Abstract and Applied Analysis

Abstract and Applied Analysis / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 395257 | https://doi.org/10.1155/2008/395257

B. M. Singh, J. Rokne, R. S. Dhaliwal, "The Study of Triple Integral Equations with Generalized Legendre Functions", Abstract and Applied Analysis, vol. 2008, Article ID 395257, 12 pages, 2008. https://doi.org/10.1155/2008/395257

The Study of Triple Integral Equations with Generalized Legendre Functions

Academic Editor: Lance Littlejohn
Received28 Apr 2008
Accepted12 Sep 2008
Published09 Nov 2008

Abstract

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.

1. 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 integral equations given in [1–5] and to develop an easier method for solving triple integral equations in general.

2. Integral Involved Generalized Legendre Functions and Some Useful Results

We first summarize some known results needed in the paper.

We find from [14, equation (21), page 330] thatξ‚™2πœ‹Ξ“ξ‚€12ξ‚„βˆ’πœ‡ξ‚ξ‚ƒsinh(𝛼𝑐)βˆ’πœ‡βˆ«βˆž0π‘ƒπœ‡βˆ’1/2+𝑖(𝜏/𝑐)cosh(𝛼𝑐)cos(𝜏π‘₯)π‘‘πœ=𝑐cosh(𝛼𝑐)βˆ’cosh(π‘₯𝑐)βˆ’πœ‡βˆ’1/2𝐻(π›Όβˆ’π‘₯),(2.1)where πœ‡<1/2 and from [4], we obtain√2πœ‹βˆ’3/2Ξ“ξ‚€12ξ‚„+πœ‡ξ‚ξ‚ƒsinh(𝛼𝑐)πœ‡Γ—βˆ«βˆž0Ξ“ξ‚€12πœβˆ’πœ‡+𝑖𝑐Γ12πœβˆ’πœ‡βˆ’π‘–π‘ξ‚sinh(π‘“πœ)sin(π‘₯𝜏)π‘ƒπœ‡βˆ’1/2+𝑖(𝜏/𝑐)cosh(𝛼𝑐)π‘‘πœ=𝑐cosh(π‘₯𝑐)βˆ’cosh(𝛼𝑐)πœ‡βˆ’1/2𝐻(π‘₯βˆ’π›Ό),(2.2)where πœ‡>βˆ’1/2 and 𝐻() denotes the Heaviside unit function. Furthermore, 𝑐=πœ‹/𝑓, 𝑓>0 and π‘ƒπœ‡βˆ’1/2+𝑖(𝜏/𝑐)(cosh𝛼𝑐) is the generalized Legendre function defined in [15, page 370]. From [4, 16], the generalized Mehler-Fock transform is defined byπœ“ξ‚€ξ‚=ξ€œcosh(𝛼𝑐)∞0π‘ƒπœ‡βˆ’1/2+𝑖(𝜏/𝑐)cosh(𝛼𝑐)𝐹(𝜏)π‘‘πœ,(2.3)and its inversion formula is𝐹(𝜏)=π‘“πœπœ‹2ξ‚€1sinh(π‘“πœ)Ξ“2πœβˆ’πœ‡+𝑖𝑐Γ12πœβˆ’πœ‡βˆ’π‘–π‘ξ‚Γ—ξ€œβˆž0π‘ƒπœ‡βˆ’1/2+𝑖(𝜏/𝑐)ξ‚ƒξ‚„πœ“ξ‚€ξ‚cosh(𝛼𝑐)cosh(𝛼𝑐)sinh(𝛼𝑐)𝑑𝛼.(2.4)Equations (2.1) and (2.2) are of form (2.3). From the inversion formula given by (2.4), (2.1), and (2.2), it follows thatcos(π‘₯𝜏)𝜏=Γsinh(π‘“πœ)Ξ“1/2βˆ’πœ‡+𝑖(𝜏/𝑐)1/2βˆ’πœ‡βˆ’π‘–(𝜏/𝑐)βˆšξ€·ξ€ΈΓ—ξ€œ2πœ‹Ξ“1/2βˆ’πœ‡βˆžπ‘₯sinh(𝛼𝑐)1+πœ‡π‘ƒπœ‡βˆ’1/2+𝑖(𝜏/𝑐)cosh(𝛼𝑐)𝑑𝛼cosh(𝛼𝑐)βˆ’cosh(π‘₯𝑐)πœ‡+1/21,πœ‡<2,(2.5)sin(π‘₯𝜏)𝜏=ξ‚™πœ‹21Ξ“ξ€·ξ€Έξ€œ1/2+πœ‡π‘₯0sinh(𝛼𝑐)1βˆ’πœ‡π‘ƒπœ‡βˆ’1/2+𝑖(𝜏/𝑐)(cosh𝛼𝑐)𝑑𝛼cosh(π‘₯𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡1,πœ‡>βˆ’2.(2.6)

