Page 155 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

STURM–LIOUVILLE TRANSFORMS 145
The lower boundary condition at x = a is then

[ (a,λ) f (a) − (a,λ) f (a)]r(a)

 
(a,λ)N [ f (a)] cos α + (a,λ)N α [ f (a)] sin α − (a,λ)N α [ f (a)] cos α

α
=   r(a)
+ (a,λ)N [ f (a)] sin α

α
(9.10)
But if (x,λ) is chosen to satisfy the Sturm–Liouville equation and the boundary con-
ditions then
N α [ (x,λ)] x=a = (a,λ)cos α + (a,λ)sin α

(9.11)
N β [ (x,λ)] x=b = (b,λ)cos β + (b,λ)sin β

and

(a,λ) = N α [ (a,λ)] cos α − N [ (a,λ)] sin α
α
(9.12)

(a,λ) = N [ (a,λ)] cos α + N α [ (a,λ)] sin α
α
and we have
[(N [ (a,λ)] cos α + N α [ f (a)] sin α)(N α [ (a,λ)] cos α + N [ (a,λ)] sin α)

α
α

− (N [ (a,λ)] cos α + N α [ (a,λ)] sin α)(N [ f (a)] cos α
α α
− N α [ f (a)] sin α)]r(a) (9.13)

={N [ f (a)]N α [ (a,λ)] − N α [ f (a)]N [ (a,λ)]}r(a)
α α
If the kernel function is chosen so that N α [ (a,λ)] = 0, for example, the lower boundary
condition is

−N α [ f (a)]N [ (a,λ)]r(a) (9.14)
α
Similarly, at x = b

(b,λ) f (b) − (b,λ) f (b) r(b) =−N β [ f (b)]N [ (b,λ)]r(b) (9.15)

β
Since (x,λ) satisfies the Sturm–Liouville equation, there are n solutions forming a set
of orthogonal functions with weight function p(x)and

2
n (x,λ n ) =−λ p(x) n (x,λ n ) (9.16)
n