Page 654 - Handbook Of Integral Equations

If conditions (21) are satisfied, then the general solution of the nonhomogeneous equation (16)
can be given by the formula (e.g., see Subsection 11.6-5)
n

ϕ(t)= f(t) – R g (t, τ; λ)f(τ) dτ + C k ϕ k (t), (22)
L
k=1
where R g (t, τ; λ) is called the generalized resolvent and the sum on the right-hand side of (22) is
the general solution of the corresponding homogeneous equation.
Now we consider an equation of the second kind with weak singularity on the contour:

M(t, τ)
ϕ(t)+ α ϕ(τ) dτ = f(t), (23)
L |τ – t|
where M(t, τ) is a continuous function and 0 < α < 1. By iterating we can reduce this equation
to a Fredholm integral equation of the second kind (e.g., see Remark 1 in Section 11.3). It has all
properties of a Fredholm equation.
For the above reasons, in the theory of singular integral equations it is customary to make no
difference between Fredholm equations and equations with weak singularity and use for them the
same notation

M(t, τ)

ϕ(t)+ λ K(t, τ)ϕ(τ) dτ =0, K(t, τ)= α , 0 ≤ α < 1. (24)
L |τ – t|
The integral equation (24) is called simply a Fredholm equation, and its kernel is called a Fredholm
kernel.
If in Eq. (24) the known functions satisfy the H¨ older condition, and M(t, τ) satisfies this
condition with respect to both variables, then each bounded integrable solution of Eq. (24) also
satisfies the H¨ older condition.
Remark 2. By the above estimates, the kernels of the regular parts of the above singular integral
equations are Fredholm kernels.

Remark 3. The complete and characteristic singular integral equations are sometimes called
singular integral equations of the second kind.
•
References for Section 13.1: F. D. Gakhov (1977), F. G. Tricomi (1985), S. G. Mikhlin and S. Pr¨ ossdorf (1986),
A. Dzhuraev (1992), N. I. Muskhelishvili (1992), I. K. Lifanov (1996).

13.2. The Carleman Method for Characteristic Equations

13.2-1. A Characteristic Equation With Cauchy Kernel
Consider a characteristic equation with Cauchy kernel:
b(t) ϕ(τ)
K [ϕ(t)] ≡ a(t)ϕ(t)+ dτ = f(t), (1)
◦
πi L τ – t
where the contour L consists of m + 1 closed smooth curves L = L 0 + L 1 + ··· + L m .
Solving Eq. (1) can be reduced to solving a Riemann boundary value problem (see Subsec-
tion 12.3-10), and the solution of the equation can be presented in a closed form.
Let us introduce the piecewise analytic function given by the Cauchy integral whose density is
the desired solution of the characteristic equation:

1 ϕ(τ)
Φ(z)= dτ. (2)
2πi τ – z
L

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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