A desktop application that uses the finite difference method to solve differential equations.
This application uses techniques from these lecture notes to compute the solution of PDEs numerically.
Numerical algorithm for solving the heat equation
We let $u(t, x)$ be the solution function for the differential equation defined by
$$ \begin{align} \text{PDE}:\quad &u_{t}(t, x) = \alpha u_{xx}(t, x) + S(t, x), 0 < x \leq L, 0 < t \leq T \\ \text{BC}:\quad &u(t, 0) = \Phi_1(t),\quad u_{t}(t, L) = \Phi_2(t) \\ \text{IC}:\quad &u(0, x) = f(x) \end{align} $$
Let $u[i][j] = u(i \Delta t, j \Delta x)$ for $i = 0$ to $i = K - 1$ and $j = 0$ to $j = N - 1$ for $\Delta t = T / (K - 1)$ and $\Delta x = L / (N - 1)$. For $\Delta t$ small enough, we can approximate $u_{t}(t, x)$ with the forward difference:
$$ u_{t}(t, x) = (u(t + \Delta t, x) - u(t, x)) / \Delta t $$
For $\Delta x$ small enough, we can approximate $u_{xx}(t, x)$ with the second order central difference:
$$ u_{xx}(t, x) = \frac{u(t, x + \Delta x) - 2 u(t, x) + u(t, x - \Delta x)}{(\Delta x)^2} $$
So after substituting and rearranging $u_{xx}$, $u_{t}$ in $u_{t}(t, x) = \alpha u_{xx}(t, x) + S(t, x)$, we get
$$ u(t + \Delta t, x) = u(t, x) + \frac{\alpha \Delta t}{(\Delta x)^2} (u(t, x + \Delta x) - 2 u(t, x) + u(t, x - \Delta x)) + S(t, x) \Delta t $$
Thus, we can compute all values of $u[i][j]$ with the following:
$$ u[i][j] = \begin{cases} u[i - 1, j] + \frac{\alpha \Delta t}{(\Delta x)^2} (u[i - 1, j + 1] - 2 u[i - 1, j] + u[i - 1, j - 1]) + S(i \Delta t, j \Delta x) \Delta t, &\text{if } 0 < i \leq K - 1, 0 < j < N - 1 \\ \Phi_1(i \Delta t), &\text{if } j = N - 1 \\ u[i, N - 2] + \Phi_2(i \Delta t) \Delta t, &\text{if } j = 0 \\ f(j \Delta x), &\text{if } i = 0 \end{cases} $$