Solving system of differential equations using matlab

Question

Solve the system of 51 differential equations: $$\frac{du_1}{dt}=0$$ $$\frac{du_k}{dt}=100(u_{k+1}-2u_k+u_{k-1})$$ $$\frac{du_{51}}{dt}=0$$

with initial condition $$u_k=\frac{k-1}{10}$$ from $t=0$ to $t=10$

I know I need to use backwards euler to solve the system since that is what we are studying. Also I must use successive over relaxation scheme to solve the matrix. To begin using backwards euler i know $$\frac{u_k^{n+1}-u_k^n}{dt}=\frac{du_k^{n+1}}{dt}$$ and the apply that to the system, but coding this into matlab is where i am stuck.

Answer 1

Construct the system matrix

D = 51
A = -2*diag(ones(D-2,1))+diag(ones(D-3,1),1)+diag(ones(D-3,1),-1)
A = 100*A
b = zeros(D-2,1)
b(1)=100*u0(1)
b(D-2)=100*u0(D)

Then your ODE system is just $\dot u = Au+b$. Solving the implicit Euler step requires to solve $$u^{n+1}+dt·A·u^{n+1} = u^n-dt·b.$$