Someone proposes to find a steady temperature in the region by giving data only along…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions

Problem

Someone proposes to find a steady temperature

u(x,y)

in the region

y > 0

by giving data only along the edge

y=0

: both the value

u(x,0)

and the normal derivative

\partial_yu(x,0)

, just as the wave equation takes a starting shape and a starting velocity. The field obeys Laplace's equation

\partial_x^2u + \partial_y^2u = 0

, with lengths measured in some fixed unit so that

x

and

y

are pure numbers. ((a)) Show that, for each positive integer

n

u_n(x,y) = n^{-2}\sin(nx)\,e^{ny}

solves Laplace's equation, and find its edge data

u_n(x,0)

and

\partial_yu_n(x,0)

. ((b)) The zero function solves the same problem with zero edge data. Give the largest size of each piece of edge data for

u_n

, and evaluate the amplitude

n^{-2}e^{n}

u_n

on the line

y=1

for

n=10

and for

n=20

. ((c)) Decide which of existence, uniqueness and continuous dependence on the data this example shows to fail, and say how the answer fits the type of Laplace's equation.

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