Page 535 - Handbook Of Integral Equations

◦
4 . Consider the dual integral equation
∞

2β
t J µ (xt)y(t) dt = f(x) for 0 < x <1,
0
(13)
∞
J µ (xt)y(t) dt =0 for 1 < x < ∞,
0
where J µ (x) is the Bessel function of order µ.
The solution of Eq. (13) can be obtained by applying the Mellin transform. For β > 0, this
solution is defined by the formulas

(2x) 1–β 1 1+β 1 µ+1 2 β–1
y(x)= t J µ+β (xt)F(t) dt, F(t)= f(tζ)ζ (1 – ζ ) dζ. (14)
Γ(β) 0 0
For β > –1, the solution of the dual equation (13) has the form

(2x) –β 1+β 1 µ+1 2 β 1 µ+1 2 β
y(x)= x J µ+β (x) t (1 – t ) f(t) dt + t (1 – t ) Φ(x, t) dt , (15)
Γ(1 + β) 0 0

1

Φ(x, t)= (xξ) 2+β J µ+β+1 (xξ)f(ξt) dξ.
0
3
Formula (15) holds for β > –1 and for –µ – 1 <2β < µ + . It can be shown that for β > 0 the
2 2
solution of Eq. (15) can be reduced to the form (14).
5 . The exact solution of the dual integral equation
◦

∞
tP 1 (cosh x)y(t) dt = f(x) for 0 < x < a,
– +it
0 2
∞
(16)
tanh(πt)P (cosh x)y(t) dt =0 for a < x < ∞,
1
– +it
0 2
2
where P µ (x) is the Legendre spherical function of the first kind (see Supplement 10) and i = –1,
can be constructed by means of the Meler–Fock integral transform (see Section 7.6) and is given by
the formula
a t
√
2 f(s) sinh s
y(x)= sin(xt) √ ds dt. (17)
π cosh t – cosh s
0 0
Note that
√
2 x cos(ts)
P – +it (cosh x)= √ ds, x >0,
1
2 π cosh x – cosh s
0
where the integral on the right-hand side is called the Meler integral.
10.6-3. Reduction of Dual Equations to a Fredholm Equation
One of the most effective methods for the approximate solution of dual integral equations of the
first kind is the method of reducing these equations to Fredholm integral equations of the second
kind (see Chapter 11). In what follows, we present some dual equations encountered in problems of
mechanics and physics and related Fredholm equations of the second kind.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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