The inversion theorem for Fourier cosine transforms and the results (2.1) and (2.2) lead toπ‘ƒπœ‡βˆ’1/2+𝑖(𝜏/𝑐)=ξ‚™cosh(𝛼𝑐)2πœ‹π‘sinhπœ‡(𝛼𝑐)Ξ“ξ€·ξ€Έξ€œ1/2βˆ’πœ‡π›Ό0cos(πœπ‘ )𝑑𝑠cosh(𝛼𝑐)βˆ’cosh(𝑠𝑐)πœ‡+1/21,πœ‡<2,𝑃(2.7)πœ‡βˆ’1/2+𝑖(𝜏/𝑐)=√cosh(𝛼𝑐)2πœ‹π‘ξ‚ƒsinhπœ‡ξ€·ξ€Έξ‚€ξ‚Ξ“ξ‚€ξ‚Γ—ξ€œ(𝛼𝑐)]Ξ“1/2+πœ‡sin(π‘“πœ)Ξ“1/2βˆ’πœ‡+𝑖(𝜏/𝑐)1/2βˆ’πœ‡βˆ’π‘–(𝜏/𝑐)βˆžπ›Όsin(πœπ‘ )𝑑𝑠cosh(𝑠𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡1,πœ‡>βˆ’2.(2.8)

If β„Ž(𝑑) is monotonically increasing and differentiable for π‘Ž<𝑑<𝑏 and β„Žξ…ž(𝑑)β‰ 0 in this interval, then the solutions of the equationsξ€œπ‘‘π‘Žπ‘“(π‘₯)𝑑π‘₯ξ‚ƒξ‚„β„Ž(𝑑)βˆ’β„Ž(π‘₯)π›Όξ€œ=𝑔(𝑑),π‘Ž<𝑑<𝑏,0<𝛼<1,(2.9)𝑏𝑑𝑓(π‘₯)𝑑π‘₯ξ‚ƒξ‚„β„Ž(π‘₯)βˆ’β„Ž(𝑑)𝛼=𝑔(𝑑),π‘Ž<𝑑<𝑏,0<𝛼<1,(2.10)are given by Sneddon [13] as𝑓(π‘₯)=sin(πœ‹π›Ό)πœ‹π‘‘ξ€œπ‘‘π‘₯π‘₯π‘Žβ„Žξ…ž(𝑑)𝑔(𝑑)π‘‘π‘‘ξ‚ƒξ‚„β„Ž(π‘₯)βˆ’β„Ž(𝑑)1βˆ’π›Ό,π‘Ž<π‘₯<𝑏,(2.11)𝑓(π‘₯)=βˆ’sin(πœ‹π›Ό)πœ‹π‘‘ξ€œπ‘‘π‘₯𝑏π‘₯β„Žξ…ž(𝑑)𝑔(𝑑)π‘‘π‘‘ξ‚ƒξ‚„β„Ž(𝑑)βˆ’β„Ž(π‘₯)1βˆ’π›Ό,π‘Ž<π‘₯<𝑏,(2.12)respectively, where the prime denotes the derivative with respect to 𝑑.

3. Triple Integral Equations with Generalized Legendre Functions: Set I

In this section, we will find solution of the following triple integral equations:ξ€œβˆž0ξ‚€1𝜏𝐴(𝜏)sinh(πœπ‘“)Ξ“2βˆ’πœ‡1𝜏+𝑖𝑐Γ12βˆ’πœ‡1πœβˆ’π‘–π‘ξ‚π‘ƒπœ‡1βˆ’1/2+𝑖(𝜏/𝑐)ξ‚ƒξ‚„ξ€œcosh(𝛼𝑐)π‘‘πœ=0,0<𝛼<π‘Ž,(3.1)∞0𝐴(𝜏)π‘ƒπœ‡2βˆ’1/2+𝑖(𝜏/𝑐)ξ‚ƒξ‚„ξ€œcosh(𝛼𝑐)π‘‘πœ=𝑓(𝛼),π‘Ž<𝛼<𝑏,(3.2)∞0ξ‚€1𝜏𝐴(𝜏)sinh(πœπ‘“)Ξ“2βˆ’πœ‡3𝜏+𝑖𝑐Γ12βˆ’πœ‡3πœβˆ’π‘–π‘ξ‚π‘ƒπœ‡3βˆ’1/2+𝑖(𝜏/𝑐)cosh(𝛼𝑐)π‘‘πœ=0,𝑏<𝛼<∞,(3.3) where 𝐴(𝜏) is an unknown function to be determined, 𝑓(𝛼) is a known function, and π‘ƒπœ‡βˆ’1/2+𝑖(𝜏/𝑐)[cosh(𝛼𝑐)] is the generalized Legendre function defined in Section 2 and βˆ’1/2<πœ‡1<1/2, βˆ’1/2<πœ‡2<1/2, πœ‡3>βˆ’1/2.

