The Study of Triple Integral Equations with Generalized Legendre Functions

A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.


Introduction
Dual integral equations involving Legendre functions have been solved by Babloian 1 . He applied these equations to problems of potential theory and to a torsion problem. Later on Pathak 2 and Mandal 3 who considered dual integral equations involving generalized Legendre functions which have more general solution than the ones considered by Babloian 1

. Recently, Singh et al. 4 considered dual integral equations involving generalized
Legendre functions, and their results are more general than those in 1-3 . In the analysis of mixed boundary value problems, we often encounter triple integral equations. Triple integral equations involving Legendre functions have been studied by Srivastava 5 . Triple integral equations involving Bessel functions have also been considered by Cooke 6-9 , Tranter 10 , Love and Clements 11 , Srivastava 12 , and most of these authors reduced the solution into a solution of Fredholm integral equation of the second kind. The relevant references for dual and triple integral equations are given in the book of Sneddon 13 . In this paper, a method is developed for solutions of two sets of triple integral equations involving generalized Legendre functions in Sections 3 and 4. Each set of triple integral equations is reduced to a Fredholm integral equation of the second kind which may be solved numerically. The aim of this paper is to find a more general solution for the type of 2 Abstract and Applied Analysis integral equations given in 1-5 and to develop an easier method for solving triple integral equations in general.

Integral involved generalized Legendre functions and some useful results
We first summarize some known results needed in the paper.

2.8
If h t is monotonically increasing and differentiable for a < t < b and h t / 0 in this interval, then the solutions of the equations are given by Sneddon 13 as respectively, where the prime denotes the derivative with respect to t.

Triple integral equations with generalized Legendre functions: set I
In this section, we will find solution of the following triple integral equations: Abstract and Applied Analysis where A τ is an unknown function to be determined, f α is a known function, and P μ −1/2 i τ/c cosh αc is the generalized Legendre function defined in Section 2 and −1/2 < The trial solution of 3.1 , 3.2 , and 3.3 can be written as where ψ t is an unknown function to be determined. On integrating 3.4 by parts, we get where the prime denotes the derivative with respect to t. Substituting 3.5 into 3.3 , interchanging the order of integrations and using 2.2 , we find that 3.3 is satisfied identically. Substituting 3.5 into 3.1 and using the integral defined by 2.2 , we obtain 3.6 Equation 3.6 is equivalent to the following integral equation: By substituting 3.4 into 3.2 , interchanging the order of integrations and using the integral defined by 2.1 we find that

3.8
For obtaining the solution of the problem, we need to solve two Abel's type integral equations 3.7 and 3.8 . We assume that B. M. Singh et al.

5
The above equation is of the same form as 3.7 and defined in a different region. Equation 3.9 is of form 2.12 . Hence, the solution of the integral equation 3.9 can be written as The solution of Abel's type integral equations 2.11 together with 3.7 and 3.9 leads to

3.12
Substituting the expression for ψ t from 3.11 and 3.10 into the first and second integral of 3.12 we obtain α a S t dt cosh αc − cosh tc Assuming that the right-hand side of 3.13 is a known function of α it has the form of 2.9 , whose solution is given by 3.17 From the integral

3.18
we then obtain I t c cos μ 2 π sinh ct π cosh ct − cosh ca

3.19
Equation 3.14 is an Abel-type equation. Hence, its solution is

3.21
Substituting the expression for φ u from 3.20 into 3.21 , integrating by parts, and finally interchanging the order of integrations in second integral, we arrive at R p c cos μ 1 π π 1 cosh ca − cosh cp

Triple integral equations with generalized Legendre functions: set II
In this section, we will find the solution of the following triple integral equations: The inversion formula for generalized Mehler-Fock transforms 2.4 together with 4.3 and 4.4 implies that

4.6
Substituting the value of A τ from 4.5 into 4.6 , interchanging the order of integrations, and using the integral 2.2 , we get where and then 2.8 and 2.2 imply that Assuming that the right-hand side of 4.14 is known function equation and 4.14 has the form of 2.10 , hence the solution of 4.14 can be written as

Conclusions
The solution of the two sets of triple integral equations involving generalized Legendre functions is reduced to the solution of Fredholm integral equations of the second kind which can be solved numerically.