Special Functions - Heat Conduction: Mathematical Models and Analytical Solutions - page 437

When c 0, a 0 and a 1 can take any value. A repeated application of Eq. (A.21)

yields a 2 , a 4 , a 6 ,

=

··· in terms of a 0 and a 3 , a 5 , a 7 , ··· in terms of a 1 . Finally, we

obtain the solution of Eq. (A.19)

y = a 0 y 1 + a 1 y 2 ,

where

n ( n + 1 ) n ( n − 2 )( n + 1 )( n + 3 )

x 2 x 4 y 1 = 1 − + −··· ,

2! 4!

− (

n − 1 )( n + 2 )

+ (

n − 1 )( n − 3 )( n + 2 )( n + 4 )

x 3 x 5 y 2 = x −··· .

3! 5!

Note that a 0 and a 1 are two arbitrary constants. Also, the y 1 and the y 2 are linearly
independent. Thus y is the general solution of Eq. (A.19). By the solution structure
of linear homogeneous equations, Y 1 = a 1 y 2 are also solutions of

Eq. (A.19). The convergence radius of series Y 1 and Y 2 is 1. The Y 1 and the Y 2 are,
however, divergent at x

a 0 y 1 and Y 2 =

= ± 1.

=

0. The xy 1 is thus a particular solution.
For an integer n , in particular, Y 1 or Y 2 becomes a polynomial. When n is positive
and even or negative and odd, Y 1 reduces into a polynomial of degree not larger
than n .When n is positive and odd or negative and even, Y 2 also reduces into a
polynomial of degree not larger than n .When c

When c 1, a 1 = 0sothat a 3 = a 5 = ··· =

= 0, to express such polynomials,

rewrite Eq. (A.21) as

( k + 2 )( k + 1 )

a k = − a k + 2 ,

( − )( + + )

n k k n 1

where k + 2 ≤ n so that k ≤ n − 2. A repeated application of the recurrence formula

thus yields a n − 2 , a n − 4 , ··· in terms of a n . For example,

n ( n − 1 )

a n − 2 = − a n .

2 ( 2 n − 1 )

( 2 n ) !

Therefore, a n becomes an arbitrary constant. Take a n =

2 ,wehave

2 n ( n ! )

( 2 n − 2 m ) !

m

a n − 2 m =( − 1 )

! ,

n − 2 m ≥ 0 .

2 n m !

( n − m ) ! ( n − 2 m )

The Y 1 or the Y 2 thus reduces into a fixed polynomial denoted by

n
2

]
m = 0 ( − 1 )

[

( 2 n − 2 m ) !

m

! x n − 2 m

P n ( x )= , (A.22)

2 n m ! ( n − m ) ! ( n − 2 m )

where n 2 stands for the maximum integer less than or equal to n 2.The P n (

in
Eq. (A.22) is called the Legendre polynomial of n-th degree or the Legendre function
of the first kind .

x )