MATLAB Script - Euler's Method - Stability of Trapezium Rule - Maths Assessment Answer

Maths Assessment Answer

Task: Discuss the stability of trapezium rule by considering y⁰ = y: Here, the method is de ned as follows: h ^Un+1 ^{= U}n ⁺ ₂ ^[f(tn^{; U}n^{) + f(t}n+1^{; U}n+1^)] for U⁰ = f(t; U); U(t₀) = U₀:

2. Consider the IVP

y⁰ + 100y = (99t + 1)e ^t;         y(0) = 1: The exact solution is y = e ^100t +te ^t so that the irst term quickly becomes negligible relative to the second term, but continues to govern stability.

1. Apply Euler's method with h = 0:1 and plot (100 points) the solution estimates versus

0 t 1: Repeat with h = 0:01 and plot versus 0 t 5: Fully discuss the results as related to the sti ness of the di erential equation.

1. Then, write a MATLAB script that employs the user-de ned function trapezium and impliciteuler to solve the same sti equation. Solve the equation by using the de ned function. The le will return comparison of the plots on same gure by labeling the gures. Take h = 0:1 and t 2 [0; 1]: What if t 2 [0; 5]?
2. Write a MATLAB script that solves the following second-order initial-value problem using Explicit Euler with h = 0:025, 4th order Runge-Kutta with h = 0:1 and implicit mid-point rule h = 0:1. The le must return a table that includes the estimated solutions, as well as the global % relative errors, at x = 1 : 0:1 : 2 for all three methods. Discuss the results.

3x2y00 + 2xy0   y = 0 Hint: To construct a table use disp(t y solution') at the beginning of the loop(count how many space between 2 strings) and for each iteration you may use fprintf('\%5.2f \%7.6f \%5.4f',time,y,sol). Here, %5.2: 5 is # of counts of the spaces up to t whereas .2: means that take 2 digits after decimal point. Then, if you wish you may customize the output up to u:)

4. a) Transform into a system of rst-order IVPs.

4. b) Write a MATLAB script that employs the user-de ned function which can solve the given system in (a), accurately. The  le must return the plot of x₁  and x₂  versus0 t 8 in the same graph. It is recommended to use at least 100 points for smoothness of curves.

Hint: Decide whether an implicit method can be applicable or not. If yes, choose one of them. If not, then choose a method, which will be the best one, among Explicit Euler, 2nd order Runge-Kutta, 4th order Runge-Kutta.

5. Write a MATLAB script that employs the user-de ned function implicitEuler to solve the system of governing equations. The le must return two separate plots (use subplot): angular velocity ! versus 0 t 10, and the current i versus 0 t 10:

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