The trial solution of (3.1), (3.2), and (3.3) can be written asξ€œπ΄(𝜏)=𝑏0πœ“(𝑑)cos(πœπ‘‘)𝑑𝑑,(3.4)where πœ“(𝑑) is an unknown function to be determined. On integrating (3.4) by parts, we get𝐴(𝜏)=πœ“(𝑏)sin(πœπ‘)πœβˆ’1πœξ€œπ‘0πœ“ξ…ž(𝑑)sin(πœπ‘‘)𝑑𝑑,(3.5)where the prime denotes the derivative with respect to 𝑑.

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 obtainπœ“(𝑏)cosh(𝑏𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡1βˆ’ξ€œπ‘π›Όπœ“ξ…ž(𝑑)𝑑𝑑cosh(𝑑𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡1=0,0<𝛼<π‘Ž.(3.6)Equation (3.6) is equivalent to the following integral equation:π‘‘ξ€œπ‘‘π›Όπ‘π›Όπ‘sinh(𝑑𝑐)πœ“(𝑑)𝑑𝑑cosh(𝑑𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡1=0,0<𝛼<π‘Ž.(3.7)By substituting (3.4) into (3.2), interchanging the order of integrations and using the integral defined by (2.1) we find thatπ‘ξ€œπ›Ό0πœ“(𝑑)𝑑𝑑cosh(𝛼𝑐)βˆ’cosh(𝑑𝑐)1/2+πœ‡2=ξ‚™2πœ‹Ξ“ξ‚€12βˆ’πœ‡2sinh(𝛼𝑐)βˆ’πœ‡2𝑓(𝛼),π‘Ž<𝛼<𝑏,πœ‡2<12.(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 thatπ‘‘ξ€œπ‘‘π›Όπ‘π›Όπ‘sinh(𝑑𝑐)πœ“(𝑑)𝑑𝑑cosh(𝑑𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡1=πœ™(𝛼),π‘Ž<𝛼<𝑏.(3.9)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 asξ‚€πœ“(𝑑)=βˆ’cosπœ‹πœ‡1ξ‚πœ‹ξ€œπ‘π‘‘πœ™(𝛼)𝑑𝛼cosh(𝑐𝛼)βˆ’cosh(𝑑𝑐)1/2+πœ‡11,βˆ’2<πœ‡1<12,π‘Ž<𝑑<𝑏.(3.10)

The solution of Abel's type integral equations (2.11) together with (3.7) and (3.9) leads toξ‚€πœ“(𝑑)=βˆ’cosπœ‹πœ‡1ξ‚πœ‹ξ€œπ‘π‘Žπœ™(𝛼)𝑑𝛼cosh(𝑐𝛼)βˆ’cosh(𝑑𝑐)1/2+πœ‡11,βˆ’2<πœ‡1<12,0<𝑑<π‘Ž.(3.11)

Equations (3.10) and (3.11) mean that (3.7) is satisfied identically. Equation (3.8) can be rewritten in the formβˆ«π‘Ž0πœ“(𝑑)𝑑𝑑cosh(𝛼𝑐)βˆ’cosh(𝑑𝑐)1/2+πœ‡2+βˆ«π›Όπ‘Žπœ“(𝑑)𝑑𝑑cosh(𝛼𝑐)βˆ’cosh(𝑑𝑐)1/2+πœ‡2=1𝑐2πœ‹Ξ“ξ‚€1βˆ’πœ‡2𝑓(𝛼)sinh(𝛼𝑐)πœ‡2,π‘Ž<𝛼<𝑏.𝑓(3.12)Substituting the expression for πœ“(𝑑) from (3.11) and (3.10) into the first and second integral of (3.12) we obtainβˆ«π›Όπ‘Žπ‘†(𝑑)𝑑𝑑cosh(𝛼𝑐)βˆ’cosh(𝑑𝑐)1/2+πœ‡2∫=𝐹(𝛼)βˆ’π‘Ž0𝑑𝑑cosh(𝛼𝑐)βˆ’cosh(𝑑𝑐)1/2+πœ‡2βˆ«π‘π‘Žπœ™(𝑒)𝑑𝑑cosh(𝑐𝑒)βˆ’cosh(𝑑𝑐)1/2+πœ‡2,π‘Ž<𝑑<𝑏,(3.13)whereξ€œπ‘†(𝑑)=π‘π‘‘πœ™(𝑒)𝑑𝑑cosh(𝑐𝑒)βˆ’cosh(𝑑𝑐)1/2+πœ‡1,βˆ’βˆš(3.14)𝐹(𝛼)=ξ‚€2πœ‹Ξ“1βˆ’πœ‡2𝑓(𝛼)sinh(𝛼𝑐)βˆ’πœ‡2ξ‚€πœ‡π‘cos1πœ‹ξ‚.(3.15)

