Page 294 - Handbook Of Integral Equations

b
29. y(x)+ A te λ|x–t| y(t) dt = f(x).
a
This is a special case of equation 4.9.37 with g(t)= At. The solution of the integral equation
can be written via the Bessel functions (or modified Bessel functions) of order 1/3.

∞
30. y(x)+ (a + b|x – t|) exp(–|x – t|)y(t) dt = f(x).
0
4
2
Let the biquadratic polynomial P(k)= k +2(a – b +1)k +2a +2b + 1 have no real roots and
let k = α + iβ be a root of the equation P(k) = 0 such that α > 0 and β > 0. In this case, the
solution has the form

∞
y(x)= f(x)+ ρ exp(–β|x – t|) cos(θ + α|x – t|)f(t) dt
0
2 2
[α +(β – 1) ] ∞
+ exp[–β(x + t)] cos[α(x – t)]f(t) dt
4α β 0
2

R ∞
+ exp[–β(x + t)] cos[ψ + α(x + t)]f(t) dt,
4α 2 0
where the parameters ρ, θ, R, and ψ are determined from the system of algebraic equations
obtained by separating real and imaginary parts in the relations
µ iψ (β – 1 – iα) 4
iθ
ρe = , Re = .
2
β – iα 8α (β – iα)
•
Reference: F. D. Gakhov and Yu. I. Cherskii (1978).

4.3. Equations Whose Kernels Contain Hyperbolic
Functions

4.3-1. Kernels Containing Hyperbolic Cosine

b

1. y(x) – λ cosh(βx)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x) = cosh(βx) and h(t)=1.

b
2. y(x) – λ cosh(βt)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = cosh(βt).

b

3. y(x) – λ cosh[β(x – t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.13 with g(x) = cosh(βx) and h(t) = sinh(βt).
Solution:

y(x)= f(x)+ λ A 1 cosh(βx)+ A 2 sinh(βx) ,
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.13.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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