Numerical Analysis of Differential Equations Boundary Value Problems (I)

Fakult¨ at f¨ ur Mathematik und Informatik 6. Juli 2012 TU Bergakademie Freiberg

W. Queck/M. Helm

Numerical Analysis of Differential Equations Boundary Value Problems (I)

============ Part B – Main Part ========================

Exercise 1

Consider the following elliptic boundary value problem of second order in the one dimensional space:

−u

(x) = −5x

, x ∈ Ω = (0, 1), u(0) = −2,

u(1) = −1.

a) Check that u(x) =

x

+

x − 2 is the solution to the above boundary value problem.

b) Discretize the boundary value problem with central finite differences on a uniform grid with mesh width ∆x = h = 1/3.

Write down the associated linear system and determine a numerical approximation to the solution of the boundary value problem by solving this linear system.

c) Compare your numerical solution with the exact one.

Exercise 2

Modify the boundary value problem from Exercise 1 to

−u

(x) = −5x

, x ∈ Ω = (0, 1), u(0) = −2,

u

(1) = 2.

a) Show, that u(x) =

x

+

x − 2 is still a solution of this boundary value problem.

b) Discretize the boundary value problem with central finite differences on a uniform grid with mesh width ∆x = h = 1/3.

Write down the associated linear system and determine a numerical approximation to the solution of the boundary value problem by solving this linear system.

c) Compare your numerical solution with the exact one.

Exercise 3

Write two programs for the boudary value problems in Exercise 1 and 2 which are working for an

arbitrary number of grid points. Try to verify the relation e = max

|u

− u(x

)| = O(h

) for

the global error in a numerical experiment.

Initial Value Problems 2 Exercise 4

Consider the following second order elliptic boundary value problem in the one dimensional space:

−u

(x) = 2, x ∈ Ω := (0, 1), u(0) = 0,

u

(1) = 0.

(a) Verify that the solution of the boundary value problem is given by u(x) = x(2 − x).

(b) Discretize the boundary value problem using central differences on a uniform grid with mesh width ∆x =

. Write down the associated linear system and determine a numerical approximation to the solution of the boundary value problem by solving this linear system.

(c) Compare your numerical solution with the exact one. What happens if the Neumann boun- dary condition is replaced by u(1) = 0? Explain your observations.

Exercise 5

Determine the approximate stationary temperature distribution in a thin quadratic metal plate with a side length of 0.5 m. Two adjacent boundaries are hold on a temperature of 0˚C. On the other boundaries the temperature should be linearly increasing from 0˚C to 100˚C.

(Hint: If x and y are the space coordinates, the problem can be described by the stationary heat equation ∆u(x, y) = 0 together with appropriate boundary value conditions. For the solution of the linear system you should use software like Matlab. For purposes of comparison notice, that the exact solution is given by u(x, y) = 400xy if the boundaries with zero boundary condition are placed along the coordinate axes.)

Exercise 6

A rectangular silver plate of 6 cm×5 cm is heated uniformly at each point with a constant energy rate q = 6.2802 W/cm

. Assume that the temperature along the boundaries is hold at

u(x, 0) = x(6 − x), u(x, 5) = 0 (0 < x < 6), u(0, y) = y(5 − y), u(6, y) = 0 (0 < y < 5).

Then, the stationary temperature distribution can be described by the Poisson equation

∆u(x, y) = − q

K for (x, y) ∈ Ω := (0, 6) × (0, 5),

together with the above boundary conditions. The constant K = 4.3543 W/(cm·K) is the thermal conductivity.

Approximate the temperature distribution by using the finite difference method with mesh widths

∆x = 0.4 and ∆y =

. Try also some other mesh widths.