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𝑆(𝑑)=cosπœ‹πœ‡2ξ‚πœ‹π‘‘ξ€œπ‘‘π‘‘π‘‘π‘Žπ‘sinh(𝑐𝛼)𝐹(𝛼)𝑑𝛼cosh(𝑐𝑑)βˆ’cosh(𝑐𝛼)1/2βˆ’πœ‡21βˆ’πΌ(𝑑),π‘Ž<𝑑<𝑏,βˆ’2<πœ‡2<12,(3.16)where𝐼(𝑑)=cosπœ‹πœ‡2ξ‚πœ‹π‘‘ξ€œπ‘‘π‘‘π‘‘π‘Žπ‘sinh(𝑐𝛼)𝑑𝛼cosh(𝑐𝑑)βˆ’cosh(𝑐𝛼)1/2βˆ’πœ‡2ξ€œπ‘Ž0𝑑𝑝cosh(𝑐𝛼)βˆ’cosh(𝑐𝑝)1/2+πœ‡2Γ—ξ€œπ‘π‘Žπœ™(𝑒)𝑑𝑒cosh(𝑐𝑒)βˆ’cosh(𝑐𝑝)1/2+πœ‡21,π‘Ž<𝑑<𝑏,βˆ’2<πœ‡2<12.(3.17)From the integralπ‘‘βˆ«π‘‘π‘‘π‘‘π‘Žπ‘sinh(𝑐𝛼)𝑑𝛼cosh(𝑐𝑑)βˆ’cosh(𝑐𝛼)1/2βˆ’πœ‡2cosh(𝑐𝛼)βˆ’cosh(𝑐𝑝)1/2+πœ‡2=𝑐sinh(𝑐𝑑)cosh(𝑐𝑑)βˆ’cosh(𝑐𝑝)cosh(π‘π‘Ž)βˆ’cosh(𝑐𝑝)1/2βˆ’πœ‡2cosh(𝑐𝑑)βˆ’cosh(π‘π‘Ž)1/2βˆ’πœ‡21,𝑝<π‘Ž<𝑑,βˆ’2<πœ‡2<12,(3.18)we then obtainξ‚€πœ‡πΌ(𝑑)=𝑐cos2πœ‹ξ‚sinh(𝑐𝑑)πœ‹ξ‚ƒξ‚„cosh(𝑐𝑑)βˆ’cosh(π‘π‘Ž)1/2βˆ’πœ‡2ξ€œπ‘Ž0cosh(π‘π‘Ž)βˆ’cosh(𝑐𝑝)1/2βˆ’πœ‡2π‘‘π‘ξ‚ƒξ‚„Γ—ξ€œcosh(𝑐𝑑)βˆ’cosh(𝑐𝑝)π‘π‘Žπœ™(𝑒)𝑑𝑒cosh(𝑐𝑒)βˆ’cosh(𝑐𝑝)1/2+πœ‡1.(3.19)Equation (3.14) is an Abel-type equation. Hence, its solution isξ‚€πœ‡πœ™(𝑒)=βˆ’cos1πœ‹ξ‚πœ‹π‘‘ξ€œπ‘‘π‘’π‘π‘’π‘sinh(𝑐𝑣)𝑆(𝑣)𝑑𝑣cosh(𝑣𝑐)βˆ’cosh(𝑒𝑐)1/2βˆ’πœ‡11,π‘Ž<𝑒<𝑏,βˆ’2<πœ‡1<12,ξ€œ(3.20)𝑅(𝑝)=π‘π‘Žπœ™(𝑒)𝑑𝑒cosh(𝑐𝑒)βˆ’cosh(𝑐𝑝)1/2+πœ‡1.(3.21)Substituting the expression for πœ™(𝑒) from (3.20) into (3.21), integrating by parts, and finally interchanging the order of integrations in second integral, we arrive atξ‚€πœ‡π‘…(𝑝)=𝑐cos1πœ‹ξ‚πœ‹ξ‚Έ1cosh(π‘π‘Ž)βˆ’cosh(𝑐𝑝)1/2+πœ‡1ξ€œπ‘π‘Žπ‘†(𝑣)sinh(𝑐𝑣)𝑑𝑣cosh(𝑐𝑣)βˆ’cosh(π‘π‘Ž)1/2βˆ’πœ‡1βˆ’ξ‚€12+πœ‡1ξ‚ξ€œπ‘π‘ŽΓ—ξ€œπ‘†(𝑣)sinh(𝑐𝑣)π‘‘π‘£π‘£π‘Žπ‘sinh(𝑐𝑒)𝑑𝑒cosh(𝑐𝑒)βˆ’cosh(𝑐𝑝)3/2+πœ‡1cosh(𝑐𝑣)βˆ’cosh(𝑐𝑒)1/2βˆ’πœ‡1ξ‚Ή.