Page 567 - Handbook Of Integral Equations

11.8-4. The Method of Model Solutions for Equations on the Entire Axis

Let us illustrate the capability of a generalized modification of the method of model solutions (see
Subsection 9.6) by an example of the equation

∞
βt
Ay(x)+ Q(x + t)e y(t) dt = f(x), (32)
–∞
where Q = Q(z) and f(x) are arbitrary functions and A and β are arbitrary constants satisfying some
constraints.
For clarity, instead of the original equation (32) we write

L [y(x)] = f(x). (33)

For a test solution, we take the exponential function

px
y 0 = e . (34)
On substituting (34) into the left-hand side of Eq. (33), after some algebraic manipulations we obtain

∞

px
L [e ]= Ae px + q(p)e –(p+β)x , where q(p)= Q(z)e (p+β)z dz. (35)
–∞
The right-hand side of (35) can be regarded as a functional equation for the kernel e px of the inverse
Laplace transform. To solve it, we replace p by –p – β in Eq. (33). We finally obtain

px
L [e –(p+β)x ]= Ae –(p+β)x + q(–p – β)e . (36)

Let us multiply Eq. (35) by A and Eq. (36) by –q(p) and add the resulting relations. This yields

2
px
L [Ae px – q(p)e –(p+β)x ]=[A – q(p)q(–p – β)]e . (37)
2
On dividing Eq. (37) by the constant A – q(p)q(–p – β), we obtain the original model solution
Ae px – q(p)e –(p+β)x px
Y (x, p)= , L [Y (x, p)] = e . (38)
A – q(p)q(–p – β)
2
Since here –∞ < x < ∞, one must set p = iu and use the formulas from Subsection 9.6-3. Then the
solution of Eq. (32) for an arbitrary function f(x) can be represented in the form

1 ∞ ∞ –iux
˜
˜
y(x)= √ Y (x, iu)f(u) du, f(u)= f(x)e dx. (39)
2π –∞ –∞
•
References for Section 11.8: M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), V. I. Smirnov (1974),
P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov and Yu. I. Cherskii (1978), A. D. Polyanin and A. V. Manzhirov
(1997, 1998).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 550