Page 139 - Handbook Of Integral Equations

2.1-6. Kernels Containing Square Roots and Fractional Powers

x √
44. y(x)+ A (x – t) ty(t) dt = f(x).
a
1
This is a special case of equation 2.1.49 with λ = .
2
x

√ √
45. y(x)+ A x – t y(t) dt = f(x).
a
1
This is a special case of equation 2.1.52 with λ = .
2
x y(t) dt
46. y(x)+ λ √ = f(x).
a x – t
Abel’s equation of the second kind. This equation is encountered in problems of heat
and mass transfer.
Solution:
x

2
y(x)= F(x)+ πλ 2 exp[πλ (x – t)]F(t) dt,
a
where
x
f(t) dt
F(x)= f(x) – λ √ .
a x – t
•
References: H. Brakhage, K. Nickel, and P. Rieder (1965), Yu. I. Babenko (1986).
x
y(t) dt
47. y(x) – λ √ = f(x), a >0, b >0.
2
0 ax + bt 2
◦
1 . The solution of the homogeneous equation (f ≡ 0) is
y(x)= Cx β (β > –1, λ > 0). (1)
Here C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation

β
1 z dz
λI(β) = 1, where I(β)= √ . (2)
0 a + bz 2
2 . For a polynomial right-hand side,
◦
N
n
f(x)= A n x
n=0
the solution bounded at zero is given by


N
A n

 n
 x for λ < λ 0 ,

1 – (λ/λ n )


n=0
y(x)=
 N
A n n β

x + Cx for λ > λ 0 and λ ≠ λ n ,



 1 – (λ/λ n )
n=0
√ 1
n
b 1 z dz
λ 0 = , λ n = , I(n)= √ .
Arsinh b/a I(n) 0 a + bz 2
Here C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation (2).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 118