(3.22)The integral βˆ«π‘£π‘Žπ‘sinh(𝑐𝑒)𝑑𝑒cosh(𝑐𝑒)βˆ’cosh(𝑐𝑝)3/2+πœ‡1cosh(𝑐𝑣)βˆ’cosh(𝑐𝑒)1/2βˆ’πœ‡1=[cosh(𝑐𝑣)βˆ’cosh(π‘π‘Ž)]1/2+πœ‡1(πœ‡11+1/2)[cosh(𝑐𝑣)βˆ’cosh(𝑐𝑝)][cosh(π‘π‘Ž)βˆ’cosh(𝑐𝑝)]𝑝<π‘Ž<𝑣,βˆ’2<πœ‡1<12(3.23)together with (3.22) leads to𝑅(𝑝)=𝑐cosπœ‹πœ‡1ξ‚πœ‹ξ‚€ξ‚cosh(π‘Žπ‘)βˆ’cosh(𝑝𝑐)1/2βˆ’πœ‡1Γ—ξ€œπ‘π‘Žπ‘†(𝜈)sinh(π‘πœˆ)π‘‘πœˆξ‚ƒξ‚„cosh(πœˆπ‘)βˆ’cosh(𝑝𝑐)cosh(πœˆπ‘)βˆ’cosh(π‘Žπ‘)1/2βˆ’πœ‡1.(3.24)From (3.19), (3.21), and (3.24), we obtainξ€œπΌ(𝑑)=π‘π‘Žπ‘†(𝜈)𝐾(𝜈,𝑑)π‘‘πœˆ,(3.25)where𝑐𝐾(𝜈,𝑑)=2ξ‚€cosπœ‹πœ‡1cosπœ‹πœ‡2sinh(𝑐𝑑)sinh(π‘πœˆ)πœ‹2cosh(𝑐𝑑)βˆ’cosh(π‘π‘Ž)1/2βˆ’πœ‡2cosh(π‘πœˆ)βˆ’cosh(π‘π‘Ž)1/2βˆ’πœ‡1Γ—ξ€œπ‘Ž0cosh(π‘π‘Ž)βˆ’cosh(𝑐𝑝)1βˆ’πœ‡1βˆ’πœ‡2𝑑𝑝.cosh(𝑐𝑑)βˆ’cosh(𝑐𝑝)cosh(π‘πœˆ)βˆ’cosh(𝑐𝑝)(3.26)From (3.25), (3.16) can be written asξ€œπ‘†(𝑑)+π‘π‘Žξ‚€π‘†(𝜈)𝐾(𝜈,𝑑)π‘‘πœˆ=cosπœ‹πœ‡2ξ‚πœ‹π‘‘ξ€œπ‘‘π‘‘π‘‘π‘Žπ‘sinh(𝑐𝛼)𝐹(𝛼)𝑑𝛼cosh(𝑐𝑑)βˆ’cosh(𝑐𝛼)1/2βˆ’πœ‡2,π‘Ž<𝑑<𝑏.(3.27)Equation (3.27) is a Fredholm integral equation of the second kind with kernel 𝐾(𝜈,𝑑). The kernel is defined by (3.26). The integral in (3.26) cannot be solved analytically, but for particular values of πœ‡1 and πœ‡2 the values of 𝐾(𝜈,𝑑) can be found numerically. Hence, the numerical solution of Fredholm integral equation (3.27) can be obtained for particular value of 𝑓(𝛼), πœ‡1, and πœ‡2 to find numerical values of 𝑆(𝑑). Making use of (3.20), (3.11), and (3.10), the numerical results for πœ“(𝑑) can be obtained. Finally, making use of (3.4) the numerical results for 𝐴(𝜏) can be obtained.

