I have a differential equation:
$$ -u''(x) + \sigma u'(x) = f(x), \quad 0<x<1$$ with boundary conditions $u(0) = \alpha$ and $u(1) = \beta$.
I've discretized equation using symmetric (finite) differences:
$$ \frac{-u_{i-1} + 2u_i – u_{i+1}}{h^2} + \sigma \frac{u_{i+1} – u_i}{h} = f_i $$
If $n$ is the number of discretization points, $h = \frac{1}{n+1}$.
My question is how would I get that in a matrix form $Ax = f$ ?
Best Answer
I'm guessing you want $u(0) = \alpha$ and $u(1) = \beta$, not $u(a) = \beta$. Anyway, in the discretized version I'm guessing it means $u_0 = \alpha$ and $u_n = \beta$. So in fact you only want to solve for $u_1,\cdots, u_{n-1}$. I'll write $N = n+1$ just to not write $+1$'s everywhere. The first equation (for $u_1$) tells you that $$ N^2(-u_0 + 2u_1 - u_2) + N \sigma (u_2 - u_1) = f_1, $$ i.e. $$ (2N^2 - \sigma N) u_1 + (-N^2 + N \sigma)u_2 = f_1 + N u_0 = f_1 + N^2 \alpha. $$ For $u_{n-1}$, you have the equation $$ N^2 (-u_{n-2} + 2u_{n-1} - u_n) + N \sigma (u_n - u_{n-1}) = f_{n-1}, $$ i.e. $$ (-N^2) u_{n-2} + (2N^2 - N \sigma) u_{n-1} = f_{n-1} + (N^2 - N \sigma) u_n = f_{n-1} + (N^2 - N \sigma) \beta. $$ For the other equations, you get $$ N^2(-u_{i-1} + 2 u_i - u_{i+1}) + N\sigma (u_{i+1} - u_i) = f_i, $$ i.e. $$ (-N^2) u_{i-1} + (2N^2 - N \sigma) u_i + (-N^2 + N \sigma) u_{i+1} = f_i. $$
To put all these equations in a clean form, let $$ A = \begin{bmatrix} 2 & -1 & 0 & \cdots & & \\ -1 & 2 & -1 & 0 & \cdots & & \\ 0 & -1 & 2 & -1 & 0 & \cdots & \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots \\ \\ & & & \ddots & \ddots & \ddots & \\ & & & & -1 & 2 & -1 \\ & & & & & -1 & 2 \end{bmatrix} $$ and $$ B = \begin{bmatrix} -1 & 1 & & & \\ & -1 & 1 & & \\ & & \ddots & \ddots & \\ & & & -1 & 1 \\ & & & & -1 \end{bmatrix}. $$ Then letting $u = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_{n-1} \end{bmatrix}$, you're looking to solve $$ (N^2 A + N \sigma B) u = F, $$ where $$ F = \begin{bmatrix} f_1 + N^2 \alpha \\ f_2 \\ \vdots \\ f_{n-2} \\ f_{n-1} + (N^2 - N \sigma) \beta \end{bmatrix}. $$ Hope that helps,