Question

Using MATLAB:

Consider the following Boundary Value Differential Equation:

y''+4y=0

y(0)=-2

y(π/4)=10

Which has the exact solution: y(x)= -2cos(2x)+10sin(2x)

Create a program that will allow the user to input the step size (in x), and two guesses for y'(0). The program will then use the Euler method along with the shooting method to solve this problem. The program should give the true error at y(π/8). Run your code with step sizes of π/400 and π/4000 and compare the errors. Chose any guesses for y'(0) that are reasonable. Also list the two errors you calculated.

Answer 1

For step size =0.007854 error in Euler method at x=pi/8 is 2.899593e-02.
For step size =0.000785 error in Euler method at x=pi/8 is 3.001448e-03.

%%Matlab code using Euler method for 2nd order differential equation
clear all
close all
%Program for Euler method for question.

f=@(x,y1,y2) y2;                %function (i)
g=@(x,y1,y2) -4*y1;             %function (ii)
%two initial guesses
in_1=0; in_2=100; step_size=[pi/400 pi/4000];
for ij=1:2
    %all guesses for Z_0 using shooting method
    a_ini=linspace(in_1,in_2,10);
    for ii=1:length(a_ini)
        x_result(1)=0;y1_result(1)=-2;y2_result(1)=a_ini(ii); %initial conditions
        h=step_size(ij);                 %step length
        x_in=x_result(1);           %Initial x
        x_max=pi/4;                 %Final x
        n=(x_max-x_in)/h; %number of steps

        %Euler 2d iterations
        for i=1:n
            x_result(i+1)= x_result(i)+h;
            y1_result(i+1)= y1_result(i)+h*double(f(x_result(i),y1_result(i),y2_result(i)));
            y2_result(i+1)= y2_result(i)+h*double(g(x_result(i),y1_result(i),y2_result(i)));

        end
        yy1(ii)=y1_result(end);

    end
    p = interp1(yy1,a_ini,10);
    %fprintf('\tThe value of y_dash(0)=z_0 using Shooting method is %f\n',p)

    clear x_result; clear y1_result; clear y2_result
    %solution using value of z_0
        x_result(1)=0;y1_result(1)=-2;y2_result(1)=p; %initial conditions
        x_max=pi/8;
        n=(x_max-x_in)/h; %number of steps

        for i=1:n
            x_result(i+1)= x_result(i)+h;
            y1_result(i+1)= y1_result(i)+h*double(f(x_result(i),y1_result(i),y2_result(i)));
            y2_result(i+1)= y2_result(i)+h*double(g(x_result(i),y1_result(i),y2_result(i)));

        end

      %function for the exact solution
      f_ext=@(x) -2*cos(2*x)+10*sin(2*x);
      err=abs(f_ext(x_result(end))-y1_result(end));
      fprintf('For step size =%f error in Euler method at x=pi/8 is %e.\n',h,err)
end
    %%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%