1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

280 Chapter 4 The Potential Equation

e.add up the series (see Section 1.10, Exercise 5a). Thenv(r,θ)is given

by the single integral (Poisson integral formula)

v(r,θ)=

1

2 π

∫π

−π

f(φ)

c^2 −r^2

c^2 +r^2 − 2 rccos(θ−φ)dφ.

9.Solve Laplace’s equation in the quarter-disk 0<θ<π/2, 0<r<c,sub-

ject to the boundary conditionsv(r, 0 )=0,v(r,π/ 2 )=0,v(c,θ)=1.

10.Generalize the results of Exercise 9 by solving this problem:

1

r

∂

∂r

(

r∂v

∂r

)

+^1

r^2

∂^2 v

∂θ^2

= 0 , 0 <θ <απ, 0 <r<c,

v(r, 0 )= 0 ,v(r,απ)= 0 , 0 <r<c,

v(c,θ)=f(θ ), 0 <θ <απ.

Here,αis a parameter between 0 and 2.

11.Suppose thatα>1 in Exercise 10. Show that there is a product solution

with the property that∂v∂r(r,θ)is not bounded asr→ 0 +.

4.6 Classification of Partial Differential Equations

and Limitations of the Product Method

By this time, we have seen a variety of equations and solutions. We have con-
centrated on three different, homogeneous equations (heat, wave, and poten-
tial) and have found the qualitative features summarized in the following table:

Equation Features
Heat Exponential behavior in time. Existence of a limiting (steady-state) solution.
Smooth graph fort>0.

Wave Oscillatory (not always periodic) behavior in time. Retention of discontinuities
fort>0.

Potential Smooth surface. Maximum principle. Mean value property.

These three two-variable equations are the most important representatives
of the three classes of second-order linear partial differential equations in two
variables. The most general equation that fits this description is

A∂

(^2) u
∂ξ^2 +B
∂^2 u
∂ξ∂η+C
∂^2 u
∂η^2 +D
∂u
∂ξ+E
∂u
∂η+Fu+G=^0 ,