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I'm trying to solve this equation: d2u/dx2 + d2u/dy2 = u +3

With the following boundary limits:

u(0, y)=0 u(1, y)=0 for 0<y<1

u(x, 0)=0 u(x, 1)=0 for 0<x<1

I began using a variable substitution: w = u + 3 but when I tried to evaluate the function with the boundary conditions I was left off with 4 constants that I couldn't find their values.

Am I headed in the right direction with this variable substitution?

spaldix
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1 Answers1

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No, I don't believe you are.

Here are a couple of approaches you might try.

  1. The complete solution is a sum of homogeneous and particular solutions. You get the homogenous solution by setting the RHS equal to zero. The solution will be a product of two functions u(x, y) = Q(x)*R(y). (This is called separation of variables if you'd like to research it more.). If you substitute that in you'll get two ordinary differential equations with trigonometric functions cosine and sine as solutions. Once you have that, the particular solution solves the RHS. Assume a polynomial. Sum the two and apply the boundary conditions for the complete solution.
  2. Laplace transform is a good approach, too. You'll apply a Laplace transform to get an ordinary differential equation. Solve that and transform back to get the complete solution.
duffymo
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  • So I woulnd't need the initial variable change before starting to use separation of variables? I was just trying to get rid of the "+ 3" before using it. If I could I would upvote you but I still needs some reputations points but thanks for your help anyways – spaldix Jun 18 '22 at 16:44
  • No, that's the wrong approach in my view. – duffymo Jun 18 '22 at 18:18