Boundary value problems exercises

5.4. Boundary value problems exercises#

Answer the following exercises based on the content from this chapter. The solutions can be found in the appendices.

Exercise 5.1

Determine which of the following BVPs have a unique solution

(a) \(y'' = - \dfrac{4}{t} y' + \dfrac{2}{t^2 } y - \dfrac{2\ln (t)}{t^3 }, \qquad y(1) = 1/2, \qquad y(2) = \ln (2)\);

(b) \(y'' = e^t + y\cos(t) - (t + 1) y', \qquad y(0) = 1, \qquad y(2) = \ln(3)\);

(c) \(y'' = (t^3 + 5)y + \sin(t), \qquad y(0) = 0,\qquad y(1) = 1\);

(d) \(y'' = (5y + \sin(3t)) e^t, \qquad y(0) = 0,\qquad y(1) = 0\).

Exercise 5.2

Consider the following boundary value problem

\[ y'' + 4y = 0, \qquad y(0) = 0, \qquad y(1) = 1. \]

Calculate the Euler method solutions using a step length of \(h=0.2\) and guess values of \(y'(0) = 1\) and \(y'(0) = 2\).

Solution

\(s = 1\):

\(t\)	\(y_1\)	\(y_2\)
0.00	0.0000	1.0000
0.20	0.2000	1.0000
0.40	0.4000	0.8400
0.60	0.5680	0.5200
0.80	0.6720	0.0656
1.00	0.6851	-0.4720

\(s = 2\):

\(t\)	\(y_1\)	\(y_2\)
0.00	0.0000	2.0000
0.20	0.4000	2.0000
0.40	0.8000	1.6800
0.60	1.1360	1.0400
0.80	1.3440	0.1312
1.00	1.3702	-0.9440

Exercise 5.3

Use the Secant method to calculate a guess value for \(y'(0)\) based on your solutions to the boundary value problem in Exercise 5.2. Calculate the solutions using the Euler method using this new guess value.

Exercise 5.4

Calculate the solution of the boundary value problem in Exercise 5.2 using the finite-difference method with a step length \(h=0.2\).

Exercise 5.5

The exact solution to the boundary value problem in Exercise 5.2 is \(y = \dfrac{\sin(2t)}{\sin(2)}\). Produce a plot comparing the solutions to this BVP using the Euler method from Exercise 5.3 and the finite-difference method from Exercise 5.4 against the exact solution.

\(t\)	\(y_1\)	\(y_2\)
0.00	0.0000	1.4596
0.20	0.2919	1.4596
0.40	0.5838	1.2261
0.60	0.8291	0.7590
0.80	0.9809	0.0957
1.00	1.0000	-0.6889