Poisson's equation on rectangular domains I
This application solves two-dimensional Poisson's equation on a rectangular domain with Dirichlet boundary condition in the form \[ \begin{align} -\Delta u(\mathbf{x}) &= f(\mathbf{x}),\quad \mathbf{x}\in\Omega \\ u(\mathbf{x}) &= 0,\quad \mathbf{x}\in \partial\Omega \end{align} \] where $\mathbf{x}$ is $(x,y)$ in the domain $\Omega=[0,a]\times[0,b]$ and $\partial\Omega$ is the domain boundary. The symbol $\Delta u$ or in some textbooks use $\nabla^2 u$ is $\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}$ in the Cartesian coordinate. Note that the minus sign in the equation above $-\Delta u=f$ is written $\Delta y=-f$ in some textbooks. In order to be solved by this calculator, the function $f(x,y)$ must be continuous in $\Omega$. In the case $f(x,y)=0$ the Poisson's equation above reduce to Laplace's equation \[-\Delta u=0\]
Data Input
The calculator requires two parts of input. The first part is function $f(x,y)$ in the equation above and it must be entered as f(x1,x2). Another part is data for $a$ and $b$ where $1\leqslant a\leqslant 10$ and $1\leqslant b\leqslant 10$.