Page 232 - Handbook Of Integral Equations

◦
3 . The right-hand side f(x) must satisfy certain conditions. To find these conditions, one
must set x = a in the integral equation and its derivatives. (Alternatively, these conditions can
be found by setting x = a and x = b in the integral equation and all its derivatives obtained by
means of double differentiation.)
b
y(t) dt

6. = f(x), 0 < k <1.
λt k
a |e λx – e |
λt
λx
The transformation z = e , τ = e , w(τ)= e –λt y(t) leads to an equation of the form 3.1.30:
B
w(τ)

|z – τ| k dτ = F(z),
A
λb
λa

where A = e , B = e , F(z)= λf 1 ln z .
λ
y(t) dt
∞

7. λx λt k = f(x), λ >0, k >0.
0 (e + e )
This equation can be rewritten as an equation with difference kernel in the form 3.8.16:
∞
w(t) dt

= g(x),
k 1

0 cosh λ(x – t)
2
1

where w(t)=2 –k exp – λkt y(t) and g(x)=exp 1 λkx f(x).
2 2
∞

8. e –zt y(t) dt = f(z).
0
The left-hand side of the equation is the Laplace transform of y(t)(z is treated as a complex
variable).
1 . Solution:
◦
1 c+i∞ zt 2
y(t)= e f(z) dz, i = –1.
2πi
c–i∞
For specific functions f(z), one may use tables of inverse Laplace transforms to calculate
the integral (e.g., see Supplement 5).
◦
2 . For real z = x, under some assumptions the solution of the original equation can be
represented in the form
n
(–1) n n+1 (n)
n
y(x) = lim f x ,
n→∞ n! x x
which is the real inversion of the Laplace transform. To calculate the solution approximately,
one should restrict oneself to a specific value of n in this formula instead of taking the limit.
•
References: H. Bateman and A. Erd´ elyi (vol. 1, 1954), I. I. Hirschman and D. V. Widder (1955), V. A. Ditkin
and A. P. Prudnikov (1965).
3.2-2. Kernels Containing Power-Law and Exponential Functions
a
λx
9. ke – k – t y(t) dt = f(x).

0
This is a special case of equation 3.8.5 with g(x)= ke λx – k.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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