Question about reformulating a difference equation as a matrix equation

The difference equation and the boundary value are: .

I know how to solve for y(n) with eigen vector to obtain homogeneous solution and particular solution, and solve the coeffeicient using the boundary value. However, I donot know how to reformulate the whole difference equation into a matrix equation. Can anyone help?

Answer 1

[MULI]

Hi William,

I understand that you are asking about the reformulation of difference equation into a matrix equation.

Here I am providing the MATLAB code that solves a specific linear difference equation with given boundary conditions by reformulating the problem into a matrix equation.

% Define the parameters 

N = 101; % Number of equations for n=0 to n=100 
A = zeros(N, N); % Initialize A matrix 
b = -1.35 * (0:N-1)'; % Correctly initialize b vector with -1.35n for n=0 to n=100 

% Populate the A matrix based on the difference equation 

A(1,1) = 1; % First equation only involves y(0) because y(-1) and y(-2) do not exist 

for n = 2:N 
    A(n, n) = 1; % y(n) coefficient 
    A(n, n-1) = -10.1; % y(n-1) coefficient 
    if n > 2 
        A(n, n-2) = -10.1; % y(n-2) coefficient, valid from n=2 onwards 
    end 
end 

% Adjust for boundary conditions 
b(N) = 101/6; % Adjust the last value of b for y(100) = 101/6 

% Solve the system Ay = b for y 
y = A\b; 

% Display specific values of y(n) 
disp(['y(0) = ', num2str(y(1))]); 
disp(['y(50) = ', num2str(y(51))]); 
disp(['y(100) = ', num2str(y(101))]); 

In summary, the code efficiently translates a linear difference equation with boundary conditions into a solvable matrix equation.