Page 513 - Handbook Of Integral Equations

Let us apply the Fourier transform to Eq. (1). In this case, taking into account the convolution
theorem (see Subsection 7.4-4), we obtain
√
˜
˜
2π K(u) ˜y(u)= f(u). (2)
Thus, by means of the Fourier transform we have reduced the solution of the original integral
equation (1) to the solution of the algebraic equation (2) for the Fourier transform of the desired
solution. The solution of the latter equation has the form

˜
1 f(u)
˜ y(u)= √ , (3)
˜
2π K(u)
˜
˜
where the function f(u)/K(u) must belong to the space L 2 (–∞, ∞).
Thus, the Fourier transform of the solution of the original integral equation is expressed via the
Fourier transforms of known functions, namely, the kernel and the right-hand side of the equation.
The solution itself can be expressed via its Fourier transform by means of the Fourier inversion
formula:
∞ ∞ ˜
1 iux 1 f(u) iux
y(x)= √ ˜ y(u)e du = e du. (4)
˜
2π –∞ 2π –∞ K(u)
10.3-2. Equations With Kernel K(x, t)= K(x/t) on the Semiaxis

The integral equation of the first kind

∞
K(x/t)y(t) dt = f(x), 0 ≤ x < ∞, (5)
0
ξ
τ
can be reduced to the form (1) by the change of variables x = e , t = e , w(τ)= ty(t). The solution
to this equation can also be obtained by straightforward application of the Mellin transform, and this
method is applied in a similar situation in the next section.

10.3-3. Equation With Kernel K(x, t)= K(xt) and Some Generalizations
1 .We first consider the equation
◦

∞
K(xt)y(t) dt = f(x), 0 ≤ x < ∞. (6)
0
ξ
By changing variables x = e and t = e –τ this equation can be reduced to the form (1), but it is more
convenient here to apply the Mellin transform (see Section 7.3). On multiplying Eq. (6) by x s–1 and
integrating with respect to x from 0 to ∞, we obtain

∞ ∞ ∞
y(t) dt K(xt)x s–1 dx = f(x)x s–1 dx.
0 0 0
We make the change of variables z = xt in the inner integral of the double integral. This implies the
relation
∞

ˆ
ˆ
–s
K(s) y(t)t dt = f(s). (7)
0
Taking into account the formula
∞

–s
y(t)t dt = ˆy(1 – s),
0
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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