Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 395257, 12 pages
doi:10.1155/2008/395257
Research Article
The Study of Triple Integral Equations with
Generalized Legendre Functions
B. M. Singh,1 J. Rokne,1 and R. S. Dhaliwal2
1
2
Department of Computer Science, University of Calgary, Calgary, AB, Canada T2N-1N4
Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N-1N4
Correspondence should be addressed to R. S. Dhaliwal, dhali.r@shaw.ca
Received 28 April 2008; Accepted 12 September 2008
Recommended by Lance Littlejohn
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.
Copyright q 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.
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
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Abstract and Applied Analysis
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
−μ ∞ μ
2
1
Γ
− μ sinhαc
P−1/2iτ/c coshαc cosτxdτ
π
2
0
−μ−1/2
Hα − x,
c coshαc − coshxc
2.1
where μ < 1/2 and from 4, we obtain
√ −3/2 1
μ
μ sinhαc
2π
Γ
2
∞ τ
τ
1
1
μ
−μi
−μ−i
× Γ
Γ
sinhfτ sinxτP−1/2iτ/c coshαc dτ
2
c
2
c
0
μ−1/2
Hx − α,
c coshxc − coshαc
2.2
where μ > −1/2 and H denotes the Heaviside unit function. Furthermore, c π/f, f > 0
μ
and P−1/2iτ/c cosh αc is the generalized Legendre function defined in 15, page 370. From
4, 16, the generalized Mehler-Fock transform is defined by
ψ coshαc ∞
0
μ
P−1/2iτ/c coshαc Fτdτ,
2.3
and its inversion formula is
Fτ fτ
τ
1
τ
1
−
μ
i
−
μ
−
i
Γ
sinhfτΓ
2
c
2
c
π2
∞
μ
×
P−1/2iτ/c coshαc ψ coshαc sinhαcdα.
2.4
0
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
cosxτ sinhfτΓ 1/2 − μ iτ/c Γ 1/2 − μ − iτ/c
√
τ
2πΓ1/2 − μ
coshαc dα
1
−1/2iτ/c
, μ< ,
×
μ1/2
2
x
coshαc − coshxc
x sinhαc1−μ P μ
cosh αcdα
sinxτ
1
π
−1/2iτ/c
,
1/2−μ
τ
2 Γ1/2 μ 0
coshxc − coshαc
2.5
∞ sinhαc1μ P μ
1
μ>− .
2
2.6
B. M. Singh et al.
3
The inversion theorem for Fourier cosine transforms and the results 2.1 and 2.2
lead to
μ
P−1/2iτ/c coshαc
2 sinhμ αc
c
π Γ1/2 − μ
α
cosτsds
1
2.7
μ1/2 , μ < 2 ,
0 coshαc − coshsc
√
2πc
μ
P−1/2iτ/c coshαc sinhμ αcΓ1/2μ sinfτΓ 1/2 −μ iτ/c Γ 1/2 −μ −iτ/c
∞
sinτsds
1
×
1/2−μ , μ > − 2 .
α coshsc − coshαc
2.8
If ht is monotonically increasing and differentiable for a < t < b and h t /
0 in this
interval, then the solutions of the equations
t
fxdx
α gt,
a ht − hx
a < t < b, 0 < α < 1,
2.9
a < t < b, 0 < α < 1,
2.10
b
fxdx
α gt,
t hx − ht
are given by Sneddon 13 as
x
h tgtdt
1−α , a < x < b,
a hx − ht
− sinπα d b h tgtdt
fx , a < x < b,
π
dx x ht − hx1−α
sinπα d
fx π
dx
2.11
2.12
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:
∞
0
1
τ
τ
1
μ1
− μ1 i
− μ1 − i
τAτ sinhτfΓ
coshαc dτ 0,
Γ
P−1/2iτ/c
2
c
2
c
∞
∞
0
0
μ2
AτP−1/2iτ/c
coshαc dτ fα,
a < α < b,
1
1
τ
τ
μ3
− μ3 i
Γ
− μ3 − i
P−1/2iτ/c
τAτ sinhτfΓ
coshαc dτ 0,
2
c
2
c
0 < α < a,
3.1
3.2
b < α < ∞,
3.3
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Abstract and Applied Analysis
where Aτ is an unknown function to be determined, fα is a known function, and
μ
P−1/2iτ/c coshαc 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
Aτ b
3.4
ψt cosτtdt,
0
where ψt is an unknown function to be determined. On integrating 3.4 by parts, we get
ψb sinτb 1
Aτ −
τ
τ
b
ψ t sinτtdt,
3.5
0
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
ψb
1/2−μ1
coshbc − coshαc
−
b
α
ψ tdt
coshtc − coshαc
1/2−μ1 0,
0 < α < a.
