Page 230 - Handbook Of Integral Equations

3.2. Equations Whose Kernels Contain Exponential
Functions

3.2-1. Kernels Containing Exponential Functions
b

1. e λ|x–t| y(t) dt = f(x), –∞ < a < b < ∞.
a
1 . Let us remove the modulus in the integrand:
◦
x b

e λ(x–t) y(t) dt + e λ(t–x) y(t) dt = f(x). (1)
a x
Differentiating (1) with respect to x twice yields
x b

2λy(x)+ λ 2 e λ(x–t) y(t) dt + λ 2 e λ(t–x) y(t) dt = f (x). (2)

xx
a x
By eliminating the integral terms from (1) and (2), we obtain the solution
1 2
y(x)= f (x) – λ f(x) . (3)

xx
2λ
2 . The right-hand side f(x) of the integral equation must satisfy certain relations. By setting
◦
x = a and x = b in (1), we obtain two corollaries
b b

λa
λt
e y(t) dt = e f(a), e –λt y(t) dt = e –λb f(b). (4)
a a
On substituting the solution y(x) of (3) into (4) and then integrating by parts, we see that
λa
λb
λb
e f (b) – e f (a)= λe f(a)+ λe f(b),
λa
x x
e –λb x –λa x –λa f(a)+ λe –λb f(b).
f (a)= λe
f (b) – e
Hence, we obtain the desired constraints for f(x):

f (a)+ λf(a)=0, f (b) – λf(b) = 0. (5)

x
x
The general form of the right-hand side satisfying conditions (5) is given by
f(x)= F(x)+ Ax + B,
1 1

A = F (a)+ F (b)+ λF(a) – λF(b) , B = – F (a)+ λF(a)+ Aaλ + A ,

x
x
x
bλ – aλ – 2 λ
where F(x) is an arbitrary bounded, twice differentiable function.
b

λ|x–t| µ|x–t|
2. Ae + Be y(t) dt = f(x), –∞ < a < b < ∞.
a
Let us remove the modulus in the integrand and differentiate the resulting equation with
respect to x twice to obtain
b

2 λ|x–t| 2 µ|x–t|

2(Aλ + Bµ)y(x)+ Aλ e + Bµ e y(t) dt = f (x). (1)
xx
a
µ|x–t|
Eliminating the integral term with e from (1) with the aid of the original integral equation,
we find that
b
2
2
2

2(Aλ + Bµ)y(x)+ A(λ – µ ) e λ|x–t| y(t) dt = f (x) – µ f(x). (2)
xx
a
For Aλ+Bµ = 0, this is an equation of the form 3.2.1, and for Aλ+Bµ ≠ 0, this is an equation
of the form 4.2.15.
The right-hand side f(x) must satisfy certain relations, which can be obtained by setting
x = a and x = b in the original equation (a similar procedure is used in 3.2.1).
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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