# The Study of Triple Integral Equations with Generalized Legendre Functions

Authors: B. M. Singh; J. Rokne; R. S. Dhaliwal
Journal: Abstract and Applied Analysis
 (2008)

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## 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.


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## Body

## 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 Sections3 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] that(2.1)2πΓ(12−μ)[sinh(αc)]−μ∫0∞P−1/2+i(τ/c)μ[cosh(αc)]cos(τx)dτ=c[cosh(αc)−cosh(xc)]−μ−1/2H(α−x),where μ<1/2 and  from [4], we obtain(2.2)2π−3/2Γ(12+μ)[sinh(αc)]μ×∫0∞Γ(12−μ+iτc)Γ(12−μ−iτc)sinh(fτ)sin(xτ)P−1/2+i(τ/c)μ[cosh(αc)]dτ=c[cosh(xc)−cosh(αc)]μ−1/2H(x−α),where μ>−1/2  and H() denotes the
Heaviside unit function. Furthermore, c=π/f, f>0 and P−1/2+i(τ/c)μ(coshαc)  is the generalized
Legendre function defined in [15, page 370]. From [4, 16], the generalized  Mehler-Fock transform is defined
by(2.3)ψ(cosh(αc))=∫0∞P−1/2+i(τ/c)μ[cosh(αc)]F(τ)dτ,and its inversion formula
is(2.4)F(τ)=fτπ2sinh(fτ)Γ(12−μ+iτc)Γ(12−μ−iτc)×∫0∞P−1/2+i(τ/c)μ[cosh(αc)]ψ(cosh(αc))sinh(αc)dα.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 that(2.5)cos(xτ)τ=sinh(fτ)Γ(1/2−μ+i(τ/c))Γ(1/2−μ−i(τ/c))2πΓ(1/2−μ)×∫x∞[sinh(αc)]1+μP−1/2+i(τ/c)μ[cosh(αc)]dα[cosh(αc)−cosh(xc)]μ+1/2,μ<12,(2.6)sin(xτ)τ=π21Γ(1/2+μ)∫0x[sinh(αc)]1−μP−1/2+i(τ/c)μ(coshαc)dα[cosh(xc)−cosh(αc)]1/2−μ,μ>−12.The inversion theorem for Fourier cosine transforms and the results (2.1) and (2.2) lead to(2.7)P−1/2+i(τ/c)μ[cosh(αc)]=2πcsinhμ(αc)Γ(1/2−μ)∫0αcos(τs)ds[cosh(αc)−cosh(sc)]μ+1/2,μ<12,(2.8)P−1/2+i(τ/c)μ[cosh(αc)]=2πc[sinhμ(αc)]Γ(1/2+μ)sin(fτ)Γ(1/2−μ+i(τ/c))Γ(1/2−μ−i(τ/c))×∫α∞sin(τs)ds[cosh(sc)−cosh(αc)]1/2−μ,μ>−12.Ifh(t) is monotonically increasing and differentiable
for a<t<b and h′(t)≠0 in this interval, then the solutions of the
equations(2.9)∫atf(x)dx[h(t)−h(x)]α=g(t),a<t<b,0<α<1,(2.10)∫tbf(x)dx[h(x)−h(t)]α=g(t),a<t<b,0<α<1,are given by Sneddon [13] as(2.11)f(x)=sin(πα)πddx∫axh′(t)g(t)dt[h(x)−h(t)]1−α,a<x<b,(2.12)f(x)=−sin(πα)πddx∫xbh′(t)g(t)dt[h(t)−h(x)]1−α,a<x<b,respectively, where the prime
denotes the derivative with respect to t.


