The Study of Triple Integral Equations with Generalized...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2008, Article ID 395257, 12 pagesdoi:10.1155/2008/395257

Research ArticleThe Study of Triple Integral Equations withGeneralized Legendre Functions

B. M. Singh,1 J. Rokne,1 and R. S. Dhaliwal2

1 Department of Computer Science, University of Calgary, Calgary, AB, Canada T2N-1N42 Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N-1N4

Correspondence should be addressed to R. S. Dhaliwal, [email protected]

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 associatedLegendre functions of imaginary arguments. The solution of each set of triple integral equationsinvolving associated Legendre functions is reduced to a Fredholm integral equation of the secondkind which can be solved numerically.

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

1. Introduction

Dual integral equations involving Legendre functions have been solved by Babloian [1]. Heapplied these equations to problems of potential theory and to a torsion problem. Later onPathak [2] and Mandal [3] who considered dual integral equations involving generalizedLegendre functions which have more general solution than the ones considered by Babloian[1]. Recently, Singh et al. [4] considered dual integral equations involving generalizedLegendre functions, and their results are more general than those in [1–3].

In the analysis of mixed boundary value problems, we often encounter triple integralequations. Triple integral equations involving Legendre functions have been studied bySrivastava [5]. Triple integral equations involving Bessel functions have also been consideredby Cooke [6–9], Tranter [10], Love and Clements [11], Srivastava [12], and most of theseauthors reduced the solution into a solution of Fredholm integral equation of the secondkind. The relevant references for dual and triple integral equations are given in the book ofSneddon [13].

In this paper, a method is developed for solutions of two sets of triple integralequations involving generalized Legendre functions in Sections 3 and 4. Each set of tripleintegral equations is reduced to a Fredholm integral equation of the second kind which maybe 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 integralequations 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

√2πΓ(12− μ

)[sinh(αc)

]−μ∫∞

0Pμ

−1/2+i(τ/c)[cosh(αc)

]cos(τx)dτ

= c[cosh(αc) − cosh(xc)

]−μ−1/2H(α − x),

(2.1)

where μ < 1/2 and from [4], we obtain

√2π−3/2Γ

(12+ μ

)[sinh(αc)

]μ

×∫∞

0Γ(12− μ + i

τ

c

)Γ(12− μ − i τ

c

)sinh(fτ) sin(xτ)Pμ−1/2+i(τ/c)

[cosh(αc)

]dτ

= c[cosh(xc) − cosh(αc)

]μ−1/2H(x − α),

(2.2)

where μ > −1/2 and H() denotes the Heaviside unit function. Furthermore, c = π/f , f > 0and Pμ−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 by

ψ(cosh(αc)

)=∫∞

0Pμ


]F(τ)dτ, (2.3)

and its inversion formula is

F(τ) =fτ

π2sinh(fτ)Γ

(12− μ + i

τ

c

)Γ(12− μ − i τ

c

)

×∫∞

0Pμ


]ψ(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 that

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+μPμ


]dα

[cosh(αc) − cosh(xc)

]μ+1/2 , μ <12,

(2.5)

sin(xτ)τ

=√π

21

Γ(1/2 + μ)

∫x

0

[sinh(αc)

]1−μPμ

−1/2+i(τ/c)(coshαc)dα[cosh(xc) − cosh(αc)

]1/2−μ , μ > −12. (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

Pμ


]=

√2πcsinhμ(αc)Γ(1/2 − μ)

∫α

0

cos(τs)ds[cosh(αc) − cosh(sc)

]μ+1/2 , μ <12, (2.7)

Pμ


]=

√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.

(2.8)

If h(t) is monotonically increasing and differentiable for a < t < b and h′(t) /= 0 in thisinterval, then the solutions of the equations

∫ t

a

f(x)dx[h(t) − h(x)]α = g(t), a < t < b, 0 < α < 1, (2.9)

∫b

t

f(x)dx[h(x) − h(t)]α = g(t), a < t < b, 0 < α < 1, (2.10)

are given by Sneddon [13] as

f(x) =sin(πα)

π

d

dx

∫x

a

h′(t)g(t)dt[h(x) − h(t)]1−α , a < x < b, (2.11)

f(x) =− sin(πα)

π

d

dx

∫b

x

h′(t)g(t)dt[h(t) − h(x)]1−α , a < x < b, (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:

∫∞

0τA(τ) sinh(τf)Γ

(12− μ1 + i

τ

c

)Γ(12− μ1 − i τ

c

)Pμ1−1/2+i(τ/c)

[cosh(αc)

]dτ = 0, 0 < α < a,

(3.1)∫∞

0A(τ)Pμ2−1/2+i(τ/c)

[cosh(αc)

]dτ = f(α), a < α < b, (3.2)

∫∞

0τA(τ) sinh(τf)Γ

(12− μ3 + i

τ

c

)Γ(12− μ3 − i τ

c

)Pμ3−1/2+i(τ/c)

[cosh(αc)

]dτ = 0, b < α <∞,

(3.3)


where A(τ) is an unknown function to be determined, f(α) is a known function, andPμ

−1/2+i(τ/c)[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

A(τ) =∫b

0ψ(t) cos(τt)dt, (3.4)

where ψ(t) is an unknown function to be determined. On integrating (3.4) by parts, we get

A(τ) =ψ(b) sin(τb)

τ− 1τ

∫b

0ψ ′(t) sin(τt)dt, (3.5)

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 integraldefined by (2.2), we obtain

ψ(b)[cosh(bc) − cosh(αc)

]1/2−μ1 −∫b

α

ψ ′(t)dt[cosh(tc) − cosh(αc)

]1/2−μ1 = 0, 0 < α < a. (3.6)

Equation (3.6) is equivalent to the following integral equation:

d

dα

∫b

α

c sinh(tc)ψ(t)dt[cosh(tc) − cosh(αc)

]1/2−μ1 = 0, 0 < α < a. (3.7)

By substituting (3.4) into (3.2), interchanging the order of integrations and using the integraldefined by (2.1) we find that

c

∫α

0

ψ(t)dt[cosh(αc) − cosh(tc)

]1/2+μ2 =

√2πΓ(12− μ2

)[sinh(αc)

]−μ2f(α), a < α < b, μ2 <12.

(3.8)

For obtaining the solution of the problem, we need to solve twoAbel’s type integral equations(3.7) and (3.8).

We assume that

d

dα

∫b

α

c sinh(tc)ψ(t)dt[cosh(tc) − cosh(αc)

]1/2−μ1 = φ(α), a < 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 as

ψ(t) = −cos(πμ1

)π

∫b

t

φ(α)dα[cosh(cα) − cosh(tc)

]1/2+μ1 , −12< μ1 <

12, a < t < b. (3.10)

The solution of Abel’s type integral equations (2.11) together with (3.7) and (3.9) leadsto

ψ(t) = −cos(πμ1

)π

∫b

a

φ(α)dα[cosh(cα) − cosh(tc)

]1/2+μ1 , −12< μ1 <

12, 0 < t < a. (3.11)

Equations (3.10) and (3.11) mean that (3.7) is satisfied identically. Equation (3.8) canbe rewritten in the form

∫a

0


]1/2+μ2 +∫α

a


]1/2+μ2

=1c

√2πΓ(1 − μ2

)f(α)

[sinh(αc)

]μ2 , a < α < b.

(3.12)

Substituting the expression for ψ(t) from (3.11) and (3.10) into the first and second integralof (3.12) we obtain

∫α

a

S(t)dt[cosh(αc) − cosh(tc)

]1/2+μ2

= F(α) −∫a

0

dt[cosh(αc) − cosh(tc)

]1/2+μ2∫b

a

φ(u)dt[cosh(cu) − cosh(tc)

]1/2+μ2 , a < t < b,

(3.13)

where

S(t) =∫b

t

φ(u)dt[cosh(cu) − cosh(tc)

]1/2+μ1 , (3.14)

F(α) =−√2πΓ

(1 − μ2

)f(α)

[sinh(αc)

]−μ2c cos

(μ1π

) . (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(t) =cos

(πμ2

)π

d

dt

∫ t

a

c sinh(cα)F(α)dα[cosh(ct) − cosh(cα)

]1/2−μ2 − I(t), a < t < b, − 12< μ2 <

12,

(3.16)


where

I(t) =cos

(πμ2

)π

d

dt

∫ t

a

c sinh(cα)dα[cosh(ct) − cosh(cα)

]1/2−μ2∫a

0

dp(cosh(cα) − cosh(cp)

)1/2+μ2

×∫b

a

φ(u)du(cosh(cu) − cosh(cp)

)1/2+μ2 , a < t < b, − 12< μ2 <

12.

