Page 657 - Handbook Of Integral Equations

2. If ν ≤ 0, then the homogeneous equation is unsolvable (has only the trivial solution).
◦
◦
3. If ν ≥ 0, then the nonhomogeneous equation is solvable for an arbitrary right-hand side f(t),
and its general solution linearly depends on ν arbitrary constants.
◦
4. If ν < 0, then the nonhomogeneous equation is solvable if and only if its right-hand side f
satisfies the –ν conditions,
k–1
t
ψ k (t)f(t) dt =0, ψ k (t)= . (16)
L Z(t)
The above properties of characteristic singular integral equations are essentially different from
the properties of Fredholm integral equations (see Subsection 13.1-3). With Fredholm equations, if
the homogeneous equation is solvable, then the nonhomogeneous equation is in general unsolvable,
and conversely, if the homogeneous equation is unsolvable, then the nonhomogeneous equation
is solvable. However, for a singular equation, if the homogeneous equation is solvable, then
the nonhomogeneous equation is unconditionally solvable, and if the homogeneous equation is
unsolvable, then the nonhomogeneous equation is in general unsolvable as well.
By analogy with the case of Fredholm equations, we introduce a parameter λ into the kernel of
the characteristic equation and consider the equation

λb(t) ϕ(τ)
a(t)ϕ(t)+ dτ =0.
πi L τ – t

As shown above, the last equation is solvable if
a(t) – λb(t)
ν = Ind >0.
a(t)+ λb(t)

The index of a continuous function changes by jumps and only for the values of λ such that
a(t) ∓ λb(t) = 0. If in the complex plane λ = λ 1 + iλ 2 we draw the curves λ = ±a(t)/b(t), then these
curves divide the plane into domains in each of which the index is constant. Thus, the characteristic
values of the characteristic integral equation occupy entire domains, and hence the spectrum is
continuous, in contrast with the spectrum of a Fredholm equation.

13.2-2. The Transposed Equation of a Characteristic Equation
The equation
1 b(τ)ψ(τ)
◦∗
K [ψ(t)] ≡ a(t)ψ(t) – dτ = g(t), (17)
πi L τ – t
which is transposed to the characteristic equation K [ϕ(t)] = f(t), is not characteristic. However,
◦
the substitution
b(t)ψ(t)= ω(t) (18)

reduces it to a characteristic equation for the function ω(t):
b(t) ω(τ)
a(t)ω(t) – dτ = b(t)g(t). (19)
πi L τ – t
From the last equation we find ω(t), by the formula obtained by adding (17) to (18), and
determine the desired function ψ(t):

1 1 ω(τ)
ψ(t)= ω(t)+ dτ + g(t) .
a(t)+ b(t) πi L τ – t

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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