#Homework 1 : The Jello Cube (Continuous Simulation Assignment

IDS6938-Simulation Techniques - University of Central Florida

This is the framework for homework #1. Please first read the homework writeup.

The assignment is due: Monday, February 27 at 11:59PM (EST)

Arash Zarmehr (*)

Part One

The following ODE can be solved easily with Laplace method. Laplace method is the easy way to reach exact solution of some ODEs.

dy/dx=y-(1/2)*e^(x/2)*sin(5x)+5e^(x/2)*cos(5x)

Solution:

ℒ:

SY-Y(0)=Y-0.5(5/((S-0.5)^2+25))+5*((S-0.5)/(((S-0.5)^2+25))

Y*(S-1)=Y(0)+5*((S-0.5)/((S-0.5)^2+25)-0.5/((S-0.5)^2+25))

Y*(S-1)=Y(0)+5*((S-1)/((S-0.5)^2+25))

One of the answer which called public answer is S=1 but this is not answer which work for us.

The second answer is:

Y=5/((S-0.5)^2+25)

Now we can get Laplace inverse to find a solution which we need.

ℒ -1

y(x)=e^0.5x*sin⁡5x

Another way to find a solution for this ODE is to do Change of variables.

If we define a µ(x)= e^(-x) and take integral we will reach the same solution.

We were able to find a exact solution for this ODE, but sometimes it is very hard to find axact solution. Alternatively, math scientists inttroduces approxiame methods which try to find a solution with numerical ways but the problem is that the solution would not be exact. accurancy of these method was always part of scientific discussions.

Here we run a solution with four different method to compare them and see the accurancy of them. For methods are exact solution, Euler method, Midpoint method and RK4.

1. Exact Solution

| undeformed jello | deformed jello |

| ------------- | ------------- |

| | |