Page 286 - Handbook Of Integral Equations

4.1-7. Singular Equations

In this subsection, all singular integrals are understood in the sense of the Cauchy principal value.

B 1 y(t) dt
65. Ay(x)+ = f(x), –1< x <1.
π –1 t – x
2
2
Without loss of generality we may assume that A + B =1.
◦
1 . The solution bounded at the endpoints:
1
B g(x) f(t) dt α 1–α
y(x)= Af(x) – , g(x) = (1 + x) (1 – x) , (1)
π –1 g(t) t – x
where α is the solution of the trigonometric equation

A + B cot(πα) = 0 (2)

1 f(t)
on the interval 0 < α < 1. This solution y(x) exists if and only if dt =0.
–1 g(t)
◦
2 . The solution bounded at the endpoint x = 1 and unbounded at the endpoint x = –1:
B 1 g(x) f(t) dt α –α
y(x)= Af(x) – , g(x) = (1 + x) (1 – x) , (3)
π –1 g(t) t – x

where α is the solution of the trigonometric equation (2) on the interval –1< α <0.

◦
3 . The solution unbounded at the endpoints:
1
B g(x) f(t) dt α –1–α
y(x)= Af(x) – + Cg(x), g(x)=(1 + x) (1 – x) , (4)
π –1 g(t) t – x
where C is an arbitrary constant and α is the solution of the trigonometric equation (2) on the
interval –1< α <0.
•
Reference: I. K. Lifanov (1996).

1 1
1
66. y(x) – λ – y(t) dt = f(x), 0 < x <1.
t – x x + t – 2xt
0
Tricomi’s equation.
Solution:
1 α α
β
1 t (1 – x) 1 1 C(1 – x)
y(x)= f(x)+ – f(t) dt + ,
α
2 2
1+ λ π x (1 – t) α t – x x + t – 2xt x 1+β
0
2 βπ
α = arctan(λπ)(–1< α < 1), tan = λπ (–2< β < 0),
π 2
where C is an arbitrary constant.
•
References: P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. G. Tricomi (1985).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 265