3.6
Equation 3.6 is equivalent to the following integral equation:
d
dα
b
c sinhtcψtdt
1/2−μ1 0,
α coshtc − coshαc
3.7
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
α
c
ψtdt
1/2μ2 0 coshαc − coshtc
−μ
1
2
Γ
− μ2 sinhαc 2 fα,
π
2
a < α < b, μ2 <
1
.
2
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
d
dα
b
c sinhtcψtdt
1/2−μ1 φα,
α coshtc − coshαc
a < α < b.
3.9
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
ψt −
cos πμ1 b
φαdα
1/2μ1 ,
π
t coshcα − coshtc
−
1
1
< μ1 < , a < t < b.
2
2
3.10
The solution of Abel’s type integral equations 2.11 together with 3.7 and 3.9 leads
to
cos πμ1 b
φαdα
ψt −
1/2μ1 ,
π
a coshcα − coshtc
−
1
1
< μ1 < , 0 < t < a.
2
2
3.11
Equations 3.10 and 3.11 mean that 3.7 is satisfied identically. Equation 3.8 can
be rewritten in the form
a
0
ψtdt
1/2μ2 α
ψtdt
1/2μ2
a coshαc − coshtc
coshαc − coshtc
μ
1 2 Γ 1 − μ2 fα sinhαc 2 ,
c π
3.12
a < α < b.
Substituting the expression for ψt from 3.11 and 3.10 into the first and second integral
of 3.12 we obtain
α
Stdt
1/2μ2
a coshαc − coshtc
b
a
φudt
dt
Fα − 1/2μ2 1/2μ2 ,
0 coshαc − coshtc
a coshcu − coshtc
a < t < b,
3.13
where
St b
t
φudt
coshcu − coshtc
3.14
1/2μ1 ,
√
−μ
− 2πΓ 1 − μ2 fα sinhαc 2
.
Fα c cos μ1 π
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
cos πμ2 d t
c sinhcαFαdα
St − It,
π
dt a coshct − coshcα1/2−μ2
a < t < b, −
1
1
< μ2 < ,
2
2
3.16
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Abstract and Applied Analysis
where
a
cos πμ2 d t
dp
c sinhcαdα
It 1/2μ2
1/2−μ
2
π
dt a coshct − coshcα
0 coshcα − coshcp
×
b
a
φudu
coshcu − coshcp
1/2μ2 ,
1
1
a < t < b, − < μ2 < .
2
2
3.17
From the integral
d
dt
t
c sinhcαdα
1/2−μ2 1/2μ2
a coshct − coshcα
coshcα − coshcp
3.18
1/2−μ2
coshca − coshcp
c sinhct
1
1
, p < a < t, − < μ2 < ,
2
2
coshct − coshcp coshct − coshca 1/2−μ2
we then obtain
1/2−μ2
a c cos μ2 π sinhct
dp
coshca − coshcp
It 1/2−μ2
coshct − coshcp
0
π coshct − coshca
b
φudu
× 1/2μ1 .
a coshcu − coshcp
3.19
Equation 3.14 is an Abel-type equation. Hence, its solution is
φu −
cos μ1 π d b
c sinhcvSvdv
,
π
du u coshvc − coshuc1/2−μ1
Rp a < u < b, −
1
1
< μ1 < ,
2
2
b
φudu
1/2μ1 .
a coshcu − coshcp
3.20
3.21
Substituting the expression for φu from 3.20 into 3.21, integrating by parts, and finally
interchanging the order of integrations in second integral, we arrive at
b
c cos μ1 π
Sv sinhcvdv
1
Rp 1/2μ1 1/2−μ1
π
a coshcv − coshca
coshca − coshcp
−
1
μ1
2
b
Sv sinhcvdv
a
c sinhcudu
× 3/2μ1 1/2−μ1 .