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

In this
section, we will find solution of the following triple integral
equations:(3.1)∫0∞τA(τ)sinh(τf)Γ(12−μ1+iτc)Γ(12−μ1−iτc)P−1/2+i(τ/c)μ1[cosh(αc)]dτ=0,0<α<a,(3.2)∫0∞A(τ)P−1/2+i(τ/c)μ2[cosh(αc)]dτ=f(α),a<α<b,(3.3)∫0∞τA(τ)sinh(τf)Γ(12−μ3+iτc)Γ(12−μ3−iτc)P−1/2+i(τ/c)μ3[cosh(αc)]dτ=0,b<α<∞, where A(τ) is an unknown function to be determined, f(α) is a known function, and P−1/2+i(τ/c)μ[cosh(αc)] is the generalized Legendre function defined
in Section 2 and −1/2<μ1<1/2, −1/2<μ2<1/2, μ3>−1/2.The trial solution of (3.1), (3.2), and (3.3) can be
written as(3.4)A(τ)=∫0bψ(t)cos(τt)dt,where ψ(t) is an unknown function to be determined. On
integrating (3.4) by parts, we get(3.5)A(τ)=ψ(b)sin(τb)τ−1τ∫0bψ′(t)sin(τt)dt,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)ψ(b)[cosh(bc)−cosh(αc)]1/2−μ1−∫αbψ′(t)dt[cosh(tc)−cosh(αc)]1/2−μ1=0,0<α<a.Equation (3.6) is equivalent to
the following integral equation:(3.7)ddα∫αbcsinh(tc)ψ(t)dt[cosh(tc)−cosh(αc)]1/2−μ1=0,0<α<a.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)c∫0αψ(t)dt[cosh(αc)−cosh(tc)]1/2+μ2=2πΓ(12−μ2)[sinh(αc)]−μ2f(α),a<α<b,μ2<12.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(3.9)ddα∫αbcsinh(tc)ψ(t)dt[cosh(tc)−cosh(αc)]1/2−μ1=ϕ(α),a<α<b.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(3.10)ψ(t)=−cos(πμ1)π∫tbϕ(α)dα[cosh(cα)−cosh(tc)]1/2+μ1,−12<μ1<12,a<t<b.The solution of Abel's type integral equations (2.11)
together with (3.7) and (3.9) leads
to(3.11)ψ(t)=−cos(πμ1)π∫abϕ(α)dα[cosh(cα)−cosh(tc)]1/2+μ1,−12<μ1<12,0<t<a.Equations (3.10) and (3.11) mean that (3.7) is satisfied identically.
Equation (3.8) can be rewritten in the form(3.12)∫0aψ(t)dt[cosh(αc)−cosh(tc)]1/2+μ2+∫aαψ(t)dt[cosh(αc)−cosh(tc)]1/2+μ2=1c2πΓ(1−μ2)f(α)[sinh(αc)]μ2,a<α<b.fSubstituting the expression for ψ(t) from (3.11) and (3.10) into the first and second
integral of (3.12) we obtain(3.13)∫aαS(t)dt[cosh(αc)−cosh(tc)]1/2+μ2=F(α)−∫0adt[cosh(αc)−cosh(tc)]1/2+μ2∫abϕ(u)dt[cosh(cu)−cosh(tc)]1/2+μ2,a<t<b,where(3.14)S(t)=∫tbϕ(u)dt[cosh(cu)−cosh(tc)]1/2+μ1,(3.15)F(α)=−2πΓ(1−μ2)f(α)[sinh(αc)]−μ2ccos(μ1π).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(3.16)S(t)=cos(πμ2)πddt∫atcsinh(cα)F(α)dα[cosh(ct)−cosh(cα)]1/2−μ2−I(t),a<t<b,−12<μ2<12,where(3.17)I(t)=cos(πμ2)πddt∫atcsinh(cα)dα[cosh(ct)−cosh(cα)]1/2−μ2∫0adp(cosh(cα)−cosh(cp))1/2+μ2×∫abϕ(u)du(cosh(cu)−cosh(cp))1/2+μ2,a<t<b,−12<μ2<12.From the
integral(3.18)ddt∫atcsinh(cα)dα[cosh(ct)−cosh(cα)]1/2−μ2[cosh(cα)−cosh(cp)]1/2+μ2=csinh(ct)[cosh(ct)−cosh(cp)][cosh(ca)−cosh(cp)]1/2−μ2[cosh(ct)−cosh(ca)]1/2−μ2,p<a<t,−12<μ2<12,we then obtain(3.19)I(t)=ccos(μ2π)sinh(ct)π[cosh(ct)−cosh(ca)]1/2−μ2∫0a(cosh(ca)−cosh(cp))1/2−μ2dp[cosh(ct)−cosh(cp)]×∫abϕ(u)du[cosh(cu)−cosh(cp)]1/2+μ1.Equation (3.14) is an Abel-type
