need help with the collocation method

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Lisa Vol
Lisa Vol 2020-5-15
编辑： Anurag Ojha 2024-8-14
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
  1 个评论
Yash
Yash 2023-6-21
Hey can you please share some more information about the problem you are facing and the results you are expecting from the code.

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回答（1 个）

Anurag Ojha
Anurag Ojha 2024-8-13
编辑：Anurag Ojha 2024-8-14
Ran in:
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');

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