1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

354 Chapter 5 Higher Dimensions and Other Coordinates

vibrating membrane, will be found inThe Physics of Musical Instruments,by
Fletcher and Rossing.

Chapter Review

See the CD for review questions and special exercises.

Miscellaneous Exercises

1.Solve the heat problem

∇^2 u=

1

k

∂u

∂t,^0 <x<a,^0 <y<b,^0 <t,

∂u

∂x(^0 ,y,t)=d^0 ,

∂u

∂x(a,y,t)=^0 ,^0 <y<b,^0 <t,

u(x, 0 ,t)= 0 , u(x,b,t)= 0 , 0 <x<a, 0 <t,

u(x,y, 0 )=Tx

a

, 0 <x<a, 0 <y<b.

2.Same as Exercise 1, but the initial condition is

u(x,y, 0 )=Ty

b

, 0 <x<a, 0 <y<b.

3.Letu(x,y,t)be the solution of the heat equation in a rectangle as stated

here. Find an expression foru(a/ 2 ,b/ 2 ,t).Writeoutthefirstthree

nonzero terms for the casea=b.

∇^2 u=^1 k∂∂ut, 0 <x<a, 0 <y<b, 0 <t,

u= 0 on all boundaries,

u(x,y, 0 )=T, 0 <x<a, 0 <y<b.

4.Find the nodal lines of the square membrane. These are loci of points

satisfyingφmn(x,y)=0, whereφmnsatisfies∇^2 φ=−λ^2 φin the square

andφ=0 on the boundary.

5.Find the solution of the boundary value problem

1

r

d

dr

(

rdu

dr

)

=− 1 , 0 <r<a,

u( 0 )bounded, u(a)= 0 ,