equation. Hence, its solution is(3.20)ϕ(u)=−cos(μ1π)πddu∫ubcsinh(cv)S(v)dv[cosh(vc)−cosh(uc)]1/2−μ1,a<u<b,−12<μ1<12,(3.21)R(p)=∫abϕ(u)du[cosh(cu)−cosh(cp)]1/2+μ1.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(3.22)R(p)=ccos(μ1π)π[1[cosh(ca)−cosh(cp)]1/2+μ1∫abS(v)sinh(cv)dv[cosh(cv)−cosh(ca)]1/2−μ1−(12+μ1)∫abS(v)sinh(cv)dv×∫avcsinh(cu)du[cosh(cu)−cosh(cp)]3/2+μ1[cosh(cv)−cosh(cu)]1/2−μ1].The integral
(3.23)∫avcsinh(cu)du[cosh(cu)−cosh(cp)]3/2+μ1[cosh(cv)−cosh(cu)]1/2−μ1=[cosh(cv)−cosh(ca)]1/2+μ1(μ1+1/2)[cosh(cv)−cosh(cp)][cosh(ca)−cosh(cp)]p<a<v,−12<μ1<12together with (3.22) leads
to(3.24)R(p)=ccos(πμ1)π(cosh(ac)−cosh(pc))1/2−μ1×∫abS(ν)sinh(cν)dν[cosh(νc)−cosh(pc)][cosh(νc)−cosh(ac)]1/2−μ1.From (3.19), (3.21), and (3.24), we
obtain(3.25)I(t)=∫abS(ν)K(ν,t)dν,where(3.26)K(ν,t)=c2cos(πμ1)cos(πμ2)sinh(ct)sinh(cν)π2[cosh(ct)−cosh(ca)]1/2−μ2[cosh(cν)−cosh(ca)]1/2−μ1×∫0a(cosh(ca)−cosh(cp))1−μ1−μ2dp[cosh(ct)−cosh(cp)][cosh(cν)−cosh(cp)].From (3.25), (3.16) can be written
as(3.27)S(t)+∫abS(ν)K(ν,t)dν=cos(πμ2)πddt∫atcsinh(cα)F(α)dα[cosh(ct)−cosh(cα)]1/2−μ2,a<t<b.Equation (3.27) is a Fredholm
integral equation of the second kind with kernel K(ν,t).
The kernel is defined by (3.26). The integral in (3.26) cannot be solved
analytically, but for particular values of μ1 and μ2 the values of K(ν,t) can be found numerically. Hence, the numerical
solution of Fredholm integral equation (3.27) can be obtained for particular
value of f(α), μ1,
and μ2 to find numerical values of S(t).
Making use of (3.20), (3.11), and (3.10), the numerical results for ψ(t) 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:(4.1)∫0∞τA(τ)P−1/2+i(τ/c)μ1[cosh(αc)]dτ=0,0<α<a,(4.2)∫0∞sinh(τf)Γ(12−μ2+iτc)Γ(12−μ2−iτc)A(τ)P−1/2+i(τ/c)μ2[cosh(αc)]dτ=f(α),a<α<b,(4.3)∫0∞τA(τ)P−1/2+i(τ/c)μ3[cosh(αc)]dτ=0,b<α, where μ1>−1/2, −1/2<μ2<1/2, −1/2<μ3<1/2.We assume that(4.4)∫0∞τA(τ)P−1/2+i(τ/c)μ3[cosh(αc)]dτ=M(α),0<α<b.The inversion formula for
generalized Mehler-Fock  transforms (2.4) together with (4.3) and (4.4) implies
that(4.5)A(τ)=fπ2sinh(fτ)Γ(12−μ3+iτc)Γ(12−μ3−iτc)×∫0bsinh(uc)P−1/2+i(τ/c)μ3[cosh(uc)]M(u)du.Multiplying (4.1) by [sinh(αc)]1−μ1/[cosh(xc)−cosh(αc)]1/2−μ1, integrating both sides from 0 to x and with respect to α,
and then using (2.6) we obtain(4.6)∫0∞A(τ)sin(xτ)dτ=0,0<x<a.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(4.7)∫0xsinh(uc)M(u)du[cosh(xc)−cosh(uc)]1/2−μ3=0,μ3>−12,0<x<a.Substituting the value of A(τ) from (4.5) into (4.2) and interchanging the
