Euler's Method/Improved Euler's Method

Having trouble working out the bugs in my Improved Euler's Method code. I previously had trouble with the normal Euler's method code, but I figured it out.

Euler's Method (working code):

syms t y
h=0.01;
N=200;
y(1)=1;
t(1)=0;
for n=1:N
    k1=1-t(n)+4*y(n);
    y(n+1)=y(n)+h*k1;
    t(n+1)=t(n)+h;
end
plot(t,y)

And here is my attempt at Improved Euler's Method:

h=0.01;
N=200;
y(1)=1;
t(1)=0;
for n=1:N
    k1=1-t(n)+4*y(n);
    k2=1-t(n+1)+4*(y(n)+h*k1);
    y(n+1)=y(n)+(h/2)*(k1+k2);
    t(n+1)=t(n)+h;
end
plot(t,y)

The error message that pops up is "Index exceeds the number of array elements (1)." I'm rather new at MATLAB, and don't know what this means, can someone help me rework this? Thank you!

[Lucas Howarth]

Here is the initial value problem: y'=1-t+4*y with y(0)=1 on the interval [0, 2] using a step size of h = 0.01

[Mike Asmanis]

Hey , how would i be able to solve this : y'(t)=cos(t + y) y(0)=0 t[0,3] exact solution y(t)=-t + 2arctan(t)

using your code?

Answer 1

[Sudhakar Shinde]

May be position of t(n+1)=t(n)+h; coulb be at the starting of loop.

[Lucas Howarth]

Thank you so much!! That worked for me!

[Sudhakar Shinde]

Welcome. Glad to help you.

[Lucas Howarth]

Sorry to bother you again, but now I have to do Runge-Kutta Method on the same ODE (step size is now h=0.1) and I'm getting the error "Array indices must be positive integers or logical values." I thought that I used similar formatting as I did in the Improved Euler problem, so I'm not sure what the issue is.

Here's my code:

clear all
close all
clc
h=0.1;
N=20;
y(1)=1;
t(1)=0;
for n=1:N
    t(n+1)=t(n)+h;
    k1=1-t(n)+4*y(n);
    k2=1-t(n+0.5)+4*(y(n)+0.5*h*k1);
    k3=1-t(n+0.5)+4*(y(n)+0.5*h*k2);
    k4=1-t(n+1)+4*(y(n)+h*k3);
    y(n+1)=y(n)+(h/6)*(k1+k2+k3+k4);
end
plot(t,y)

[Sudhakar Shinde]

Is any specific reason to use t(n+0.5)?

I am not sure about mathematical equation but if t(n+0.5) can be replaced with t(n+1), your error will get resolved.

[Lucas Howarth]

For the Runge-Kutta Method for approximation, k2 and k3 are done with the "t" value halfway between the current step and the next step. I'm not sure how to do this in MATLAB and still keeping integer values.

[Sudhakar Shinde]

have a look on:

https://www.mathworks.com/matlabcentral/answers/483679-how-to-make-a-function-that-uses-runge-kutta-method

Answer 2

[J. Alex Lee]

The error is telling you that at the first step of your loop (n=1), you are trying to access the n=2nd element of t and y, but at the stage, t and y are only scalars (arrays with only 1 element) variables. You are trying to access an element of the "arrays" that doesn't exist.

if you are trying to implement implicit Euler, your problem is math, not coding.

[J. Alex Lee]

my bad, i didn't look very closely. i guess you are doing a 2 step RK, and it is probably right according to Sudhakar's answer.

[Sudhakar Shinde]

yup.