Help With Finite Difference Method in Solving Second Order Differenial Equation

[abooaboo110]

Homework Statement

y'' + 3y' + 2y = 0, y(0) = 1, y'(0) = 0

Homework Equations

Finite Difference Approximations:
y'' = (y(ii+1) - 2y(ii) + y(ii-1))/h^2
y' = (y(ii+1) - y(ii-1))/(2h)
where h is the finite difference.

The Attempt at a Solution

I wrote the MATLAB code (just to try out finite differences):

x = linspace(0,1,20);
h = x(2) - x(1);
N = length(x);
y0 = 1;
yp0 = 0;
ynext = y0 + h*yp0;

a = ones(size(x))*(1/h^2 - 3/(2*h))
b = ones(size(x))*(-2/h^2 + 2);
c = ones(size(x))*(1/h^2 + 3/(2*h))
r = zeros(size(x));

a(1) = 0; c(1) = 0; b(1) = 1; r(1) = y0;
a(2) = 0; c(2) = 0; b(2) = 1; r(2) = ynext;
%a(2) = -1/(2*h); b(2) = 0; c(2) = 1/(2*h); r(2) = yp0;

A = diag(a(2:N),-1) + diag(b,0) + diag(c(1:(N-1)),1)
y = inv(A)*r';
yt = 2*exp(-x) - exp(-2*x);
plot(x,y,'o',x,yt);

Basically setting up a tridiagonal matrix involving terms y(ii).
I can solve the problem by doing a two step forward iteration but do not see why solving the system does not give me the same(correct plot). yt gives the exact solution.

 

[KataKoniK]

Hello,

Thank you for sharing your MATLAB code and approach to solving this problem. It looks like you are on the right track with using finite difference approximations to solve for y'' and y'. However, I noticed that in your code, you are using a two-step forward iteration to solve for y, rather than using the tridiagonal matrix approach you mentioned. This could be the reason why your plot is not matching the exact solution.

I would suggest trying to solve the system using the tridiagonal matrix approach, as it will give you a more accurate solution. Also, make sure to double check your boundary conditions and ensure that they are being properly incorporated into your code.

Overall, your approach seems sound and I believe with a few adjustments, you will be able to obtain the correct plot. Keep up the good work!