I want to solve the following PDE using Fourier series:

  • $u(x,y): \Omega \to \mathbb{R}$,
  • $\Omega=(0,\pi)\times (0,2\pi)$
  • $u-3u_{xx}-u_{yy}= 3\sin(2x)-\sin(5x)$

$u_{xx}$ and $u_{yy}$ are second derivatives with respect to $x$ and $y$ respectively. The boundary conditions of the problem are:

  • Dirichlet boundary condition in $x$ direction: $u(x=0,y)=u(x=\pi,y)=0$
  • Neumann boundary condition in $y$ direction: $u_y(x,y=0)=u_y(x,y=2\pi)=0$

I chose my base functions for $u$ such that they satisfy boundary conditions: $\sin(nx)\cos(my/2)$. Therefore, the left-hand side of the equation would be: $$\sum a(n,m)\left[1+3n^2+\frac{m^2}{4}\right]\sin(nx)\cos(my/2)$$ My problem is that I should be able to write the right-hand side of the equation in form of $\sum b(n,m)\sin(nx)\cos(my/2)$, so that I can obtain $a(n,m)$. Any suggestions?

  • 1
    $\begingroup$ Please edit the question, instead of using * to bold the text, use \$ text \$. For example: \$\pi\$ renders as $\pi$. $\endgroup$ – fosho Jan 30 '16 at 13:55
  • $\begingroup$ See this page for info on how to typeset math formulas using mathjax/latex. It's really easy to learn. $\endgroup$ – Winther Jan 30 '16 at 14:07
  • $\begingroup$ Thanks for the tips and also the edit. $\endgroup$ – Eman Jan 30 '16 at 14:13
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    $\begingroup$ $m=0$ is allowed, which means $\sin(2x),\sin(5x) \in \{ \sin(nx)\cos(my) : 1 \le n < \infty, \;\; 0 \le m <\infty \}$ $\endgroup$ – DisintegratingByParts Jan 31 '16 at 0:11

You have to project the left-hand and right-hand sides of the PDE on the basis $\sin(nx)\cos(my/2)$, that is you have to compute $$\int_0^\pi\int_0^{2\pi} \Bigl(\sum_{n,m}a(n,m)[1+3n^2+m^2/4]\sin(nx)\cos(my/2)-3\sin(2x)+\sin(5x)\Bigr)\sin(nx)\cos(my/2)\, \mathrm{d}y\,\mathrm{d}x,\quad \forall n,m$$ It will give you the missing terms so that you can solve in $a(n,m)$.


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