Page 271 - Handbook Of Integral Equations

3 . In the special case a = 0 and A > 0, the solution of the integral equation is given by
◦
formula (4) with

I s (1 + cos λ) – I c (λ + sin λ) I s sin λ + I c (1 + cos λ)
C 1 = k , C 2 = k ,
2+2 cos λ + λ sin λ 2 + 2 cos λ + λ sin λ
b b
√
k = 2A, λ = bk, I s = sin[k(b – t)]f(t) dt, I c = cos[k(b – t)]f(t) dt.
0 0

4.1-2. Kernels Quadratic in the Arguments x and t

b

2
2
7. y(x) – λ (x + t )y(t) dt = f(x).
a
The characteristic values of the equation:
1 1
λ 1 = , λ 2 = .
1 (b – a )+ 1 (b – a )(b – a) 1 (b – a ) – 1 (b – a )(b – a)
3
5
5
3
5
3
3
5
3 5 3 5
1 . Solution with λ ≠ λ 1,2 :
◦
2
y(x)= f(x)+ λ(A 1 x + A 2 ),
where the constants A 1 and A 2 are given by
1 1 1
f 1 – λ f f 2 – λ f
5
A 1 = 1 3 1 ∆ 3 – f 2 ∆ 1 , A 2 = 1 3 2 ∆ 3 – f 1 ∆ 5 ,

2
2
2
2
λ 2 1 ∆ – ∆ 1 ∆ 5 – λ∆ 3 +1 λ 2 1 ∆ – ∆ 1 ∆ 5 – λ∆ 3 +1
9 3 5 3 9 3 5 3
b b
n
n
2
f 1 = f(x) dx, f 2 = x f(x) dx, ∆ n = b – a .
a a
◦
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:

5
b – a 5
2
y(x)= f(x)+ Cy 1 (x), y 1 (x)= x + ,
5(b – a)
where C is an arbitrary constant and y 1 (x) is an eigenfunction of the equation corresponding
to the characteristic value λ 1 .
3 . Solution with λ = λ 2 ≠ λ 1 and f 1 = f 2 =0:
◦

5
b – a 5
2
y(x)= f(x)+ Cy 2 (x), y 2 (x)= x – .
5(b – a)
where C is an arbitrary constant and y 2 (x) is an eigenfunction of the equation corresponding
to the characteristic value λ 2 .
◦
4 . The equation has no multiple characteristic values.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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