Page 565 - Handbook Of Integral Equations

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11.8-2. An Equation With the Kernel K(x, t)= t Q(x/t) on the Semiaxis
Here we consider the following equation on the semiaxis:

x
∞ 1
y(x) – Q y(t) dt = f(x). (16)
0 t t
To solve this equation we apply the Mellin transform which is defined as follows (see also Sec-
tion 7.3):

∞
ˆ
f(s)= M{f(x), s}≡ f(x)x s–1 dx, (17)
0
ˆ
where s = σ + iτ is a complex variable (σ 1 < σ < σ 2 ) and f(s) is the transform of the function f(x).
In what follows, we briefly denote the Mellin transform by M{f(x)}≡ M{f(x), s}.
ˆ
For known f(s), the original function can be found by means of the Mellin inversion formula

1 c+i∞
f(x)= M {f(s)}≡ f(s)x ds, σ 1 < c < σ 2 , (18)
ˆ
–s
–1 ˆ
2πi
c–i∞
where the integration path is parallel to the imaginary axis of the complex plane s and the integral
is understood in the sense of the Cauchy principal value.
On applying the Mellin transform to Eq. (16) and taking into account the fact that the integral
with such a kernel is transformed into the product by the rule (see Subsection 7.3-2)

∞
1 x
ˆ
M Q y(t) dt = Q(s) ˆy(s),
0 t t
we obtain the following equation for the transform ˆy(s):
ˆ
ˆ
ˆ y(s) – Q(s) ˆy(s)= f(s).
The solution of this equation is given by the formula
ˆ
f(s)
ˆ y(s)= . (19)
ˆ
1 – Q(s)
On applying the Mellin inversion formula to Eq. (19) we obtain the solution of the original integral
equation
ˆ

1 c+i∞ f(s) –s
y(x)= x ds. (20)
ˆ
2πi c–i∞ 1 – Q(s)
This solution can also be represented via the resolvent in the form
∞ 1

x
y(x)= f(x)+ N f(t) dt, (21)
0 t t
where we have used the notation
ˆ
Q(s)
ˆ
–1
ˆ
N(x)= M {N(s)}, N(s)= . (22)
ˆ
1 – Q(s)
Under the application of this analytical method of solution, the following technical difficulties
can occur: (a) in the calculation of the transform for a given kernel K(x) and (b) in the calculation
of the solution for the known transform ˆy(s). To find the corresponding integrals, tables of direct
and inverse Mellin transforms are applied (e.g., see Supplements 8 and 9). In many cases, the
relationship between the Mellin transform and the Fourier and Laplace transforms is first used:
x
x
–x
M{f(x), s} = F{f(e ), is} = L{f(e ), –s} + L{f(e ), s}, (23)
and then tables of direct and inverse Fourier transforms and Laplace transforms are applied (see
Supplements 4–7).
Remark 1. The equation

x
∞
α –α–1
y(x) – H x t y(t) dt = f(x) (24)
0 t
α
can be rewritten in the form of Eq. (16) under the notation K(z)= z H(z).
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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