Higher Engineering Mathematics, Sixth Edition

An introduction to partial differential equations 527

from which,X=Acospx+Bsinpx

and Y=Cepy+De−py

or Y=Ccoshpy+Dsinhpy

or Y=Esinhp(y+φ)

Hence u(x,y)=XY

={Acospx+Bsinpx}{Esinhp(y+φ)}

or u(x,y)

={Pcospx+Qsinpx}{sinhp(y+φ)}

whereP=AEandQ=BE.

The first boundary condition is: u( 0 ,y)= 0 , hence
0 =Psinhp(y+φ)from which,P=0.
Hence,u(x,y)=Qsinpxsinhp(y+φ).
The second boundary condition is:u( 1 ,y)= 0 ,hence
0 =Qsinp( 1 )sinhp(y+φ)from which,
sinp= 0 ,hence,p=nπ forn= 1 , 2 , 3 ,...
The third boundary condition is:u(x, 0 )= 0 ,hence,
0 =Qsinpxsinhp(φ)from which,
sinhp(φ)=0andφ=0.
Hence,u(x,y)=Qsinpxsinhpy.
Since there are many solutions for integer values ofn,

u(x,y)=

∑∞

n= 1

Qnsinpxsinhpy

=

∑∞

n= 1

Qnsinnπxsinhnπy (a)

The fourth boundary condition is:u(x, 1 )= 4 =f(x),

hence,f(x)=

∑∞

n= 1

Qnsinnπxsinhnπ( 1 ).

From Fourier series coefficients,

Qnsinhnπ= 2 ×the mean value of

f(x)sinnπxfromx=0tox= 1

i.e. =

2

1

∫ 1

0

4sinnπxdx

= 8

[

−

cosnπx

nπ

] 1

0

=−

8

nπ

(cosnπ−cos0)

=

8

nπ

( 1 −cosnπ)

= 0 (for even values ofn),

=

16

nπ

(for odd values ofn)

Hence, Qn=

16

nπ(sinhnπ)

=

16

nπ

cosechnπ

Hence, from equation (a),

u(x,y)=

∑∞

n= 1

Qnsinnπxsinhnπy

=

16

π

∑∞

n(odd)= 1

1

n

(cosechnπsinnπxsinhnπy)

Now try the following exercise

Exercise 203 Further problemson the

Laplace equation

1. A rectangular plate is bounded by the
   linesx= 0 ,y= 0 ,x=1andy= 3 .Apply the

Laplace equation

∂^2 u

∂x^2

+

∂^2 u

∂y^2

=0 to determine

thepotentialdistributionu(x,y)overtheplate,

subject to the following boundary conditions:

u=0whenx= 00 ≤y≤2,

u=0whenx= 10 ≤y≤ 2 ,

u=0wheny= 20 ≤x≤ 1 ,

u=5wheny= 30 ≤x≤1.

⎡

⎣u(x,y)=^20

π

∑∞

n(odd)= 1

1

n

cosechnπsinnπxsinhnπ(y− 2 )

⎤

⎦

2. A rectangular plate is bounded by the
   linesx= 0 ,y= 0 ,x= 3 ,y= 2 .Determine the
   potential distributionu(x,y)over the rec-
   tangle using the Laplace equation
   ∂^2 u
   ∂x^2

+

∂^2 u

∂y^2

=0, subject to the following

boundary conditions:

u( 0 ,y)= 00 ≤y≤2,

u( 3 ,y)= 00 ≤y≤2,

u(x, 2 )= 00 ≤x≤3,

u(x, 0 )=x( 3 −x) 0 ≤x≤3.

⎡

⎣u(x,y)=^216

π^3

∑∞

n(odd)= 1

1

n^3 cosech

2 nπ

3 sin

nπx

3 sinh

nπ

3 (^2 −y)

⎤

⎦