1. Using the RK4 method, give the numerical approximation for the IVP: y' = ty; y(0) = e; with a step size of h = 0.5 for the point at t = 20. How does this compare to Euler's Method (e.g. simply using the ki values to approximate this ODE)?. 2. The exact solution to the IVP above is y(t) = et²/2+1. In a single figure, plot the results of RK4, Euler methods and the exact function (thus, your plot should contain 3 lines). To show the full Y-range, the Y-axis should be plotted using a log-scale, however the X-axis should remain linear. To do this, type “set(gca, 'YScale','log’)" on the line below that used to create the plot. Make sure that your figure has a legend that describes what each line corresponds to.
1. Using the RK4 method, give the numerical approximation for the IVP: y' = ty; y(0) = e; with a step size of h = 0.5 for the point at t = 20. How does this compare to Euler's Method (e.g. simply using the ki values to approximate this ODE)?. 2. The exact solution to the IVP above is y(t) = et²/2+1. In a single figure, plot the results of RK4, Euler methods and the exact function (thus, your plot should contain 3 lines). To show the full Y-range, the Y-axis should be plotted using a log-scale, however the X-axis should remain linear. To do this, type “set(gca, 'YScale','log’)" on the line below that used to create the plot. Make sure that your figure has a legend that describes what each line corresponds to.
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Use python or Matlab
Transcribed Image Text:1. Using the RK4 method, give the numerical approximation for the IVP: y' = ty;
y(0) = e; with a step size of h = 0.5 for the point at t = 20. How does this compare
to Euler's Method (e.g. simply using the ki values to approximate this ODE)?.
2. The exact solution to the IVP above is y(t) = et"/2+1. In a single figure, plot the
results of RK4, Euler methods and the exact function (thus, your plot should contain
3 lines). To show the full Y-range, the Y-axis should be plotted using a log-scale,
however the X-axis should remain linear. To do this, type "set(gca, 'YScale','log')"
on the line below that used to create the plot. Make sure that your figure has a
legend that describes what each line corresponds to.
Transcribed Image Text:1
Overview
In this lab you will implement a Runge-Kutta routine within Matlab. This iterative method
will allow you to approximate solutions for ordinary differential equations. The Runge-
Kutta Method is an extension of the first-order approximation (Euler Method).
1.1
RK4 Method
Suppose we have the initial value problem (IVP):
ý = f(t, y); y(to) = Yo
where ý is the time derivative of the function y, i.e. ý = 4. y is an unknown function
of time t, which we would like to approximate; we are told that ý, the rate at which y
changes, is a function of t and of y itself. At the initial time t, the corresponding y value
is yo. The function f and the data to, Yo are given.
We want to 'step' through the function to find a numerical approximation at some time,
t. Choose a step size, h, such that h > 0 and define:
Yn+1 = Yn + (ki + 2k2 + 2k3 + k4)
tn+1 = tn + h
for n = 0, 1,2, 3, ..., using:
f(tn, Yn)
ki)
k1
k2 = f(tn +5, Yn +
k3 = f(tn + , Yn + k2)
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