Vector calculus and numerical mathematics Worksheet 9

Fakult¨at Mikrosystemtechnik Sommersemester 2020

PD Dr. Michael Seidl 30 July 2020

Vector calculus and numerical mathematics Worksheet 9

Problem 1: Finite difference method (FDM) for a PDE

For a given functionS(x, y), we wish to solve (exactly or approximately) the PDE¹ ∂²

∂x² + ∂²

∂y²

T(x, y) = −S(x, y)

for the unknown functionT(x, y)≡T(r), subject to the Dirichlet boundary condition T(r) = 0 r∈∂Σ

,

where∂Σ is the rim (”boundary”) of a given region Σ ⊂R² in the xy-plane.

We consider the simple caseS(x, y) =S of a constant S, and several different regionsΣ, Σ₁ = the disk with radius R >0, centered at (x|y) = (0|0),

Σ₂ = the square with diagonal D >0 and corners ^D₂ 0

, 0 ^D₂

, −^D₂ 0

, ..., Σ₃ = the square with side L >0 and corners ^L₂

^L₂

, −^L₂ ^L₂

, ^L₂ − ^L₂

, ..., Σ₄ = the square with side L >0 and corners (0|0), (L|0), (0|L), (L|L) Σ₅ = the trianglewith side L >0 and corners (0|0), (L|0), (0|L).

(a) Can you solve the problem exactly (analytically) in some of these five cases?

(b) In each case, find a proper mesh of discrete points r_nm = r₀ + h·(ne_x + me_y) ≡

a + n·h b + m·h

(n, m∈Z), with some step size h >0 and some reference point r₀ = ^a_b

, in such a way that:

(1) For each mesh point r_nm∈Σ, also its four next neighbors are ∈Σ or ∈∂Σ. (2) There are sufficiently many mesh points ∈∂Σ.

(In typical exercise problems, such a mesh will be given from the beginning.) (c) Using the mesh found in part (b), plus an initial guess T⁽⁰⁾(r_nm) for all r_nm ∈ Σ,

find improved values T⁽ⁱ⁾(r_nm) (withi= 1,2,3, ...) by iterating a FDM.

1Written as ∇²T(r) = −¹_λs(r), this is the heat conduction equation, where T(r) is a steady-state temperature distribution,λis a thermal conductivity, ands(r) is a heating source density.

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Problem 2: ODEs

Find infinitely many solutions for each one of the following ODEs.

(a) f⁰⁰(x) + 8f⁰(x) + 65f(x) + 13 = 0.

(b) f⁰⁰(x) + 8f⁰(x) + 65 = 0.

(c) f⁰⁰(x) + 8f⁰(x)² = 0.

Problem 3: Average Temperature

A piece of solid material has the shape of one half of a cylinder Ω with radius R and height H,

Ω = n

r(x, y, z)

x² +y² ≤R², x≥0, 0≤z ≤Ho .

This solid shall have a non-uniform temperature distribution, T(x, y, x) = T₀ x

R. Compute the average temperature hT(r)i_r∈Ω of this solid:

(a) Using cartesian coordinates (x, y, z), (b) Using cylindrical coordinates (s, φ, z).

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