Question

MATLAB help please!!!!!

1. Use the forward Euler method Vi+,-Vi + (ti+1-tinti , yi) for i=0.1, 2, , taking yo-y(to) to be the initial condition, to approximate the solution at 2 of the IVP y'=y-t2 + 1, 0 2, y(0) = 0.5. t Use N 2k, k2,...,20 equispaced timesteps so to 0 and t-1 2) Make a convergence plot computing the error by comparing with the exact solution, y: t (t+1)2 exp(t)/2, and plotting the error as a function of the step-size. Challenge (ungraded): This ODE is of a very special type that can be solved analytically. Find a procedure to analytically solve ODEs of the form y' + Py Q, where y, P, Q are univariate functions. 2. Use Matlab's ODE solver, ode45, to approximate y(2). How many timesteps did it take? How accurate was its approximation?

Answer 1

`Hey,

Note: Brother in case of any queries, just comment in box I would be very happy to assist all your queries

clc
clear all
k=1:20;
N=2.^k;
f=@(t,y) y-t^2+1;
err=[];
ex=(3)^2-exp(2)/2;
for i=1:length(N)
t=linspace(0,2,N(i));
h(i)=t(2)-t(1);
y=[0.5];
for j=2:length(t)
y(j)=y(j-1)+h(i)*f(t(j-1),y(j-1));
end
err(i)=abs(ex-y(end));
end
plot(h,err)
xlabel('Step size');
ylabel('Error');

​​​​​​​

Figure 1 File Edit View srt Iools Desktop Window Help 1.8 1.6 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.2 1.6 1.8 Step size O Type here to search ^徊溟 ENG 11:22 AM IN 04/25/19

2)

close all
clear
clc
tspan = [0 2];
y0 = 0.5;
dydt = @(t,y) y - t^2 + 1;
y_exact = @(t) (t + 1)^2 - exp(t)/2;
[t, y] = ode45(dydt, tspan, y0);
fprintf('Approximate y(2): %.8f\n', y(end))
fprintf('Number of timesteps: %d\n', length(t))
fprintf('Error in the Approx. y(2): %.8f\n', y(end) - (y_exact(2)))

MATLAB R2018a Search Documentation Log In EDITOR Find Files Compare Print Run Section New Open Save Advance Run and Find Indent Time Advance C: Users PRADEEP KALRA 、Documents MATLAB Editor-CAUsers\PRADEEP KALRA Current Folder TLABsaveee.m Name mathlab pradeep.mnewtonrapsonsbypradeep.msecantbypradeep.mlagrange nterp.m x+ saveee.m × lose all New folder a1.m span- [o 21: exact = e(t) {t + 1)^2 - exp(t)/2; t, yl-ode45 (dydt, tspan, yo): tprint f('Approximate y{2): % .8f\n', y(end)) fprintf ('Number of timesteps: dn', length (t)) tprint f ["Error 1n the Approx. y [2): %.8f\n', y(end) a4.m a5.m A1.dat A1.mat A2.dat A3.dat y exact (2) 12 Details Command Window New to MATLAB? See resources for Getting Started. Name ▲ Value Approximate y (2): 5.30547235 Number of timesteps: 45 Error in the Approx. y (2): 0.00000040 dydt @ityly-t 2+1 45x1 double 10,2) 45x1 double 0.5000 it)lt 1)A2-expit2 y exact Ln 12 Col 1 Click and drag to move bisectionmethodbypradeep.m or its tab.. ENG 842 AM IN 04/30/19 O Type here to search

Kindly revert for any queries

Thanks.