4. Triple Integral Equations with Generalized Legendre Functions: Set II

In this section, we will find the solution of the following triple integral equations:ξ€œβˆž0𝜏𝐴(𝜏)π‘ƒπœ‡1βˆ’1/2+𝑖(𝜏/𝑐)ξ‚ƒξ‚„ξ€œcosh(𝛼𝑐)π‘‘πœ=0,0<𝛼<π‘Ž,(4.1)∞0ξ‚€1sinh(πœπ‘“)Ξ“2βˆ’πœ‡2𝜏+𝑖𝑐Γ12βˆ’πœ‡2πœβˆ’π‘–π‘ξ‚π΄(𝜏)π‘ƒπœ‡2βˆ’1/2+𝑖(𝜏/𝑐)ξ‚ƒξ‚„ξ€œcosh(𝛼𝑐)π‘‘πœ=𝑓(𝛼),π‘Ž<𝛼<𝑏,(4.2)∞0𝜏𝐴(𝜏)π‘ƒπœ‡3βˆ’1/2+𝑖(𝜏/𝑐)cosh(𝛼𝑐)π‘‘πœ=0,𝑏<𝛼,(4.3) where πœ‡1>βˆ’1/2, βˆ’1/2<πœ‡2<1/2, βˆ’1/2<πœ‡3<1/2.

We assume thatξ€œβˆž0𝜏𝐴(𝜏)π‘ƒπœ‡3βˆ’1/2+𝑖(𝜏/𝑐)cosh(𝛼𝑐)π‘‘πœ=𝑀(𝛼),0<𝛼<𝑏.(4.4)The inversion formula for generalized Mehler-Fock transforms (2.4) together with (4.3) and (4.4) implies that𝑓𝐴(𝜏)=πœ‹2ξ‚€1sinh(π‘“πœ)Ξ“2βˆ’πœ‡3𝜏+𝑖𝑐Γ12βˆ’πœ‡3πœβˆ’π‘–π‘ξ‚Γ—ξ€œπ‘0sinh(𝑒𝑐)π‘ƒπœ‡3βˆ’1/2+𝑖(𝜏/𝑐)cosh(𝑒𝑐)𝑀(𝑒)𝑑𝑒.(4.5)

Multiplying (4.1) by [sinh(𝛼𝑐)]1βˆ’πœ‡1/[cosh(π‘₯𝑐)βˆ’cosh(𝛼𝑐)]1/2βˆ’πœ‡1, integrating both sides from 0 to π‘₯ and with respect to 𝛼, and then using (2.6) we obtainξ€œβˆž0𝐴(𝜏)sin(π‘₯𝜏)π‘‘πœ=0,0<π‘₯<π‘Ž.(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 getξ€œπ‘₯0sinh(𝑒𝑐)𝑀(𝑒)𝑑𝑒cosh(π‘₯𝑐)βˆ’cosh(𝑒𝑐)1/2βˆ’πœ‡3=0,πœ‡31>βˆ’2,0<π‘₯<π‘Ž.(4.7)Substituting the value of 𝐴(𝜏) from (4.5) into (4.2) and interchanging the order of integrations we arrive atξ€œπ‘0sinh(𝑒𝑐)𝑀(𝑒)𝐾2(𝑒,𝛼)𝑑𝑒=𝑓(𝛼),π‘Ž<𝛼<𝑏,(4.8)where𝐾2ξ€œ(𝑒,𝛼)=∞0π‘“πœ‹2Ξ“ξ‚€12βˆ’πœ‡2πœβˆ’π‘–π‘ξ‚Ξ“ξ‚€12βˆ’πœ‡2𝜏+𝑖𝑐Γ12βˆ’πœ‡3𝜏+𝑖𝑐Γ12βˆ’πœ‡3πœβˆ’π‘–π‘ξ‚Γ—sinh2(π‘“πœ)π‘ƒπœ‡3βˆ’1/2+𝑖(𝜏/𝑐)𝑃cosh(𝑒𝑐)πœ‡2βˆ’1/2+𝑖(𝜏/𝑐)cosh(𝛼𝑐)π‘‘πœ,(4.9)and then (2.8) and (2.2) imply that𝐾2(𝑒,𝛼)=π‘πœ‹Ξ“ξ‚€1/2+πœ‡2Γ1/2+πœ‡3sinh(𝛼𝑐)πœ‡2sinh(𝑒𝑐)πœ‡3Γ—ξ€œβˆžmax(𝛼,𝑒)𝑑𝑠cosh(𝑠𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡2cosh(𝑠𝑐)βˆ’cosh(𝑒𝑐)1/2βˆ’πœ‡3,πœ‡31>βˆ’2,πœ‡21>βˆ’2.(4.10)

