Computer Assignment

Extend your MATLAB code which you wrote last session to solve (1).

• Start by computing the matrices $\mathchoice{\hbox{\boldmath $\displaystyle A$}} {\hbox{\boldmath $\textstyle A$}} {\hbox{\boldmath $\scriptstyle A$}} {\hbox{\boldmath $\scriptscriptstyle A$}}$ and $\mathchoice{\hbox{\boldmath $\displaystyle M$}} {\hbox{\boldmath $\textstyle M$}} {\hbox{\boldmath $\scriptstyle M$}} {\hbox{\boldmath $\scriptscriptstyle M$}}$ by hand using the basis functions

  $$\varphi_1(x) = \left\{ \begin{array}{ll} 3x, & 0 < x < 1/3 2-3... ...0 < x < 1/3 0, & 1/3 < x < 2/3 3x-2, & 2/3 < x < 1 \end{array} \right . .$$ (15)

  
  
  Once the matrices have been computed you have to implement the Backward Euler Method. It might look something like this

      
      T = 1; % final time
      t = 0; % current time
      k = 0.05; % time step
      
      while t < T
      
        xi = (M + k * A) \ (M * xi);
        t = t + k;
        U = zeros(N+1,1);
        U(2:N+1) = xi; 
        plot(x,U) 
        pause(0.1)
      
      end
  

  To get a nicer looking animation of the solution you might want to check out the command axis.

• Improve the solution approximation $U$ by increasing the number of basis functions $\varphi_j(x)$.

• There are other ways to discretize $\mathchoice{\hbox{\boldmath $\displaystyle M$}} {\hbox{\boldmath $\textstyle M... ...} {\hbox{\boldmath $\scriptstyle 0$}} {\hbox{\boldmath $\scriptscriptstyle 0$}}$, for instance

  $${\mathchoice{\hbox{\boldmath$\displaystyle M$}} {\hbox{\boldmath$... ...}} {\hbox{\boldmath$\scriptstyle 0$}} {\hbox{\boldmath$\scriptscriptstyle 0$}},$$ (16)

  
  
  which is called the Crank-Nicolson Method. Implement this iteration scheme instead of the Backward Euler Method.