A NOTE ON BESSEL FUNCTION DUAL INTEGRAL EQUATION WITH WEIGHT FUNCTION

An elementary procedure based on Sonine's integrals has been used to reduce dual integral equations with Bessel functions of different orders as kernels and an arbitrary weight function to a Fredholm integral equation of the second kind. The result obtained here encompasses many results concerning dual integral equations with Bessel functions as kernels known in the literature.

B.N. MANDAL Earlier Noble [2] used the 'multiplying-factor method' based on Sonine's integrals to a class of dual integral equations with Bessel functions of same order ( ) as kernels to a Fredholm integral equation of second kind.Schmeltzer and   Lewin [3] also used this multiplying-factor method to reduce another class of dual integral equations involving first kind Bessel functions of different orders and a particular weight function to dual integral equations with trigonometric sine function as kernels in closed form by the function-theoretic method.Later Ross [4] gave a simplified method of construction of explicit solution to the class of dual integral equations considered in [3].Also Rose and de Hoog [5] considered another class of dual integral equations with some particular weight functions for explicit solution.
These dual integral equations arose when standard integral transform approach was used to solve some mixed boundary value problems were solved by using a complex variable technique.The explicit solution to the dual integral equations were then deduced from the solutions to the boundary value problems.Except for these special class of dual integral equations considered in [3,4,5], the general class of dual integral equations (I.I) does not appear to admit of explicit solution.References to most of the works on different dual integral equations with Bessel function kernels can be found in [1,2,4,6].
While in [2] dual integral equations with first kind Bessel functions of the same order as kernels were considered, in the present paper as well as in [I], integral equations with Bessel functions of different orders as kernels are considered.
Here we employ the multiplying-factor method of Noble [2] to reduce (I.I) to a single integral equation with Bessel function as kernel.Invoking Hankel inversion, this single integral equation is then reduced to a Fredholm integral equation of second kind.The result is obtained for all arbitrary values of the parameters ,u,a,B.
From this result many known results concerning dual integral equations with Bessel function kernels can be deduced.As special cases, three sets of values of the para- meters are considered.The general result obtained here then reduces to the known results for these special cases.

REDUCTION TO A FREDHOLM INTEGRAL EQUATION OF SECOND KIND.
We assume that the parameters ,,a,B appearing in (I.I) are most general.By where p,q are non negative integers, it is always possible to change the orders , to ',' respectively such that ' I, ' > -3/2 (of course, a,B are then also changed) by choosing p suitably.Thus without any loss of generality, we can assume that the orders , of the Bessel functions in (I.I) satisfy the restrictions > -I, > -3/2.These restrictions are necessary in the analysis that follows.
we obtain Using (2.2) (with q replaced by and x replaced by r) in (2.4), we obtain f t-2a--l+E Jv++l- where m is a nonnegative integer.
In (2.7) and (2.8) we now equate the powers of t and the orders of the Bessel functions.This gives two equations to determine ,q as g(x) dx, r > I, (2.8) (2.9) The requirements that and n must be greater than -I (cf (2.3) and (2.5)) can be satisfied by choosing the nonnegative integers and m appropriately.Thus we obtain the single integral equation and F(r) and G(r) are given by the right sides in (2.7) and (2.8) respectively.Now in the single integral equation (2.10) we require % > -I for Hankel inver- sion.In general, this requirement may not be satisfied.To overcome this difficulty, we use (2.1) (with p replaced by s and x replaced by r) in (2.10) to obtain f t-Y+s J%+s (rt) (t) dt 0 Fl(r)-(-l)S of t-Y+s J+s (rt) w(t) (t)dt, 0 < r < G (r) r > (2.12) where F l(r) (-l)S rl+S r s G l(r) (-l)S rl+S r s ( [r -G(r)], (2.13)   and s is a nonnegative integer.Whatever be a,B we can always choose %+s > -I.
Then the Hankel inversion can be invoked in (2.12) to give a Fredholm integral equa- tion of second kind in (x) as (x) x Y-s+l o r F l(r) Jx+s(rX)dr + f r l(r) J)t+s(rX) dr -y+s u J%+s+l(u) J%+s(X) xJ%+s+ (x) J%+s(U) where I, are given in (2.11) and s is a nonnegative integer to be chosen appropriately so as to make t+s > -1.
Putting g(x) O, w(u) O, we obtain the explicit solution of the dual integral and we may mention here that the restriction 0 was not stated explicitly in [I].Unless this restriction holds good, it is not possible to use Hankel inversion with the order of the Bessel function as .To make > -I, n > -I, we choose I, m in (2.9).Also as > -I, we choose s 0 in (2.14) so that the dual integral equations (2.1) in this case reduce to # This result does not seem to coincide with the general result given in [I].However, the known results for the following two special cases given in [6] as well as in [I] are deduced from (3.5) implying that (3.5) is also correct.

6 )
This result can also be deduced directly from (2.14) by choosing s integral equation for #(x) can similarly be deduced from(3.5).In this case I7) can also be deduced directly from (2.14) by choosing s 0, O, m I.