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Sheet2/ExC_Num.m

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+% Parameters
+N = 501;             % number of grid points
+x = linspace(0,1,N); % grid
+dx = x(2)-x(1);      % grid spacing
+p_values = [-10, -3, -1, 0, 1, 3, 10];
+
+% Analytical solution function
+u_analytical = @(x,p) (abs(p) < 1e-12).*x + (abs(p) >= 1e-12).*((exp(p.*x)-1)./(exp(p)-1));
+
+figure;
+hold on;
+
+for k = 1:length(p_values)
+    p = p_values(k);
+    
+    e = ones(N-2,1);
+    % Second derivative: -u'' 
+    D2 = spdiags([e -2*e e], -1:1, N-2, N-2)/dx^2;
+    % First derivative: p*u'
+    D1 = spdiags([-e*0.5 e*0.5], [-1 1], N-2, N-2)/dx; 
+    L = -D2 + p*D1;
+    
+    % Right-hand side
+    b = zeros(N-2,1);
+    
+    % Incorporate boundary conditions
+    b(1) = b(1) + 0/dx^2;      % u(0)=0
+    b(end) = b(end) + 1/dx^2;  % u(1)=1
+    
+    % Solve linear system
+    u_inner = L\b;
+    u_num = [0; u_inner; 1]; % add boundary values
+    
+    % Plot numerical and analytical solution
+    plot(x, u_num, '--', 'LineWidth', 2, 'DisplayName', sprintf('num, p=%g',p));
+    plot(x, u_analytical(x,p), '-', 'LineWidth', 2, 'DisplayName', sprintf('ana, p=%g',p));
+end
+
+xlabel('x');
+ylabel('u(x)');
+title('Numerical vs Analytical solution of -u'''' + p u'' = 0');
+legend show;
+grid on;
+hold off;