0
$\begingroup$

Is the following PDE solvable in mathematica 9?enter image description here

When i solve it, the DSolve command does not do anything.

 eqn = y*D[u[x, y], x] + (x^3 + x - u[x, y])*D[u[x, y], y] == u[x, y]^2 + u[x, y];
 sol = DSolve[eqn, u[x, y], {x, y}]
$\endgroup$
  • 1
    $\begingroup$ Please include the Mathematica code that you are using. $\endgroup$ – Mr.Wizard Feb 1 '16 at 10:16
  • $\begingroup$ z := u[x, y] p := D[u[x, y], x] q := D[u[x, y], y] eqn = y*p + (x^3 +x-z)*q == z^2 + z; sol = DSolve[eqn, z, {x, y}] $\endgroup$ – Salman Zaffar Feb 1 '16 at 10:44
  • 1
    $\begingroup$ Please edit your question accordingly, and take your time to learn how to format your code (short version: four spaces indent, more: mathematica.stackexchange.com/editing-help). $\endgroup$ – Yves Klett Feb 1 '16 at 10:53
  • $\begingroup$ I think the equation and its Mathematica code are both ok now....please help..... $\endgroup$ – Salman Zaffar Feb 1 '16 at 10:55
  • $\begingroup$ MMA 10.3 also DSolve can not solve it. $\endgroup$ – Mariusz Iwaniuk Feb 2 '16 at 18:28
1
$\begingroup$

Only numerically,and that under certain circumstances.

eqn = y*D[u[x, y], x] + (x^3 + x - u[x, y])*D[u[x, y], y] == u[x, y]^2 + u[x, y];
sol = NDSolve[{eqn, u[x, 1] == -1/2, u[1, y] == -1/2}, u[x, y], {x, 1, 4}, {y, 1, 4}, PrecisionGoal -> 10, 
 MaxStepSize -> 0.001];

 Plot3D[Evaluate[u[x, y] /. sol], {x, 1, 4}, {y, 1, 4},PlotRange -> All]

enter image description here

Example2:

sol2 = With[{eps = 0.01}, 
NDSolve[{y*D[u[x, y], x] + (x^3 + x - u[x, y])*D[u[x, y], y] == 
u[x, y]^2 + u[x, y], u[x, 2] == eps, u[eps, y] == eps}, 
u, {x, eps, 2}, {y, eps, 2}]];
With[{eps = 0.01}, 
Plot3D[Evaluate[u[x, y] /. sol2], {x, eps, 2}, {y, eps, 2}, 
PlotRange -> All]]

enter image description here

Example3:

 sol3 = NDSolve[{y*D[u[x, y], x] + (x^3 + x - u[x, y])*D[u[x, y], y] ==
 u[x, y]^2 + u[x, y], u[1, y] == 1, u[x, 1] == 1}, 
 u, {x, 1, 2}, {y, 1, 2}, Method -> "Shooting", 
 PrecisionGoal -> 2]; Plot3D[u[x, y] /. sol3, {x, 1, 2}, {y, 1, 2}, 
 PlotRange -> All, PlotPoints -> 200]

enter image description here

Maple a little more able to.

enter image description here

enter image description here

| improve this answer | |
$\endgroup$
  • $\begingroup$ Thank you for the answer...How did you come up with the initial (Cauchy) data? $\endgroup$ – Salman Zaffar Feb 3 '16 at 3:33
  • $\begingroup$ I throw random initial data until only works.You can say a brute-force search. $\endgroup$ – Mariusz Iwaniuk Feb 3 '16 at 11:58
0
$\begingroup$

You need to specify at least two reasonable boundary conditions, such as equations for u(x,0) and u(0,y) or u(0,0) and u(L,L).

| improve this answer | |
$\endgroup$
  • $\begingroup$ It does not work now either. $\endgroup$ – Salman Zaffar Feb 2 '16 at 3:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.