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points) Show that the solution of the above problem with initial data of the form: is given by (2)Here is a even function of

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臺灣大學數學系

九十一學年度第一學期碩博士班資格考試題 偏微分方程(Differential)

Sept 11, 2002

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Total score: points .

( points) Solve the following Cauchy problem for : (a)

( points) Equation with initial condition

. (b)

( points) Equation with initial condition .

.

( points)(a) ( points) Show that for the general solution of the wave equation:

with spherical symmetry about the origin has the form:

with suitable and . Here is a positive constant. (b) ( points) Show that the solution of the above problem with initial data of the form:

is given by

(2)

Here is a even function of . .

(20 points) Let be harmonic in a domain D. Show that has partial derivatives of all orders in .

.

( points) Consider the following one-dimensional diffusion equation in the semi-infinite interval :

if

where is a positive constant and is a non-negative constant.

(a)

(10 points) Assume the solution of the problem take the form:

where

Show that satisfies the conditions:

(b)

(10 points) Find the solution of the above ordinary differential equation, and hence the solution of the original diffusion problem.

.

( points) Solve the following one-dimensional diffusion equation in the unit interval :

if

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