1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

290 Chapter 4 The Potential Equation

on the assumption thatis small anduis much smaller thanU 0. Using

this boundary condition and the condition thatu(x,y)→0asy→∞,

set up and solve a complete boundary value problem forφ,assuming

M<1.

29.By superposition of solutions (αranging from 0 to∞) find the flow past

awallwhoseequationisy=f(x). Hint: Use the boundary condition

v(x, 0 )=U 0 f′(x)=

∫∞

0

[

A(α)cos(αx)+B(α)sin(αx)

]

dα.

30.In hydrodynamics, the velocity vector in a fluid isV=−grad(u),where

uis a solution of the potential equation. The normal component of ve-

locity,∂u/∂n, is 0 at a wall. Thus the problem

∇^2 u= 0 , 0 <x< 1 , 0 <y< 1 ,

∂u

∂x

( 0 ,y)= 0 , ∂u

∂x

( 1 ,y)=− 1 , 0 <y< 1 ,

∂u

∂y

(x, 0 )= 0 , ∂u

∂y

(x, 1 )= 1 , 0 <x< 1 ,

represents a flow around a corner: flow inward at the top, outward at the

right, with walls at left and bottom. Explain why, in a fluid flow problem,

it must be true that ∫

C

∂u

∂nds=0(∗)

ifuis a solution of the potential equation in a regionR,∂u/∂nis the

outward normal derivative,Cis the boundary of the region, andsis arc

length.

31.Under the conditions stated in Exercise 30, prove the validity of (∗).

Hint: Use Green’s theorem.

32.The Neumann problem consists of the potential equation in a regionR

and conditions on∂u/∂nalongC, the boundary ofR. Show (a) that

∫

C

∂u

∂nds=^0

is a necessary condition for a solution to exist, and (b) ifuis a solution

of the Neumann problem, so isu+c(cis constant).

33.Show thatu(x,y)=^12 (y^2 −x^2 )is a solution of the problem in Exercise 30.

34. a. Show that the given function is a solution of the potential equation.