Fakultät für Mathematik und Informatik 1. Juli 2016 TU Bergakademie Freiberg
Dr. M. Helm/Dr. habil. U. Prüfert
Numerical Analysis of Differential Equations Boundary Value Problems
Exercise 1
Consider the following elliptic boundary value problem of second order in the one dimensional space:
−u00(x) =−5x3, x∈Ω = (0,1), u(0) =−2,
u(1) =−1.
a) Check thatu(x) = 14x5+34x−2is 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
Consider the following second order elliptic boundary value problem in the one dimensional space:
−u00(x) = 2, x∈Ω := (0,1), u(0) = 0,
u0(1) = 0.
(a) Verify that the solution of the boundary value problem is given byu(x) =x(2−x).
(b) Discretize the boundary value problem using central differences on a uniform grid with mesh width ∆x= 13. 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 byu(1) = 0? Explain your observations.
Exercise 3
Determine the approximate stationary temperature distribution in a thin quadratic metal plate with a side length of0.5 m. Two adjacent boundaries are hold on a temperature of 0◦C. On the other boundaries the temperature should be linearly increasing from0◦C to100◦C.
(Hint:Ifxandy 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 byu(x, y) = 400xyif the boundaries with zero boundary condition are placed along the coordinate axes.)
