522 DIFFERENTIAL EQUATIONS

=

30

nπ

( 1 −cosnπ)

=0 (whennis even) and

60

nπ

(whennis odd)

Hence, the required solution is:

u(x,t)=

∑∞

n= 1

{

Qne−p

(^2) c (^2) t
sinnπx
}

60
π
∑∞
n(odd)= 1
1
n
(sinnπx)e−n
(^2) π (^2) c (^2) t
Now try the following exercise.
Exercise 203 Further problems on the heat
conduction equation

1. A metal bar, insulated along its sides, is 4 m
   long. It is initially at a temperature of 10◦C
   and at timet=0, the ends are placed into
   ice at 0◦C. Find an expression for the tem-
   perature at a pointPat a distancexm from
   one end at any timetseconds aftert= 0.
   ⎡

⎣u(x,t)=^40

π

∑∞

n(odd)= 1

1

n

e−

n^2 π^2 c^2 t

(^16) sin
nπx
4
⎤
⎦

2. An insulated uniform metal bar, 8 m long,
   has the temperature of its ends maintained
   at 0◦C, and at timet=0 the temperature
   distributionf(x) along the bar is defined by
   f(x)=x(8−x). Ifc^2 =1, solve the heat con-

duction equation

∂^2 u

∂x^2

=

1

c^2

∂u

∂t

to determine

the temperatureuat any point in the bar at

timet.

⎡

⎣u(x,t)=

(

8

π

) (^3) ∑∞
n(odd)= 1
1
n^3
e−
n^2 π^2 t
(^64) sin
nπx
8
⎤
⎦

3. The ends of an insulated rodPQ, 20 units
   long, are maintained at 0◦C. At timet=0,
   the temperature within the rod rises uniformly
   from each end reaching 4◦C at the mid-point
   ofPQ.Find an expression for the temperature
   u(x,t) at any point in the rod, distantxfrom
   Pat any timetaftert=0. Assume the heat

conduction equation to be

∂^2 u

∂x^2

=

1

c^2

∂u

∂t

and

takec^2 = 1.

⎡

⎣u(x,t)=^320

π^2

∑∞

n(odd)= 1

1

n^2

sin

nπ

2

sin

nπx

20

e

−

(n (^2) π (^2) t
400
)
⎤
⎦
53.8 Laplace’s equation
The distribution of electrical potential, or tempera-
ture, over a plane area subject to certain boundary
conditions, can be described by Laplace’s equation.
The potential at a pointPin a plane (see Fig. 53.6)
can be indicated by an ordinate axis and is a function
of its position, i.e.z=u(x,y), whereu(x,y)isthe
solution of the Laplace two-dimensional equation
∂^2 u
∂x^2

• ∂^2 u
  ∂y^2
  =0.
  The method of solution of Laplace’s equation is
  similar to the previous examples, as shown below.
  Figure 53.7 shows a rectangleOPQRbounded
  by the linesx=0,y=0,x=a, andy=b, for which
  we are required to find a solution of the equation
  ∂^2 u
  ∂x^2


• 
  

  ∂^2 u
  ∂y^2
  =0. The solutionz=(x,y) will give, say,
  y
  P
  0 x
  z
  Figure 53.6
  0 x^ = a
  y = b
  x
  u
  (x
  ,y
  )
  R
  y
  Q
  P
  z
  Figure 53.7