Page 502 - Handbook Of Integral Equations

where the parameters B k and µ k of the solution are related to the parameters A k and λ k of the
equation by algebraic relations.
For the solution of Eq. (1) with n=2, see Section 2.2 of the first part of the book (equation 2.2.10).

9.7-2. Equations With Kernel Containing a Sum of Hyperbolic Functions

1
β
β
–β
–β
1
By means of the formulas cosh β = (e +e ) and sinh β = (e –e ), any equation with difference
2 2
kernel of the form
x
y(x)+ K(x – t)y(t) dt = f(x),
a
m s (3)

K(x)= A k cosh(λ k x)+ B k sinh(µ k x),
k=1 k=1
can be represented in the form of Eq. (1) with n =2m+2s, and hence these equations can be reduced
to linear nonhomogeneous ordinary differential equations with constant coefficients.
9.7-3. Equations With Kernel Containing a Sum of Trigonometric Functions
Equations with difference kernel of the form
x m

y(x)+ K(x – t)y(t) dt = f(x), K(x)= A k cos(λ k x), (4)
a
k=1
x m

y(x)+ K(x – t)y(t) dt = f(x), K(x)= A k sin(λ k x), (5)
a
k=1
can also be reduced to linear nonhomogeneous ordinary differential equations of order 2m with
constant coefficients (see equations 2.5.4 and 2.5.15 in the first part of the book).
In a wide range of the parameters A k and λ k , the solution of Eq. (5) can be represented in the
form
x m

y(x)= f(x)+ R(x – t)f(t) dt, R(x)= B k sin(µ k x), (6)
a
k=1
where the parameters B k and µ k of the solution are related to the parameters A k and λ k of the
equation by algebraic relations.
Equations with difference kernels containing both cosines and sines can also be reduced to linear
nonhomogeneous ordinary differential equations with constant coefficients.

9.7-4. Equations Whose Kernels Contain Combinations of Various Functions
Any equation with difference kernel that contains a linear combination of summands of the form

(x – t) m (m =0, 1, 2, ...), exp α(x – t) ,
(7)

cosh β(x – t) , sinh γ(x – t) , cos λ(x – t) , sin µ(x – t) ,
can also be reduced by differentiation to a linear nonhomogeneous ordinary differential equation
with constant coefficients, where exponential, hyperbolic, and trigonometric functions can also be
n
multiplied by (x – t) (n =1, 2, ... ).
Remark. The method of differentiation can be successfully used to solve more complicated
equations with nondifference kernel to which the Laplace transform cannot be applied (see, for
instance, Eqs. 2.9.5, 2.9.28, 2.9.30, and 2.9.34 in the first part of the book).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 484