Higher Engineering Mathematics

524 DIFFERENTIAL EQUATIONS

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)=0 andφ=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

4 sinnπxdx

= 8

[

−

cosnπx

nπ

] 1

0

=−

8

nπ

(cosnπ−cos 0)

=

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 204 Further problems on the

Laplace equation

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

Laplace equation

∂^2 u

∂x^2

+

∂^2 u

∂y^2

=0 to deter-

mine the potential distributionu(x,y) over

the plate, subject to the following boundary

conditions:

u=0 whenx= 00 ≤y≤2,

u=0 whenx= 10 ≤y≤2,

u=0 wheny= 20 ≤x≤1,

u=5 wheny= 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 distribution u(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)

⎤

⎦