Page 156 - Handbook Of Integral Equations

x

2. y(x) – A cosh(λt)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A and h(t) = cosh(λt).
Solution:

x
A
y(x)= f(x)+ A cosh(λt)exp sinh(λx) – sinh(λt) f(t) dt.
a λ
x
3. y(x)+ A cosh[λ(x – t)]y(t) dt = f(x).
a
This is a special case of equation 2.9.28 with g(t)= A. Therefore, solving the original integral
equation is reduced to solving the second-order linear nonhomogeneous ordinary differential
equation with constant coefficients

2
2
y + Ay – λ y = f xx – λ f, f = f(x),

x
xx
under the initial conditions

y(a)= f(a), y (a)= f (a) – Af(a).

x x
Solution:
x
y(x)= f(x)+ R(x – t)f(t) dt,
a

A 2
1 2 1 2
R(x)=exp – Ax sinh(kx) – A cosh(kx) , k = λ + A .
2 2k 4
x n

4. y(x)+ A k cosh[λ k (x – t)] y(t) dt = f(x).
a
k=1
This equation can be reduced to an equation of the form 2.2.19 by using the identity
cosh z ≡ 1 e + e –z . Therefore, the integral equation in question can be reduced to a
z
2
linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients.
x
cosh(λx)
5. y(x) – A y(t) dt = f(x).
a cosh(λt)
Solution:
x
cosh(λx)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a cosh(λt)
x cosh(λt)
6. y(x) – A y(t) dt = f(x).
a cosh(λx)
Solution:
x cosh(λt)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a cosh(λx)
x

m
k
7. y(x) – A cosh (λx) cosh (µt)y(t) dt = f(x).
a
k m
This is a special case of equation 2.9.2 with g(x)= A cosh (λx) and h(t) = cosh (µt).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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