2018 Practice Test 1

%Constants
y0 = 0;
y1 = 1;
n = 2:2:10;
x_range = 0:1/1000:1;

%Creates a vector y_correct with the values for the analytical solution,
%using dx = 1/1000
for j = length(x_range)
    y_correct = ones(1, length(x_range));
    y_correct(j) = 1.3888*exp(x_range(j)) + 0.6112*exp(-x_range(j)) - x_range(j)^2 -2;
end

for i = n
    BVPsolver(i);
end

plot(x_range, y_correct)

legend('n=2', 'n=4', 'n=6', 'n=8', 'n=10', 'Analytical')
xlabel('x')
ylabel('y(x)')
title('BVP Problem Solution')

function BVPsolver(n)
    %Creating constants for current value of n
    dx = 1/n;
    range = 0:dx:1;
    %Creating matrix b out of a ones array with height of n-1, width 1
    b = ones(n-1, 1);
    %Multiplying all ones in b by dx^4
    b = b.*(dx^4);
    %The pattern found in b, for y(nΔx), b = n^2*Δx^4, so we're just
    %multiplying by n^2 here
    for j = 1:(n-1)
        b(j) = b(j)*j^2;
    end
    %Subtracting y1 from the last value of b
    b(end) = b(end)-1;
    %Constants for the diagonals of matrix A
    k1 = 1;
    k2 = -2-dx^2;
    k3 = 1;
    %Creating a matrix with diagonals of the constants above, and all other
    %values = 0
    A = diag(ones(1, n-2).*k1, 1) + diag(ones(1, n-1).*k2) + diag(ones(1, n-2).*k1, -1);
    y = [0;mldivide(A, b);1];
    plot(range, y, 'DisplayName',strcat('n=',num2str(n)))
    grid on
    hold on
end