clear; close all; clc;
p = 70;
Nvals = [10 20 30 40 70];

figure; hold on; box on;

for N = Nvals
    h = 1/N;
    x = linspace(0,1,N+1)'; 
    A = zeros(N-1);
    b = zeros(N-1,1);

    for i = 1:N-1
        A(i,i) = 2/h;

        if i > 1
            A(i,i-1) = -1/h - p/2;
        end

        if i < N-1
            A(i,i+1) = -1/h + p/2;
        end
    end
    b(end) = (1/h - p/2);
    uh_inner = A\b;
    uh = [0; uh_inner; 1];

    plot(x, uh, 'o-', 'LineWidth', 2, ...
         'DisplayName', ['FEM, N=', num2str(N)]);
end

x_exact = linspace(0,1,2000)';
u_exact = (exp(p*x_exact) - 1)/(exp(p) - 1);
plot(x_exact, u_exact, 'k', 'LineWidth', 1, 'DisplayName', 'Exact');

title("FEM vs Analytical Solution for -u'' + p u' = 0,  p=70");
xlabel('x'); ylabel('u(x)');
legend('Location','best');