(3.17)

From the integral

d

dt

∫ t

a

c sinh(cα)dα[cosh(ct) − cosh(cα)

]1/2−μ2[cosh(cα) − cosh(cp)]1/2+μ2

=c sinh(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,

(3.18)

we then obtain

I(t) =c cos

(μ2π

)sinh(ct)

π[cosh(ct) − cosh(ca)

]1/2−μ2∫a

0

(cosh(ca) − cosh(cp)

)1/2−μ2dp[cosh(ct) − cosh(cp)

]

×∫b

a

φ(u)du[cosh(cu) − cosh(cp)

]1/2+μ1 .(3.19)

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

φ(u) = −cos(μ1π

)π

d

du

∫b

u

c sinh(cv)S(v)dv[cosh(vc) − cosh(uc)

]1/2−μ1 , a < u < b, − 12< μ1 <

12, (3.20)

R(p) =∫b

a

φ(u)du[cosh(cu) − cosh(cp)

]1/2+μ1 . (3.21)

Substituting the expression for φ(u) from (3.20) into (3.21), integrating by parts, and finallyinterchanging the order of integrations in second integral, we arrive at

R(p) =c cos

(μ1π

)π

[1[

cosh(ca) − cosh(cp)]1/2+μ1

∫b

a

S(v) sinh(cv)dv[cosh(cv) − cosh(ca)

]1/2−μ1

−(12+ μ1

)∫b

a

S(v) sinh(cv)dv

×∫v

a

c sinh(cu)du[cosh(cu) − cosh(cp)

]3/2+μ1[cosh(cv) − cosh(cu)]1/2−μ1

].

(3.22)


The integral

∫v

a

c sinh(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 <

12

(3.23)

together with (3.22) leads to

R(p) =c cos

(πμ1

)π

(cosh(ac) − cosh(pc)

)1/2−μ1

×∫b

a

S(ν) sinh(cν)dν[cosh(νc) − cosh(pc)

][cosh(νc) − cosh(ac)

]1/2−μ1 .(3.24)

From (3.19), (3.21), and (3.24), we obtain

I(t) =∫b

a

S(ν)K(ν, t)dν, (3.25)

where

K(ν, t) =c2 cos

(πμ1

)cos

(πμ2

)sinh(ct) sinh(cν)

π2[cosh(ct) − cosh(ca)

]1/2−μ2[cosh(cν) − cosh(ca)]1/2−μ1

×∫a

0

(cosh(ca) − cosh(cp)

)1−μ1−μ2dp[cosh(ct) − cosh(cp)

][cosh(cν) − cosh(cp)

] .(3.26)

From (3.25), (3.16) can be written as

S(t) +∫b

a

S(ν)K(ν, t)dν =cos

(πμ2

)π

d

dt

∫ t

a

c sinh(cα)F(α)dα[cosh(ct) − cosh(cα)

]1/2−μ2 , a < t < b. (3.27)

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 forparticular values of μ1 and μ2 the values of K(ν, t) can be found numerically. Hence, thenumerical solution of Fredholm integral equation (3.27) can be obtained for particular valueof f(α), μ1, and μ2 to find numerical values of S(t). Making use of (3.20), (3.11), and (3.10), thenumerical results for ψ(t) can be obtained. Finally, making use of (3.4) the numerical resultsfor 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:

∫∞

0τA(τ)Pμ1−1/2+i(τ/c)

[cosh(αc)

]dτ = 0, 0 < α < a, (4.1)

∫∞

0sinh(τf)Γ

(12− μ2 + i

τ

c

)Γ(12− μ2 − i τ

c

)A(τ)Pμ2−1/2+i(τ/c)

[cosh(αc)

]dτ = f(α), a < α < b,

(4.2)∫∞


[cosh(αc)

]dτ = 0, b < α, (4.3)

where μ1 > −1/2, −1/2 < μ2 < 1/2, −1/2 < μ3 < 1/2.We assume that

∫∞


[cosh(αc)

]dτ =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 that

A(τ) =f

π2sinh(fτ)Γ

(12− μ3 + i

τ

c

)Γ(12− μ3 − i τ

c

)

×∫b

0sinh(uc)Pμ3−1/2+i(τ/c)

[cosh(uc)

]M(u)du.

