need help with the collocation method

Hi, I need help with the collocation method.

n=numel (x);
A=zeros(n,n);
x(0)=a;
for i=1:n
    w=prod((x-x(i)));
    x(i)=(a+b)/2-((b-a)/2)*cos((i*pi)/n);
    for j=1:n
        l=w/((x-x(j))*diff(w(j)));
        d(i,j)=diff
    end
end

[Yash]

Hey can you please share some more information about the problem you are facing and the results you are expecting from the code.

Answer 1

[Anurag Ojha]

Hi Lisa

For collocation method, kindly refer to the below code. I have taken certain assumptions like

• Second-Order Differential Equation: The problem is assumed to be y′′(x)=f(x).
• Homogeneous Boundary Conditions: Boundary conditions are assumed to be y(-1) = 0 and y(1) = 0
• Chebyshev Nodes: Collocation points are calculated using Chebyshev nodes.
• Interval: The interval is assumed to be [−1,1]
• Function f(x): The right-hand side function is assumed to be f(x) = sin(πx)..
• Matrix A Construction: The collocation matrix A is constructed using a simplified finite difference approach.
• Visualization: The coefficients c represent the function values at the collocation points.

Kindly make changes to the code as per your use case.

% Define the number of collocation points
n = 5; % You can change this value

% Define the interval [a, b]
a = -1;
b = 1;

% Initialize matrices and vectors
A = zeros(n, n); % Collocation matrix
x = zeros(1, n); % Collocation points

% Calculate Chebyshev nodes
for i = 1:n
    x(i) = (a + b)/2 - ((b - a)/2) * cos((i * pi) / (n));
end

% Define the function f(x) (Right-hand side of the differential equation)
f = @(x) sin(pi * x); % Example function f(x) = sin(pi * x)

% Construct the collocation matrix A and the vector b
for i = 1:n
    for j = 1:n
        if i == j
            A(i,j) = 2 / ((b-a)^2); % Second derivative of the basis function at x(i)
        else
            A(i,j) = (-1)^(i+j) / ((x(i) - x(j))^2); % Based on Lagrange basis function properties
        end
    end
end

% Calculate the right-hand side vector (function values at collocation points)
b = f(x)';

% Solve for the coefficients c
c = A \ b;

% Display the results
disp('Collocation points (x):');

Collocation points (x):

disp(x);

   -0.8090   -0.3090    0.3090    0.8090    1.0000

disp('Coefficients (c):');

Coefficients (c):

disp(c);

    1.0725
    0.0197
   -1.3843
   -0.0912
    0.1653

% Plot the resulting approximate solution
plot(x, c, '-o');
xlabel('x');
ylabel('y');
title('Collocation Method Approximation');