Page 25 - Handbook Of Integral Equations

Chapter 1

Linear Equations of the First Kind

With Variable Limit of Integration

Notation: f = f(x), g = g(x), h = h(x), K = K(x), and M = M(x) are arbitrary functions (these
may be composite functions of the argument depending on two variables x and t); A, B, C, D, E,
a, b, c, α, β, γ, λ, and µ are free parameters; and m and n are nonnegative integers.

Preliminary remarks. For equations of the form
x

K(x, t)y(t) dt = f(x), a ≤ x ≤ b,
a
where the functions K(x, t) and f(x) are continuous, the right-hand side must satisfy the following
conditions:
◦
1 .If K(a, a) ≠ 0, then we must have f(a) = 0 (for example, the right-hand sides of equations 1.1.1
and 1.2.1 must satisfy this condition).

2 .If K(a, a)= K (a, a)= ··· = K (n–1) (a, a)=0, 0< K (n) (a, a) < ∞, then the right-hand side
◦

x x x

of the equation must satisfy the conditions
f(a)= f (a)= ··· = f x (n) (a)=0.

x
For example, with n = 1, these are constraints for the right-hand side of equation 1.1.2.

◦
3 .If K(a, a)= K (a, a)= ··· = K (n–1) (a, a)=0, K x (n) (a, a)= ∞, then the right-hand side of the
x
x
equation must satisfy the conditions

f(a)= f (a)= ··· = f (n–1) (a)=0.
x x
For example, with n = 1, this is a constraint for the right-hand side of equation 1.1.30.
For unbounded K(x, t) with integrable power-law or logarithmic singularity at x = t and con-
tinuous f(x), no additional conditions are imposed on the right-hand side of the integral equation
(e.g., see Abel’s equation 1.1.36).
◦
In Chapter 1, conditions 1 –3 are as a rule not specified.
◦
1.1. Equations Whose Kernels Contain Power-Law
Functions
1.1-1. Kernels Linear in the Arguments x and t
x

1. y(t) dt = f(x).
a

Solution: y(x)= f (x).
x
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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