(4.5)

Multiplying (4.1) by [sinh(αc)]1−μ1/[cosh(xc) − cosh(αc)]1/2−μ1 , integrating both sidesfrom 0 to x and with respect to α, and then using (2.6) we obtain

∫∞

0A(τ) sin(xτ)dτ = 0, 0 < x < a. (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

∫x

0

sinh(uc)M(u)du[cosh(xc) − cosh(uc)

]1/2−μ3 = 0, μ3 > −12, 0 < x < a. (4.7)

Substituting the value ofA(τ) from (4.5) into (4.2) and interchanging the order of integrationswe arrive at

∫b

0sinh(uc)M(u)K2(u, α)du = f(α), a < α < b, (4.8)


where

K2(u, α) =∫∞

0

f

π2Γ(12− μ2 − i τ

c

)Γ(12− μ2 + i

τ

c

)Γ(12− μ3 + i

τ

c

)Γ(12− μ3 − i τ

c

)

× sinh2(fτ)Pμ3−1/2+i(τ/c)[cosh(uc)

]Pμ2−1/2+i(τ/c)

[cosh(αc)

]dτ,

(4.9)

and then (2.8) and (2.2) imply that

K2(u, α) =cπ

Γ(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 > −1

2.

(4.10)

Equation (4.7) is an Abel-type equation and has the form (2.9). Hence, the solution of(4.7) is

M(u) = 0, 0 < u < a. (4.11)

Using (4.10) and (2.5), (4.8) can be written in the form

∫b

a

[sinh(uc)

]1−μ3M(u)du∫∞

max(α,u)


]1/2−μ2[cosh(sc) − cosh(uc)]1/2−μ3

=Γ(1/2 + μ2

)Γ(1/2 + μ2

)[sinh(αc)

]μ3cπ

f(α) = F1(α), say, a < α < b.

(4.12)

Using the formula

∫b

a

du

∫∞

max(α,u)ds =

∫b

α

ds

∫s

a

du +∫∞

b

ds

∫b

a

du, (4.13)

we can write (4.12) in the form

∫b

α

S1(s)ds[cosh(sc) − cosh(αc)

]1/2−μ2 = F1(α) −∫∞

b


]1/2−μ2

×∫b

a

M(u)[sinh(uc)

]1−μ3du[cosh(sc) − cosh(uc)

]1/2−μ2 , a < α < b,

(4.14)


where

S1(s) =∫ s

a

M(u)[sinh(uc)

]1−μ3du[cosh(sc) − cosh(αc)

]1/2−μ2 , a < s < b. (4.15)

Assuming that the right-hand side of (4.14) is known function equation and (4.14) hasthe form of (2.10), hence the solution of (4.14) can be written as

S1(s) = − cπ

cos(πμ2

) dds

∫b

s

F1(α) sinh(αc)dα[cosh(αc) − cosh(sc)

]1/2+μ2 + I1(s), a < s < b, − 12< μ2 <

12,

(4.16)

where

I1(s) =c

πcos

(πμ2

) dds

∫b

s

sinh(αc)dα[cosh(αc) − cosh(sc)

]1/2+μ2

×∫∞

b

dp[cosh(pc) − cosh(αc)

]1/2−μ2∫b

a

M(u)[sinh(cu)

]1−μ3dα[cosh(pc) − cosh(uc)

]1/2−μ3 , a < s < b.

(4.17)

Equation (4.17) is simplified to

I1(s) =c cos

(πμ2

)π

sinh(sc)[cosh(bc) − cosh(sc)

]1/2+μ2∫∞

b

[cosh(cp) − cosh(bc)

]1/2+μ2dp[cosh(sc) − cosh(cp)

]

×∫b

a

M(u)[sinh(cu)

]1−μ3du[cosh(cp) − cosh(cu)

]1/2−μ3 , a < s < b.