a coshcu − coshcp
coshcv − coshcu
v
3.22
B. M. Singh et al.
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The integral
v
c sinhcudu
3/2μ1 1/2−μ1
a coshcu − coshcp
coshcv − coshcu
coshcv − coshca1/2μ1
μ1 1/2 coshcv − coshcp coshca − coshcp
p < a < v,
−
3.23
1
1
< μ1 <
2
2
together with 3.22 leads to
c cos πμ1 1/2−μ1
Rp coshac − coshpc
π
b
Sν sinhcνdν
× 1/2−μ1 .
a coshνc − coshpc coshνc − coshac
3.24
From 3.19, 3.21, and 3.24, we obtain
It b
3.25
SνKν, tdν,
a
where
c2 cos πμ1 cos πμ2 sinhct sinhcν
Kν, t 1/2−μ2 1/2−μ1
π 2 coshct − coshca
coshcν − coshca
×
a
0
1−μ1 −μ2
3.26
dp
coshca − coshcp
.
coshct − coshcp coshcν − coshcp
From 3.25, 3.16 can be written as
cos πμ2 d t
c sinhcαFαdα
St SνKν, tdν ,
π
dt a coshct − coshcα1/2−μ2
a
b
a < t < b.
3.27
Equation 3.27 is a Fredholm integral equation of the second kind with kernel Kν, 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 Kν, t can be found numerically. Hence, the
numerical solution of Fredholm integral equation 3.27 can be obtained for particular value
of fα, μ1 , and μ2 to find numerical values of St. Making use of 3.20, 3.11, and 3.10, the
numerical results for ψt can be obtained. Finally, making use of 3.4 the numerical results
for Aτ can be obtained.
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Abstract and Applied Analysis
4. Triple integral equations with generalized Legendre functions: set II
In this section, we will find the solution of the following triple integral equations:
∞
∞
0
0
μ1
τAτP−1/2iτ/c
coshαc dτ 0,
0 < α < a,
1
1
τ
τ
μ2
− μ2 i
Γ
− μ2 − i
AτP−1/2iτ/c
sinhτfΓ
coshαc dτ fα,
2
c
2
c
∞
0
μ3
τAτP−1/2iτ/c
coshαc dτ 0,
4.1
a < α < b,
4.2
b < α,
4.3
0 < α < b.
4.4
where μ1 > −1/2, −1/2 < μ2 < 1/2, −1/2 < μ3 < 1/2.
We assume that
∞
0
μ3
τAτP−1/2iτ/c
coshαc dτ Mα,
The inversion formula for generalized Mehler-Fock transforms 2.4 together with 4.3 and
4.4 implies that
Aτ f
τ
τ
1
1
−
μ
−
μ
sinhfτΓ
i
−
i
Γ
3
3
2
c
2
c
π2
b
μ3
× sinhucP−1/2iτ/c
coshuc Mudu.
4.5
0
Multiplying 4.1 by sinhαc1−μ1 /coshxc − coshαc1/2−μ1 , integrating both sides
from 0 to x and with respect to α, and then using 2.6 we obtain
∞
Aτ sinxτdτ 0,
0 < x < a.
4.6
0
Substituting the value of Aτ from 4.5 into 4.6, interchanging the order of integrations,
and using the integral 2.2, we get
x
sinhucMudu
1/2−μ3 0,
0 coshxc − coshuc
1
μ3 > − , 0 < x < a.
2
4.7
Substituting the value of Aτ from 4.5 into 4.2 and interchanging the order of integrations
we arrive at
b
0
sinhucMuK2 u, αdu fα,
a < α < b,
4.8
B. M. Singh et al.
9
where
K2 u, α ∞
0
f
1
τ
τ
τ
τ
1
1
1
−
μ
−
μ
−
μ
−
μ
Γ
−
i
i
i
−
i
Γ
Γ
Γ
2
2
3
3
2
c
2
c
2
c
2
c
π2
μ3
μ2
× sinh2 fτP−1/2iτ/c
coshuc P−1/2iτ/c
coshαc dτ,
4.9
and then 2.8 and 2.2 imply that
cπ
K2 u, α μ μ
Γ 1/2 μ2 Γ 1/2 μ3 sinhαc 2 sinhuc 3
×
∞
maxα,u
ds
1/2−μ2 1/2−μ3 ,
coshsc − coshαc
coshsc − coshuc
1
1
μ3 > − , μ2 > − .
