Page 145 - Handbook Of Integral Equations

2.2. Equations Whose Kernels Contain Exponential
Functions

2.2-1. Kernels Containing Exponential Functions

x

1. y(x)+ A e λ(x–t) y(t) dt = f(x).
a
Solution:
x

y(x)= f(x) – A e (λ–A)(x–t) f(t) dt.
a
x

2. y(x)+ A e λx+βt y(t) dt = f(x).
a
For β = –λ, see equation 2.2.1. This is a special case of equation 2.9.2 with g(x)= –Ae λx
βt
and h(t)= e .
Solution:
x

y(x)= f(x) – R(x, t)f(t) dt, R(x, t)= Ae λx+βt exp A e (λ+β)t – e (λ+β)x .
a λ + β
x

3. y(x)+ A e λ(x–t) – 1 y(t) dt = f(x).
a
1 . Solution with D ≡ λ(λ – 4A)>0:
◦
2Aλ x 1 √
1
y(x)= f(x) – √ R(x – t)f(t) dt, R(x)=exp 2 λx sinh 2 Dx .
D a
2 . Solution with D ≡ λ(λ – 4A)<0:
◦
x
2Aλ 1

1
y(x)= f(x) – √ R(x – t)f(t) dt, R(x)=exp λx sin |D| x .
|D| a 2 2
3 . Solution with λ =4A:
◦
x

y(x)= f(x) – 4A 2 (x – t)exp 2A(x – t) f(t) dt.
a
x

λ(x–t)
4. y(x)+ Ae + B y(t) dt = f(x).
a
This is a special case of equation 2.2.10 with A 1 = A, A 2 = B, λ 1 = λ, and λ 2 =0.
1 . The structure of the solution depends on the sign of the discriminant
◦
2
D ≡ (A – B – λ) +4AB (1)
of the square equation
2
µ +(A + B – λ)µ – Bλ = 0. (2)
◦
2 .If D > 0, then equation (2) has the real different roots
√ √
1
1
µ 1 = (λ – A – B)+ 1 D, µ 2 = (λ – A – B) – 1 D.
2 2 2 2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 124