Question

In Matlab please

da? Solve the boundary value problem: 0.1-0.057 subject to the conditions:y(0)=0 and y(10)=100. Find the maximum value of y over this region using the shooting method. Round your answer to TWO decimal points rounding up for 5 or more.

Answer 1

dly du² 0.1 dy - 0.05 y dze Ylo)-0, y(10)=100 assurone, wa dy dhe dy में dw da d2y du? 11 0,1 dy - 0.05 an 0.1 W -0.05% Hence, dy dx 11 ylo)= 0 dw de = 0olW-0.05% wlo) = unknown, ylio) - ylo) Assume W1o) = Ivo O 10 10-0

clc
clear all

xf = 10; % final time
y0 = 0; % initial value of y
yf = 100; % final value of y

%% define defferential equation
%- assume dy/dx = w
dy_dx =@(x,y,w) w;
dw_dx =@(x,y,w) 0.1*w - 0.05*y;
y(1,1) = y0;
% initial guess for w(0)
w(1,1) = (yf - y0)/(xf -0);

% solutin for first guess using euler method
h = 0.001; % stepsize
x = 0:h:xf;
for i = 1:length(x)-1
w(1,i+1) = w(1,i) + h*dw_dx(x(i),y(1,i),w(1,i));
y(1,i+1) = y(1,i) + h*dy_dx(x(i),y(1,i),w(1,i));
end
% second guess of w
w(2,1) = 3*(yf - y0)/(xf -0);
y(2,1) = y0; % remains same

% solution for second guess using euler method
for i = 1:length(x)-1
w(2,i+1) = w(2,i) + h*dw_dx(x(i),y(2,i),w(2,i));
y(2,i+1) = y(2,i) + h*dy_dx(x(i),y(2,i),w(2,i));
end
%% Refinement of solution using interpolation and further solution using
% Euler method
% next guess of w using interpolation technique
w(3,1) = w(1,1) + (yf-y(1,end))*(w(2,1)-w(1,1))/(y(2,end)-y(1,end));
y(3,1) = y0;
% Final solution using euler method
for i = 1:length(x)-1
w(3,i+1) = w(3,i) + h*dw_dx(x(i),y(3,i),w(3,i));
y(3,i+1) = y(3,i) + h*dy_dx(x(i),y(3,i),w(3,i));
end
%% Maximum value of y
ymax = max(y(end,1:end));
fprintf('\nMaximum value of y = %0.2f\n',ymax)

Command Window Maximum value of y = 108.80 fx »|

note:

thankyou:)