Page 154 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

144 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
We wish to transform these differential forms into algebraic forms. First we write the
differential operator in standard form. Let

x
B(ξ)

r(x) = exp dξ
A(ξ)
a
r(x)
p(x) = (9.4)
A(x)
q(x) =−p(x)C(x)

Then
1
1
D[ f (x)] = (rf ) − qf = [ f (x)] (9.5)
p(x) p(x)
where is the Sturm–Liouville operator.
Let the kernel function K(x,λ)in Eq. (9.3) be

K(x,λ) = p(x) (x,λ) (9.6)

Then
b

T[D[ f (x)]] = (x,λ) [ f (x)]dx
a
b

= f (x) [ (x,λ)]dx + [( f x − x f )r(x)] b a (9.7)
a

while

N α [ f (a)] = f (a)cos α + f (a)sin α
d

N [ f (a)] = f (a)cos α + f (a)sin α (9.8)
α
dα

=− f (a)sin α + f (a)cos α
so that

f (a) = N α [ f (a)] cos α − N [ f (a)] sin α
α
(9.9)

f (a) = N [ f (a)] cos α + N α [ f (a)] sin α
α
where the prime indicates differentiation with respect to α.