Equation (4.7) is an Abel-type equation and has the form (2.9). Hence, the solution of (4.7) is𝑀(𝑒)=0,0<𝑒<π‘Ž.(4.11)

Using (4.10) and (2.5), (4.8) can be written in the formβˆ«π‘π‘Žξ‚ƒξ‚„sinh(𝑒𝑐)1βˆ’πœ‡3βˆ«π‘€(𝑒)π‘‘π‘’βˆžmax(𝛼,𝑒)𝑑𝑠cosh(𝑠𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡2cosh(𝑠𝑐)βˆ’cosh(𝑒𝑐)1/2βˆ’πœ‡3=Ξ“ξ‚€1/2+πœ‡2Γ1/2+πœ‡2sinh(𝛼𝑐)πœ‡3π‘πœ‹π‘“(𝛼)=𝐹1(𝛼),say,π‘Ž<𝛼<𝑏.(4.12)Using the formulaξ€œπ‘π‘Žξ€œπ‘‘π‘’βˆžmax(𝛼,𝑒)ξ€œπ‘‘π‘ =π‘π›Όξ€œπ‘‘π‘ π‘ π‘Žξ€œπ‘‘π‘’+βˆžπ‘ξ€œπ‘‘π‘ π‘π‘Žπ‘‘π‘’,(4.13)we can write (4.12) in the formξ€œπ‘π›Όπ‘†1(𝑠)𝑑𝑠cosh(𝑠𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡2=𝐹1ξ€œ(𝛼)βˆ’βˆžπ‘π‘‘π‘ ξ‚ƒξ‚„cosh(𝑠𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡2Γ—ξ€œπ‘π‘Žξ‚ƒξ‚„π‘€(𝑒)sinh(𝑒𝑐)1βˆ’πœ‡3𝑑𝑒cosh(𝑠𝑐)βˆ’cosh(𝑒𝑐)1/2βˆ’πœ‡2,π‘Ž<𝛼<𝑏,(4.14)where𝑆1ξ€œ(𝑠)=π‘ π‘Žξ‚ƒξ‚„π‘€(𝑒)sinh(𝑒𝑐)1βˆ’πœ‡3𝑑𝑒cosh(𝑠𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡2,π‘Ž<𝑠<𝑏.(4.15)

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 as𝑆1𝑐(𝑠)=βˆ’πœ‹ξ‚€cosπœ‹πœ‡2ξ‚π‘‘ξ€œπ‘‘π‘ π‘π‘ πΉ1(𝛼)sinh(𝛼𝑐)𝑑𝛼cosh(𝛼𝑐)βˆ’cosh(𝑠𝑐)1/2+πœ‡2+𝐼11(𝑠),π‘Ž<𝑠<𝑏,βˆ’2<πœ‡2<12,(4.16)where𝐼1𝑐(𝑠)=πœ‹ξ‚€cosπœ‹πœ‡2ξ‚π‘‘ξ€œπ‘‘π‘ π‘π‘ sinh(𝛼𝑐)𝑑𝛼cosh(𝛼𝑐)βˆ’cosh(𝑠𝑐)1/2+πœ‡2Γ—ξ€œβˆžπ‘π‘‘π‘ξ‚ƒξ‚„cosh(𝑝𝑐)βˆ’cosh(𝛼𝑐)1/2βˆ’πœ‡2ξ€œπ‘π‘Žξ‚ƒξ‚„π‘€(𝑒)sinh(𝑐𝑒)1βˆ’πœ‡3𝑑𝛼cosh(𝑝𝑐)βˆ’cosh(𝑒𝑐)1/2βˆ’πœ‡3,π‘Ž<𝑠<𝑏.(4.17)

