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ClearAll["Global`*"];
pdeset = {Derivative[1, 0][U][t, x] == Derivative[0, 2][U][t, x], 
Derivative[1, 0][T][t, x] == Derivative[0, 2][T][t, x] + E Derivative[0, 1][U][t, x]^2}
ics = {U[0, x] == 0, T[0, x] == 0};
bcs = {U[t, 0] == cos[t], T[t, 0] == 1, U[t, 10] == 0, T[t, 10] == 0 };
bcAll = Flatten[{ics, bcs}, 1];
  • For E=0.1, when I try to solve

    sol = NDSolve[{pdeset, bcAll}, {U, T}, {t, 0, 5}, {x, 0, 4 Pi}]

I get this error NDSolve::bcedge: "Boundary condition U[t,10]==0 is not specified on a single edge of the boundary of the computational domain"

  • How to plot Derivative[0, 1][T][t, x] as x->0 vs t (0...Pi) for E=0.1,0.2,0.3?
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2  
You asked a question, accepted an answer, then modified the question and unaccepted the answer. How many times do you plan to do that? –  belisarius Mar 10 at 19:59

2 Answers 2

up vote 1 down vote accepted
+50

fixing the Cos typo and making a bold decision that the xrange ends at 4 Pi not 10

 ClearAll["Global`*"];
 pdeset = {Derivative[1, 0][U][t, x] == Derivative[0, 2][U][t, x], 
 Derivative[1, 0][T][t, x] == 
 Derivative[0, 2][T][t, x] + E Derivative[0, 1][U][t, x]^2}
 ics = {U[0, x] == 0, T[0, x] == 0};
 bcs = {U[t, 0] == Cos[t], U[t, 4 Pi] == 0,T[t, 0] == 1, T[t, 4 Pi] == 0};
 bcAll = Flatten[{ics, bcs}, 1];
 sol = NDSolve[{pdeset, bcAll}, {U, T}, {t, 0, 5}, {x, 0, 4 Pi}];

(* NDSolve::ibcinc: Warning: boundary and initial conditions are inconsistent. *)

The warning is because you have specified both U and T to be simultaneously 0 and 1 at the origin. You get a solution but it obviously doesn't satisfy both conditions.

 Plot3D[ (T /. First@sol)[t, x] , {x, 0, 4 Pi}, {t, 0, 5}, 
      PlotRange -> All]

this figure shows but U and T with red lines indicating the prescribed boundary conditions.

enter image description here

here is your plot:

 Plot[Evaluate[  (D[(T /. First@sol)[t, x], x]) /. x -> 0 ], {t, 0, 5}]

enter image description here

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After correcting some syntax errors and setting consistent boundary conditions:

ClearAll["Global`*"];
e = 0.1;
pdeset =
 {Derivative[1, 0][U][t, x] == Derivative[0, 2][U][t, x], 
  Derivative[1, 0][T][t, x] == Derivative[0, 2][T][t, x] + e Derivative[0, 1][U][t, x]^2}
ics =
  {U[0, x] == 0,
   T[0, x] == 0};
bcs =
  {U[t, 0] == Sin[t],
   T[t, 0] == 0,
   U[t, 10] == 0,
   T[t, 10] == 0};
bcAll = Flatten[{ics, bcs}, 1];

sol = NDSolve[{pdeset, bcAll}, {U, T}, {t, 0, 5}, {x, 0, 10}]

VectorPlot[{U[t, x], T[t, x]} /. sol[[1]], {t, 0, 5}, {x, 0, 10}, 
 VectorScale -> {Small, Scaled[.5], None}]

Mathematica graphics

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Thx dear but why change bc T[t,0]=1 and how plot Derivative[0, 1][T][t, x] as x->0 vs t (0...Pi) for e=0.1,0.2,0.3? –  MMM Nov 27 '13 at 3:14
    
@MMM The ics and the bcs should be coherent –  belisarius Nov 27 '13 at 3:20
    
Dear @belisarius, I am still hoping to learn how to plot Derivative[0, 1][T][t, x] as x->0 vs t (0...Pi) for e=0.1,0.2,0.3? –  MMM Nov 28 '13 at 4:45
    
i am still waiting for your @belisarius response? –  MMM Dec 2 '13 at 5:07

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