Page 497 - Handbook Of Integral Equations

9.6. The Method of Model Solutions

9.6-1. Preliminary Remarks*
Consider a linear equation, which we briefly write out in the form

L [y(x)] = f(x), (1)

where L is a linear (integral) operator, y(x) is an unknown function, and f(x) is a known function.
We first define arbitrarily a test solution

y 0 = y 0 (x, λ), (2)

which depends on an auxiliary parameter λ (it is assumed that the operator L is independent of λ
and y 0 /≡ const). By means of Eq. (1) we define the right-hand side that corresponds to the test
solution (2):
f 0 (x, λ)= L [y 0 (x, λ)].
Let us multiply Eq. (1), for y = y 0 and f = f 0 , by some function ϕ(λ) and integrate the resulting
relation with respect to λ over an interval [a, b]. We finally obtain

L [y ϕ (x)] = f ϕ (x), (3)

where
b b

y ϕ (x)= y 0 (x, λ)ϕ(λ) dλ, f ϕ (x)= f 0 (x, λ)ϕ(λ) dλ. (4)
a a
It follows from formulas (3) and (4) that, for the right-hand side f = f ϕ (x), the function y = y ϕ (x)
is a solution of the original equation (1). Since the choice of the function ϕ(λ) (as well as of the
integration interval) is arbitrary, the function f ϕ (x) can be arbitrary in principle. Here the main
problem is how to choose a function ϕ(λ) to obtain a given function f ϕ (x). This problem can be
solved if we can find a test solution such that the right-hand side of Eq. (1) is the kernel of a known
inverse integral transform (we denote such a test solution by Y (x, λ) and call it a model solution).

9.6-2. Description of the Method
Indeed, let P be an invertible integral transform that takes each function f(x) to the corresponding
transform F(λ) by the rule
F(λ)= P{f(x)}. (5)

–1
Assume that the inverse transform P has the kernel ψ(x, λ) and acts as follows:
b

–1
–1
P {F(λ)} = f(x), P {F(λ)}≡ F(λ)ψ(x, λ) dλ. (6)
a
The limits of integration a and b and the integration path in (6) may well lie in the complex plane.
Suppose that we succeeded in finding a model solution Y (x, λ) of the auxiliary problem for
–1
Eq. (1) whose right-hand side is the kernel of the inverse transform P :

L [Y (x, λ)] = ψ(x, λ). (7)
* Before reading this section, it is useful to look over Section 9.5.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 479