order of integrations we arrive at(4.8)∫0bsinh(uc)M(u)K2(u,α)du=f(α),a<α<b,where(4.9)K2(u,α)=∫0∞fπ2Γ(12−μ2−iτc)Γ(12−μ2+iτc)Γ(12−μ3+iτc)Γ(12−μ3−iτc)×sinh2(fτ)P−1/2+i(τ/c)μ3[cosh(uc)]P−1/2+i(τ/c)μ2[cosh(αc)]dτ,and then (2.8) and (2.2) imply
that(4.10)K2(u,α)=cπΓ(1/2+μ2)Γ(1/2+μ3)[sinh(αc)]μ2[sinh(uc)]μ3×∫max(α,u)∞ds[cosh(sc)−cosh(αc)]1/2−μ2[cosh(sc)−cosh(uc)]1/2−μ3,μ3>−12,μ2>−12.Equation (4.7) is an Abel-type equation and has the form (2.9). Hence,
the  solution of (4.7) is(4.11)M(u)=0,0<u<a.Using (4.10) and (2.5), (4.8) can be
written in the form(4.12)∫ab[sinh(uc)]1−μ3M(u)du∫max(α,u)∞ds[cosh(sc)−cosh(αc)]1/2−μ2[cosh(sc)−cosh(uc)]1/2−μ3=Γ(1/2+μ2)Γ(1/2+μ2)[sinh(αc)]μ3cπf(α)=F1(α),say,a<α<b.Using the
formula(4.13)∫abdu∫max(α,u)∞ds=∫αbds∫asdu+∫b∞ds∫abdu,we can write (4.12) in the
form(4.14)∫αbS1(s)ds[cosh(sc)−cosh(αc)]1/2−μ2=F1(α)−∫b∞ds[cosh(sc)−cosh(αc)]1/2−μ2×∫abM(u)[sinh(uc)]1−μ3du[cosh(sc)−cosh(uc)]1/2−μ2,a<α<b,where(4.15)S1(s)=∫asM(u)[sinh(uc)]1−μ3du[cosh(sc)−cosh(αc)]1/2−μ2,a<s<b.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(4.16)S1(s)=−cπcos(πμ2)dds∫sbF1(α)sinh(αc)dα[cosh(αc)−cosh(sc)]1/2+μ2+I1(s),a<s<b,−12<μ2<12,where(4.17)I1(s)=cπcos(πμ2)dds∫sbsinh(αc)dα[cosh(αc)−cosh(sc)]1/2+μ2×∫b∞dp[cosh(pc)−cosh(αc)]1/2−μ2∫abM(u)[sinh(cu)]1−μ3dα[cosh(pc)−cosh(uc)]1/2−μ3,a<s<b.Equation (4.17) is simplified to(4.18)I1(s)=ccos(πμ2)πsinh(sc)[cosh(bc)−cosh(sc)]1/2+μ2∫b∞[cosh(cp)−cosh(bc)]1/2+μ2dp[cosh(sc)−cosh(cp)]×∫abM(u)[sinh(cu)]1−μ3du[cosh(cp)−cosh(cu)]1/2−μ3,a<s<b.Let(4.19)R1(p)=∫abM(u)[sinh(cu)]1−μ3du[cosh(pc)−cosh(uc)]1/2−μ3.Equation (4.15) is of the form of
(2.9). Hence, its solution is(4.20)M(u)[sinh(cu)]1−μ3=ccos(πμ3)πddu∫auS1(s)sinh(sc)ds[cosh(uc)−cosh(sc)]1/2+μ3,a<u<b.Substituting the expression forM(u) from (4.20) into (4.19) and integrating by parts
and then using the following integral:(4.21)∫sbcsinh(uc)du(cosh(pc)−cosh(uc))3/2−μ3(cosh(uc)−cosh(sc))1/2+μ3=−[cosh(bc)−cosh(cs)]1/2−μ3(1/2−μ3)[cosh(cs)−cosh(cp)][cosh(cp)−cosh(cb)]1/2−μ3,s<b<p,−12<μ2<12,we find that(4.22)R1(p)=−ccos(μ3π)π(cosh(cp)−cosh(bc))1/2+μ3×∫abS1(u)sinh(cu)du[cosh(cp)−cosh(uc)][cosh(bc)−cosh(cu)]1/2+μ3.Making use of (4.18), (4.19), and
(4.22), we find that(4.23)I1(s)=−∫abS1(u)K2(u,s)du,where(4.24)K2(u,s)=c2cos(πμ2)cos(πμ3)sinh(sc)sinh(uc)π2[cosh(bc)−cosh(sc)]1/2+μ2[cosh(bc)−cosh(cu)]1/2+μ3×∫b∞[cosh(cp)−cosh(bc)]1+μ2+μ3dp[cosh(sc)−cosh(cp)][cosh(cp)−cosh(cu)].Using (4.17) and (4.23), (4.16) can be
written in the form(4.25)S1(s)+∫abS1(u)K2(u,s)du=−cπcos(πμ2)dds∫sbF1(α)sinh(αc)dα[cosh(αc)−cosh(sc)]1/2+μ2,a<s<b.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), μ2,
and μ3.


## 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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