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) = et5.
(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 = et3/3.
(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.
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