How to solve a PDE on Mathematica

Questions
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1

Hello everyone! so I want to know how to plot/(get the answer also) to this Second order partial differential equation on mathematica, I can’t seem to get the input right

Solve: Uxx + Uyy = 0 for 0 < x < L and 0 < y < L

At x = 0, U = 0

At y = 0, U = 0

At x = L, U = 0

At y = L, U = f(x)

being Uxx = d^2u/dx^2 and Uyy=d^2u/dy^2

MANYYYY MANY thanks

2

This is very similar to an example in the documentation for DSolve > Examples > Scope > Elliptic Partial Differential Equations (with Mma 10.3.1):

leqn = {Laplacian[u[x, y], {x, y}] == 0};
bc = {u[0, y] == 0, u[x, 0] == 0, u[L, y] == 0, u[x, L] == f[x]};
DSolve[{leqn, bc}, u[x, y], {x, y}]
3

Just Go here this is Best answer for this question -

Thanks , :slight_smile:

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4

Many thanks! I’ll give it a try now!

5

Thanks a lot! :slight_smile:

6

Hey, thanks very much for the code! it was extremely useful! I have now to input a very similar problem but the boundary conditions change and I’m kinda stuck there so I was wondering if I could ask you again! :slight_smile: so now my new boundary conditions are:

Solve: Uxx + Uyy = 0 for 0 < x < L and 0 < y < L

At x = 0, dU/dx = 0

At y = 0, U = 0

At x = a, dU/dx = 0

At y = b, U = f(x)

As you can figure, the problem is inserting the derivativs dU/dx in the boundary conditions I don’t really know how to do that, please help!

7

To enter partial derivatives in non-generic points you can use the syntax Derivative[n1,n2][u]:

DSolve[{Laplacian[u[x, y], {x, y}] == 0,
  Derivative[1, 0][u][0, y] == 0,
  u[x, 0] == 0,
  Derivative[1, 0][u][a, y] == 0,
  u[x, b] == f[x]},
 u[x, y],
 {x, y}]