Equation (4.17) is simplified to𝐼1ξ‚€(𝑠)=𝑐cosπœ‹πœ‡2ξ‚πœ‹sinh(𝑠𝑐)cosh(𝑏𝑐)βˆ’cosh(𝑠𝑐)1/2+πœ‡2ξ€œβˆžπ‘ξ‚ƒξ‚„cosh(𝑐𝑝)βˆ’cosh(𝑏𝑐)1/2+πœ‡2π‘‘π‘ξ‚ƒξ‚„Γ—ξ€œcosh(𝑠𝑐)βˆ’cosh(𝑐𝑝)π‘π‘Žξ‚ƒξ‚„π‘€(𝑒)sinh(𝑐𝑒)1βˆ’πœ‡3𝑑𝑒cosh(𝑐𝑝)βˆ’cosh(𝑐𝑒)1/2βˆ’πœ‡3,π‘Ž<𝑠<𝑏.(4.18)Let𝑅1ξ€œ(𝑝)=π‘π‘Žξ‚ƒξ‚„π‘€(𝑒)sinh(𝑐𝑒)1βˆ’πœ‡3𝑑𝑒cosh(𝑝𝑐)βˆ’cosh(𝑒𝑐)1/2βˆ’πœ‡3.(4.19)Equation (4.15) is of the form of (2.9). Hence, its solution is𝑀(𝑒)sinh(𝑐𝑒)1βˆ’πœ‡3=𝑐cosπœ‹πœ‡3ξ‚πœ‹π‘‘ξ€œπ‘‘π‘’π‘’π‘Žπ‘†1(𝑠)sinh(𝑠𝑐)𝑑𝑠cosh(𝑒𝑐)βˆ’cosh(𝑠𝑐)1/2+πœ‡3,π‘Ž<𝑒<𝑏.(4.20)

Substituting the expression for 𝑀(𝑒) from (4.20) into (4.19) and integrating by parts and then using the following integral:βˆ«π‘π‘ π‘sinh(𝑒𝑐)𝑑𝑒cosh(𝑝𝑐)βˆ’cosh(𝑒𝑐)3/2βˆ’πœ‡3cosh(𝑒𝑐)βˆ’cosh(𝑠𝑐)1/2+πœ‡3=βˆ’ξ‚ƒξ‚„cosh(𝑏𝑐)βˆ’cosh(𝑐𝑠)1/2βˆ’πœ‡3ξ‚€1/2βˆ’πœ‡3cosh(𝑐𝑠)βˆ’cosh(𝑐𝑝)cosh(𝑐𝑝)βˆ’cosh(𝑐𝑏)1/2βˆ’πœ‡3,1𝑠<𝑏<𝑝,βˆ’2<πœ‡2<12,(4.21)we find that𝑅1ξ‚€πœ‡(𝑝)=βˆ’π‘cos3πœ‹ξ‚πœ‹ξ‚€ξ‚cosh(𝑐𝑝)βˆ’cosh(𝑏𝑐)1/2+πœ‡3Γ—ξ€œπ‘π‘Žπ‘†1(𝑒)sinh(𝑐𝑒)𝑑𝑒cosh(𝑐𝑝)βˆ’cosh(𝑒𝑐)cosh(𝑏𝑐)βˆ’cosh(𝑐𝑒)1/2+πœ‡3.(4.22)Making use of (4.18), (4.19), and (4.22), we find that𝐼1ξ€œ(𝑠)=βˆ’π‘π‘Žπ‘†1(𝑒)𝐾2(𝑒,𝑠)𝑑𝑒,(4.23)where𝐾2𝑐(𝑒,𝑠)=2ξ‚€cosπœ‹πœ‡2cosπœ‹πœ‡3sinh(𝑠𝑐)sinh(𝑒𝑐)πœ‹2cosh(𝑏𝑐)βˆ’cosh(𝑠𝑐)1/2+πœ‡2cosh(𝑏𝑐)βˆ’cosh(𝑐𝑒)1/2+πœ‡3Γ—ξ€œβˆžπ‘ξ‚ƒξ‚„cosh(𝑐𝑝)βˆ’cosh(𝑏𝑐)1+πœ‡2+πœ‡3𝑑𝑝.cosh(𝑠𝑐)βˆ’cosh(𝑐𝑝)cosh(𝑐𝑝)βˆ’cosh(𝑐𝑒)(4.24)Using (4.17) and (4.23), (4.16) can be written in the form𝑆1ξ€œ(𝑠)+π‘π‘Žπ‘†1(𝑒)𝐾2(𝑒,𝑠)𝑑𝑒=βˆ’π‘πœ‹ξ‚€cosπœ‹πœ‡2ξ‚π‘‘ξ€œπ‘‘π‘ π‘π‘ πΉ1(𝛼)sinh(𝛼𝑐)𝑑𝛼cosh(𝛼𝑐)βˆ’cosh(𝑠𝑐)1/2+πœ‡2,π‘Ž<𝑠<𝑏.(4.25)

Equation (4.25) is a Fredholm integral equation of the second kind with kernel defined by (4.24). The Fredholm integral equation (4.25) may be solved to find numerical values of 𝑆1(𝑠) for particular values of 𝑓(𝛼). And hence from (4.20) and (4.5), the numerical values for 𝐴(𝜏) can be obtained for particular values of 𝑓(𝛼), πœ‡2, and πœ‡3.

5. 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.

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Copyright © 2008 B. M. Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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