Description

Question1 (8 points)
Recall if Φ is continuous and satisfies a Lipschitz condition in y on the set
D = {(t,y) | t0 ≤ t ≤ T,−∞ < y < ∞}
then
y˙ = Φ(t,y), y(t0) = y0, where t0 ≤ t ≤ T
has a unique solution.
(a) (2 points) Show Φ satisfies a Lipschitz condition in y on A with Lipschitz constant c if A is convex and there exists a c > 0 such that
for all (t,y) ∈ A
(c) (2 points) Use the above to show the following IVP has a unique solution.
, where 0 ≤ t ≤ 1
(d) (2 points) Do you think it is a good idea to solve the following IVP numerically?
y˙ = 1 + y2, y(0) = 0, where 0 ≤ t ≤ 3
Justify your answer. Show Euler’s method is going to fail miserably for this IVP.
Question2 (6 points)
Consider the following IVP
y˙ = arctan(y), y(0) = y0, where t0 ≤ t ≤ T
(a) (2 points) Find a Lipschitz constant for arctan(y).
(b) (2 points) Find an upper bound on |y¨| without solving the IVP.
(c) (2 points) Find an upper bound on the absolute global error
|ek| = |yˆk − y(tk)|, where ˆyk is the Euler’s approximation to y(tk), in terms of step size and tk.
Question3 (13 points)
Solve the following IVP using the step size h = 1
y˙ = (2 + 0.01t2)y, y(0) = 4, where 0 ≤ t ≤ 15
(a) (1 point) By Euler’s method.
(b) (2 points) By the backward Euler’s method.
(c) (2 points) By the second-order Taylor’s method.
(d) (1 point) By the Heun’s method.
(e) (1 point) By the two-step Adams-Bashforth method.
(f) (2 points) It was mentioned in class that Heun’s method, which is derived by applying the trapezoidal rule

is one the simplest form of Runge-Kutta method. The other simple second-order Runge-Kutta method, which is also known as the modified Euler’s method, uses the mid-point rule

Use this information to derive this second-order Runge-Kutta method. Write a piece of pseudocode for it, then implement it to solve the above IVP.
(g) (1 point) The most widely used Runge-Kutta method is a fourth-order Runge-Kutta method, which uses four sequential evaluations of Φ during each time step, that is, it has four stages. Similar to the previous two Runge-Kutta, it can be understood from a quadrature rule. In this case, Simpson’s rule:

This scheme proceeds as follows:
yˆ0 = y0

where Φ1 = Φ(tk−1,yˆk−1)

Φ4 = Φ(tk−1 + h,yˆk−1 + hΦ3)
Use this fourth-order Runge-Kutta method to solve the above IVP.
(h) (1 point) Compare all of the above approximations to the exact solution by plotting them on the same graph.
(i) (2 points) Use the approximation from Euler’s method to find the value of y at
t = 9.625
by interpolation in Newton’s form.
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Question4 (3 points)
Use the classic fourth-order Runge-Kutta method to find the numerical solution of the following higher-order differential equation, and compare the results to the exact solution.
t3…y + t2y¨− 2ty˙ + 2y = 8t3 − 2, y(1) = 2, y˙(1) = 8, y¨(1) = 6 for 1 ≤ t ≤ 2 with h = 0.1. The exact solution is

Question5 (4 points)
Solve the following boundary value problem on the domain with

(a) (2 points) By the shoot method.
(b) (2 points) By the Finite-difference method.
Question6 (4 points)
Solve the following BVP on the domain [0,1] by using its variational form
y¨+ y˙ + t = 0; y(0) = 0 y(1) = 0
(a) (2 points) Assume a linear hat basis for the solution.
(b) (2 points) Assume a cubic polynomial basis for the solution.
Question7 (2 points)
Find the following integral using the 4-order Runge Kutta method.

Is equally spaced xk the best option for this problem?
Question8 (0 points)
Consider the following BVP on the domain [1,3]
x3y(4) + 6x2y(3) + 6xy00 − 10x = 0
The boundary conditions are
y(1) = y(3) = y0(1) = y0(3) = 0
(a) (1 point (bonus)) Find its variational form.
(b) (3 points (bonus)) Solve it using its variational form.
(c) (1 point (bonus)) Compare your solution and the derivative of your solution with the exact solution and its derivative obtained by writing the differential equation as

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