Page 447 - Handbook Of Integral Equations

The notion of an integrable function of several variables is similar. For instance, a function f(x, t)
is said to be square integrable in a domain S = {a ≤ x ≤ b, a ≤ t ≤ b} if f(x) is measurable and

b b
f ≡ f (x, t) dx dt < ∞.
2
2
a a
Here f denotes the norm of the function f(x, t), as above.
7.1-2. The Structure of Solutions to Linear Integral Equations
A linear integral equation with variable integration limit has the form
x

βy(x)+ K(x, t)y(t) dt = f(x), (1)
a
where y(x) is the unknown function.
A linear integral equation with constant integration limits has the form
b
βy(x)+ K(x, t)y(t) dt = f(x). (2)
a
For β = 0, Eqs. (1) and (2) are called linear integral equations of the first kind, and for β ≠ 0,
linear integral equations of the second kind.*
Equations of the form (1) and (2) with specific conditions imposed on the kernels and the
right-hand sides form various classes of integral equations (Volterra equations, Fredholm equations,
convolution equations, etc.), which are considered in detail in Chapters 8–12.
For brevity, we shall sometimes represent the linear equations (1) and (2) in the operator form
L [y]= f(x). (3)
A linear operator L possesses the properties
L [y 1 + y 2 ]= L [y 1 ]+ L [y 2 ],
L [σy]= σL [y], σ = const .
A linear equation is called homogeneous if f(x) ≡ 0 and nonhomogeneous otherwise.
An arbitrary homogeneous linear integral equation has the trivial solution y ≡ 0.
If y 1 = y 1 (x) and y 2 = y 2 (x) are particular solutions of a linear homogeneous integral equation,
then the linear combination C 1 y 1 + C 2 y 2 with arbitrary constants C 1 and C 2 is also a solution (in
physical problems, this property is called the linear superposition principle).
The general solution of a linear nonhomogeneous integral equation (3) is the sum of the general
solution Y = Y (x) of the corresponding homogeneous equation L [Y ] = 0 and an arbitrary particular
solution ¯y = ¯y(x) of the nonhomogeneous equation L [ ¯y]= f(x), that is,
y = Y + ¯y. (4)
If the homogeneous integral equation has only the trivial solution Y ≡ 0, then the solution of the
corresponding nonhomogeneous equation is unique (if it exists).
Let ¯y 1 and ¯y 2 be solutions of nonhomogeneous linear integral equations with the same left-hand
sides and different right-hand sides, L [ ¯y 1 ]= f 1 (x) and L [ ¯y 2 ]= f 2 (x). Then the function ¯y = ¯y 1 + ¯y 2
is a solution of the equation L [ ¯y]= f 1 (x)+ f 2 (x).
The transformation
x = g(z), t = g(τ), y(x)= ϕ(z)w(z)+ ψ(z), (5)

where g(z), ϕ(z), and ψ(z) are arbitrary continuous functions (g ≠ 0), reduces Eqs. (1) and (2) to
z
linear equations of the same form for the unknown function w = w(z). Such transformations are
frequently used for constructing exact solutions of linear integral equations.
* In Chapters 1–4, which deal with equations with variable and constant limits of integration, we sometimes consider
more general equations in which the integrand contains the unknown function y(z), where z = z(x, t), instead of y(t).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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