Matrix Computations

Question 1-b

MATLAB CODE

a = 1;

f_x = @(x) 1;

alpha = 0;

beta = 0;

nv = [25, 50, 100];

epv = [0.1, 0.01];

exact_solution = @(x, epsilon) (exp(-a*x/epsilon) - 1) / (1 - exp(-a/epsilon));

for n = nv

dx = 1 / (n - 1);

x = linspace(0, 1, n)';

for epsilon = epv

A = zeros(n, n);

b = zeros(n, 1);

for i = 2:n-1

A(i, i-1) = -epsilon / dx^2 - a / (2*dx);

A(i, i) = 2 * epsilon / dx^2;

A(i, i+1) = -epsilon / dx^2 + a / (2*dx);

b(i) = f_x(x(i));

end

A(1, 1) = 1; b(1) = alpha;

A(n, n) = 1; b(n) = beta;

u_approx = A \ b;

u_exact = exact_solution(x, epsilon);

figure;

plot(x, u_approx, 'r--', 'LineWidth', 1.5, 'DisplayName', 'Approximate Solution');

hold on;

plot(x, u_exact, 'b--', 'LineWidth', 1.5, 'DisplayName', 'Exact Solution');

xlabel('x'); ylabel('u(x)');

, title(['n = ', num2str(n), ', \epsilon = ', num2str(epsilon)]);

legend;

grid on;

end

end

The Output

Graph 1




Graph 2