Reference no: EM133055914

FD methods of ODEs

Question 1. Apply Euler Method with step size h = 1/10 to find the approximate solution for the initial value problem:

y' = 5t⁴y, t ∈(0, 1]
y(0) = 1

(a) Show that the exact solution is: y = y(t) = e^t5.
(b) Write down the formula for the Euler Method and provide with the Matlab programs by the method.
(c) Plot the approximate solutions given by the Euler Methods on [0, 1] with step size h = 0.1 along with the exact solution.
(d) Provide with a table of the t values, approximations, and global trun- cation errors of the approximate solutions over the points, t₀, t₁, t₂.

Question 2. Apply the second-order Taylor Method to the initial value problems in Exercise 1. Using step size h = 1/4, calculate the approximation on the interval [0, 1]. Compare with the correct solution and find the error at t = 1.

Question 3. Use the the Heun Method to solve the IVP: y' = t²y,   y(0) = 1 with t ∈ [0, 1] and the exact solution: y = et^3/3.

(a) Plot the global truncation error of the Heun Method at t = 1 as a function of h = 0.1 2-k for 0 k 5 and compute the numerical convergence orders.

(b) Plot the Heun Method approximate solution on [0, 1], along with the exact solution. Use the step sizes h = 0.1.

Question 4. Apply Runge-Kutta method of order 4 to the IVP in Exercise 1. Give the general formula of the RK4 method and the Matlab codes. Print a table of the t variable values, approximations and global truncation error at each step on a grid of step size h = 0.1. Plot the approximate solutions for the step sizes h = 0.1 along with the exact solution.