.

I used

 A = 0.005;
      B = 2;
      c = 0.002;
      b = 3;
    a = 0.003;
    d = 0.004;
      heqn = D[u[x, t], t] == c*D[u[x, t], {x, 2}] - A*u[x, t] - B;
      ic = u[x, 0] ==exp(-Ax) + a*exp(Ax)-d;
      bc = {u[0, t] == 0, u[500, t] == 0};
      sol = NDSolve[{heqn, ic, bc}, u, {x, 0, 1}, {t, 0, 1}]
      Plot3D[u[x, t] /. sol, {x, 0, 1}, {t, 0, 1}, AxesLabel -> Automatic]

Thanks,
Vishal

4

Hello.

You make syntax mistakes:

 u[x, 0] ==exp(-Ax) + a*exp(Ax)-d;(*its Wrong - Check Documentation Center or Help *)
 (* it should be:*)
 u[x, 0] == Exp[-A x] + a*Exp[A x] - d;

Corrected code:

 A = 0.005;
 B = 2;
 c = 0.002;
 b = 3;
 a = 0.003;
 d = 0.004;
 heqn = D[u[x, t], t] == c*D[u[x, t], {x, 2}] - A*u[x, t] - B;
 ic = u[x, 0] == Exp[-A x] + a*Exp[A x] - d;
 bc = {u[0, t] == 0, u[500, t] == 0};
 sol = NDSolve[{heqn, ic, bc}, u, {x, 0, 1}, {t, 0, 1}]
 Plot3D[u[x, t] /. sol, {x, 0, 1}, {t, 0, 1}, AxesLabel -> Automatic]

Regards,Mariusz

5

Thanks, Mariusz. The code works now. Can you post the maple analytical solution for the above initial and boundary conditions. Sorry for troubling you for this.

Regards,
Vishal

6

Hi

123456789.png
123456789.png1339×459 27.6 KB

Maple code:

  u(x, t) = Sum((1/250)*(Int((exp(-A*tau1)+a*exp(A*tau1)-d)*sin((1/500)*n*Pi*tau1), tau1 = 0 .. 500))*sin((1/500)*n*Pi*x)*exp(-
  (1/250000)*Pi^2*n^2*c*t-A*t), n = 1 .. infinity)+Int(Sum(-(1/250)*B*(Int(sin((1/500)*n*Pi*x), x = 0 .. 
  500))*sin((1/500)*n*Pi*x)*exp(-(1/250000)*(t-tau1)*(Pi^2*c*n^2+250000*A)), n = 1 .. infinity), tau1 = 0 .. t)

Mathematica code:

 Sum[(1/250)*(Integrate[(Exp[-A*tau1] + a*Exp[A*tau1] - d)*
       Sin[(1/500)*n*Pi*tau1], {tau1, 0, 500}])*Sin[(1/500)*n*Pi*x]*
    Exp[-(1/250000)*Pi^2*n^2*c*t - A*t], {n, Infinity}] + 
  Integrate[
   Sum[-(1/250)*B*(Integrate[Sin[(1/500)*n*Pi*x], {x, 0, 500}])*
     Sin[(1/500)*n*Pi*x]*
     Exp[-(1/250000)*(t - tau1)*(Pi^2*c*n^2 + 250000*A)], {n, 1, 
     Infinity}], {tau1, 0, t}]