Page 278 - Handbook Of Integral Equations

4.1-4. Kernels Containing Higher-Order Polynomials in x and t

b
n
n
31. y(x) – λ (x + t )y(t) dt = f(x), n =1, 2, ...
a
The characteristic values of the equation:
1 1 n+1 n+1
λ 1,2 = √ , where ∆ n = (b – a ).
∆ n ± ∆ 0 ∆ 2n n +1
1 . Solution with λ ≠ λ 1,2 :
◦
n
y(x)= f(x)+ λ(A 1 x + A 2 ),
where the constants A 1 and A 2 are given by
f 1 – λ(f 1 ∆ n – f 2 ∆ 0 ) f 2 – λ(f 2 ∆ n – f 1 ∆ 2n )
A 1 = , A 2 = ,
2
2
2
2
λ (∆ – ∆ 0 ∆ 2n ) – 2λ∆ n +1 λ (∆ – ∆ 0 ∆ 2n ) – 2λ∆ n +1
n
n
b b 1

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

n
y(x)= f(x)+ Cy 1 (x), y 1 (x)= x + ∆ 2n /∆ 0 ,
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:
◦

n
y(x)= f(x)+ Cy 2 (x), y 2 (x)= x – ∆ 2n /∆ 0 ,
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.
b
n
n
32. y(x) – λ (x – t )y(t) dt = f(x), n =1, 2, ...
a
The characteristic values of the equation:
–1/2
1 n+1 n+1 2 1 2n+1 2n+1
λ 1,2 = ± 2 (b – a ) – (b – a )(b – a) .
(n +1) 2n +1
◦
1 . Solution with λ ≠ λ 1,2 :
n
y(x)= f(x)+ λ(A 1 x + A 2 ),
where the constants A 1 and A 2 are given by
f 1 + λ(f 1 ∆ n – f 2 ∆ 0 ) –f 2 + λ(f 2 ∆ n – f 1 ∆ 2n )
A 1 = , A 2 = ,
2
2
2
2
λ (∆ 0 ∆ 2n – ∆ )+1 λ (∆ 0 ∆ 2n – ∆ )+1
n n
b b
n 1 n+1 n+1
f 1 = f(x) dx, f 2 = x f(x) dx, ∆ n = (b – a ).
n +1
a a
2 . Solution with λ = λ 1 ≠ λ 2 and f 1 = f 2 =0:
◦
n
y(x)= f(x)+ Cy 1 (x), y 1 (x)= x + 1 – λ 1 ∆ n ,
λ 1 ∆ 0
where C is an arbitrary constant and y 1 (x) is an eigenfunction of the equation corresponding
to the characteristic value λ 1 .
◦
3 . The solution with λ = λ 2 ≠ λ 1 and f 1 = f 2 = 0 is given by the formulas of item 2 in
◦
which one must replace λ 1 and y 1 (x)by λ 2 and y 2 (x), respectively.
4 . The equation has no multiple characteristic values.
◦
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 257