2
2
4.10
Equation 4.7 is an Abel-type equation and has the form 2.9. Hence, the solution of
4.7 is
Mu 0,
4.11
0 < u < a.
Using 4.10 and 2.5, 4.8 can be written in the form
b
a
1−μ3
∞
Mudu
ds
1/2−μ2 1/2−μ3
maxα,u coshsc − coshαc
coshsc − coshuc
μ
Γ 1/2 μ2 Γ 1/2 μ2 sinhαc 3
fα F1 α, say, a < α < b.
cπ
4.12
sinhuc
Using the formula
b
∞
ds du
a
b
maxα,u
s
∞ b
ds du ds du,
α
a
b
4.13
a
we can write 4.12 in the form
b
S1 sds
1/2−μ2 F1 α −
α coshsc − coshαc
×
b
∞
b
ds
1/2−μ2
coshsc − coshαc
1−μ3
Mu sinhuc
du
1/2−μ2 ,
a coshsc − coshuc
a < α < b,
4.14
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Abstract and Applied Analysis
where
1−μ3
Mu sinhuc
du
S1 s 1/2−μ2 ,
a coshsc − coshαc
s
4.15
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
S1 s −
d
c
cos πμ2
π
ds
b
F1 α sinhαcdα
1/2μ2 I1 s,
s coshαc − coshsc
a < s < b, −
1
1
< μ2 < ,
2
2
4.16
where
I1 s d
c
cos πμ2
π
ds
×
∞
b
b
sinhαcdα
1/2μ2
s coshαc − coshsc
1−μ3
Mu sinhcu
dα
1/2−μ3 ,
a coshpc − coshuc
4.17
b
dp
1/2−μ2
coshpc − coshαc
a < s < b.
Equation 4.17 is simplified to
1/2μ2
∞ c cos πμ2
dp
coshcp − coshbc
sinhsc
I1 s 1/2μ2
π
coshsc − coshcp
b
coshbc − coshsc
×
b
1−μ3
Mu sinhcu
du
1/2−μ3 ,
a coshcp − coshcu
4.18
a < s < b.
Let
R1 p 1−μ3
Mu sinhcu
du
1/2−μ3 .
a coshpc − coshuc
b
4.19
Equation 4.15 is of the form of 2.9. Hence, its solution is
Mu sinhcu
1−μ3
c cos πμ3 d u
S1 s sinhscds
,
π
du a coshuc − coshsc1/2μ3
a < u < b.
4.20
B. M. Singh et al.
11
Substituting the expression for Mu from 4.20 into 4.19 and integrating by parts
and then using the following integral:
b
s
c sinhucdu
3/2−μ3 1/2μ3
coshpc − coshuc
coshuc − coshsc
1/2−μ3
− coshbc − coshcs
1/2−μ3 ,
1/2 − μ3 coshcs − coshcp coshcp − coshcb
s < b < p,
−
4.21
1
1
< μ2 < ,
2
2
we find that
−c cos μ3 π 1/2μ3
R1 p coshcp − coshbc
π
b
S1 u sinhcudu
× 1/2μ3 .
a coshcp − coshuc coshbc − coshcu
4.22
Making use of 4.18, 4.19, and 4.22, we find that
I1 s −
b
S1 uK2 u, sdu,
4.23
a
where
c2 cos πμ2 cos πμ3 sinhsc sinhuc
K2 u, s 1/2μ2 1/2μ3
π 2 coshbc − coshsc
coshbc − coshcu
×
∞
b
1μ2 μ3
dp
coshcp − coshbc
.
coshsc − coshcp coshcp − coshcu
4.24
Using 4.17 and 4.23, 4.16 can be written in the form
b
d
−c
S1 s S1 uK2 u, sdu cos πμ2
π
ds
a
b
s
F1 α sinhαcdα
1/2μ2 ,
coshαc − coshsc
a < 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 fα. And hence from 4.20 and 4.5, the numerical values for
Aτ can be obtained for particular values of fα, μ2 , and μ3 .
12
Abstract and Applied Analysis
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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