Using Shooting Method on Euler Method to Solve Second Order BVP

% Project 3 Shooting Method Using Euler Method
%y''+(1-x)y'+xy=x, y(0) =0, y(1) =2
clear; clc
%parameters
x0 = 0;
y0 = 0;
xf = 1;
u0 = 1;%input('yp value at intial x value:\n');
y1 = 2; %input('Input second y condition value:\n');
n = 10;
x(1) = x0;
y(1) = y0;
u(1) = u0;
h = (xf-x0)/n;
t = x0:h:xf ;

for i = 2:length(t)  
    
    x(i) = x(i-1)+h;
    y(i) = y(i-1)+u(i-1)*h;
    u(i) = u(i-1)+h*(x(i-1)-(x(i-1)*y(i-1))-(1-x(i-1)*u(i-1)));
    
end

A1 = [x;y];
fprintf('\n%8.4f %8.4f\n', A1)
plot(x,y,'r')

This code numerically solves the second ODE using shooting method on Euler Method. However, I'm trying to figure out how to get matlab to automatically test y' values at x0 (I have this variable as u0 in my code) in the iterations until it finds a y' value that satisfies the second boundry condition at y(1). I tried using linspace for u0 to test a string of numbers but it isn't working.

Answer 1

[Bjorn Gustavsson]

To do that you should make functions out of this script. That way you will have a much cleaner programme, that is far easier to generalize and extend.

For this example you could wrap your Euler-integrator into one such function:

function [x,y] = your_Euler_odeint(x0,xf,y0,dydx0,n)
x(1) = x0;
y(1) = y0;
u(1) = dydx0;
h = (xf-x0)/n;
t = x0:h:xf ;
 
for i = 2:length(t)  
   
    x(i) = x(i-1)+h;
    y(i) = y(i-1)+u(i-1)*h;
    u(i) = u(i-1)+h*(x(i-1)-(x(i-1)*y(i-1))-(1-x(i-1)*u(i-1)));
       
end
end

That function should be possible to call from your script:

x0 = 0;
y0 = 0;
xf = 1;
dydx0 = 1;
y1 = 2;
n = 10;
[x,y] = your_Euler_odeint(x0,xf,y0,dydx0,n);
y_diff_from_target_at_end = y(end)-y1;

Once you've done this you can see the shooting-method as an equation-solving task. Matlab has a large suite of functions to use for that, but from the scope of this task it seems as you should do that yourself from very basic programming. So you could simply do some test and adjustment, Newton-solving style, just test some dydx0 and see what value that gives, test a slightly larger or smaller dydx0 and see if that takes you closer to the solution, and then make the Newton-guess for where the solution might be. Here's a brief description: Newton's method

HTH

[Andrew Each]

Thank you

[Bjorn Gustavsson]

Your'e welcome. Happy that it helped.