Page 36 - Handbook Of Integral Equations

x

λ µ λ+β µ–β λ+γ µ–γ
54. Ax t + Bx t – (A + B)x t y(t) dt = f(x).
a
This is a special case of equation 1.9.47 with g(x)= x.
x

µ λ
σ
µ
55. t (x – t ) y(t) dt = f(x), σ > –1, µ >0, λ > –1.
a
µ
µ
The transformation τ = t , z = x , w(τ)= t σ–µ+1 y(t) leads to an equation of the form 1.1.42:
z
λ
(z – τ) w(τ) dτ = F(z),
A
µ
where A = a and F(z)= µf(z 1/µ ).
Solution with –1< λ <0:
µ sin(πλ) d x µ–1 µ µ –1–λ
y(x)= – t (x – t ) f(t) dt .
πx σ dx a
x
y(t) dt
56. = f(x).
(x + t) µ
0
This is a special case of equation 1.1.57 with λ = 1 and a = b =1.
The transformation
1 2τ
1 2z
x = e , t = e , y(t)= e (µ–2)τ w(τ), f(x)= e –µz g(z)
2 2
leads to an equation with difference kernel of the form 1.9.26:
z
w(τ) dτ
= g(z).
µ
cosh (z – τ)
–∞
x
y(t) dt
57. = f(x), a >0, a + b >0.
λ µ
λ
0 (ax + bt )
1 . The substitution t = xz leads to a special case of equation 3.8.45:
◦
1
y(xz) dz λµ–1

= x f(x). (1)
λ µ
(a + bz )
0
m
2 . For a polynomial right-hand side, f(x)= n A m x , the solution has the form
◦
m=0
n 1
A m m z m+λµ–1 dz
λµ–1
y(x)= x x , I m = λ µ .
I m 0 (a + bz )
m=0
The integrals I m are supposed to be convergent.
3 . The solution structure for some other right-hand sides of the integral equation may be
◦
obtained using (1) and the results presented for the more general equation 3.8.45 (see also
equations 3.8.26–3.8.32).
4 .For a = b, the equation can be reduced, just as equation 1.1.56, to an integral equation
◦
with difference kernel of the form 1.9.26.
√ √
2λ √ √
2λ
x x + x – t + x – x – t
58. √ y(t) dt = f(x).
a 2t λ x – t
The equation can be rewritten in terms of the Gaussian hypergeometric functions in the form
x
x
1
(x – t) γ–1 F λ, –λ, γ;1 – y(t) dt = f(x), where γ = .
2
a t
See 1.8.86 for the solution of this equation.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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