(4.18)

Let

R1(p) =∫b

a

M(u)[sinh(cu)

]1−μ3du[cosh(pc) − cosh(uc)

]1/2−μ3 . (4.19)

Equation (4.15) is of the form of (2.9). Hence, its solution is

M(u)[sinh(cu)

]1−μ3 = c cos(πμ3

)π

d

du

∫u

a

S1(s) sinh(sc)ds[cosh(uc) − cosh(sc)

]1/2+μ3 , a < u < b. (4.20)


Substituting the expression for M(u) from (4.20) into (4.19) and integrating by partsand then using the following integral:

∫b

s

c sinh(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,

(4.21)

we find that

R1(p) =−c cos

(μ3π

)π

(cosh(cp) − cosh(bc)

)1/2+μ3

×∫b

a

S1(u) sinh(cu)du[cosh(cp) − cosh(uc)

][cosh(bc) − cosh(cu)

]1/2+μ3 .(4.22)

Making use of (4.18), (4.19), and (4.22), we find that

I1(s) = −∫b

a

S1(u)K2(u, s)du, (4.23)

where

K2(u, s) =c2 cos

(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)

] .(4.24)

Using (4.17) and (4.23), (4.16) can be written in the form

S1(s) +∫b

a

S1(u)K2(u, s)du =−cπ

cos(πμ2

) dds

∫b

s

F1(α) sinh(αc)dα[cosh(αc) − cosh(sc)

]1/2+μ2 , a < s < b.

(4.25)

Equation (4.25) is a Fredholm integral equation of the second kind with kernel definedby (4.24). The Fredholm integral equation (4.25) may be solved to find numerical values ofS1(s) for particular values of f(α). And hence from (4.20) and (4.5), the numerical values forA(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 Legendrefunctions is reduced to the solution of Fredholm integral equations of the second kind whichcan be solved numerically.

References

[1] A. A. Babloian, “Solutions of certain dual integral equations,” Prikladnaya Matematika i Mekhanika, vol.28, no. 6, pp. 1015–1023, 1964, English translation in Journal of Applied Mathematics and Mechanics, vol.28, no. 6, pp. 1227–1236, 1964.

[2] R. S. Pathak, “On a class of dual integral equations,” Koninklijke Nederlandse Akademie vanWetenschappen, vol. 40, no. 4, pp. 491–501, 1978.

[3] B. N. Mandal, “A note on dual integral equations involving associated Legendre function,”International Journal of Mathematics and Mathematical Sciences, vol. 15, no. 3, pp. 601–604, 1992.

[4] B. M. Singh, J. Rokne, and R. S. Dhaliwal, “The study of dual integral equations with generalizedLegendre functions,” Journal of Mathematical Analysis and Applications, vol. 304, no. 2, pp. 725–733,2005.

[5] K. N. Srivastava, “On some triple integral equations involving Legendre functions of imaginaryargument,” Journal of Maulana Azad College of Technology, vol. 1, pp. 54–67, 1968.

[6] J. C. Cooke, “Triple integral equations,” The Quarterly Journal of Mechanics and Applied Mathematics,vol. 16, no. 2, pp. 193–203, 1963.

[7] J. C. Cooke, “Some further triple integral equation solutions,” Proceedings of the EdinburghMathematicalSociety, vol. 13, pp. 303–316, 1963.

[8] J. C. Cooke, “The solution of triple integral equations in operational form,” The Quarterly Journal ofMechanics and Applied Mathematics, vol. 18, no. 1, pp. 57–72, 1965.

[9] J. C. Cooke, “The solution of triple and quadruple integral equations and Fourier-Bessel series,” TheQuarterly Journal of Mechanics and Applied Mathematics, vol. 25, no. 2, pp. 247–263, 1972.

[10] C. J. Tranter, “Some triple integral equations,” Proceedings of the Glasgow Mathematical Association, vol.4, pp. 200–203, 1960.

[11] E. R. Love and D. L. Clements, “A transformation of Cooke’s treatment of some triple integralequations,” Journal of the Australian Mathematical Society. Series B, vol. 19, no. 3, pp. 259–288, 1976.

[12] N. Srivastava, “On triple integral equations involving Bessel function as kernel,” Journal of MaulanaAzad College of Technology, vol. 21, pp. 39–50, 1988.

[13] I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, TheNetherlands, 1966.

[14] A. Erdelyi,W.Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms. Vol. II, McGrawHill, New York, NY, USA, 1954.

[15] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms. Vol. I, McGrawHill, New York, NY, USA, 1954.

[16] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions ofMathematical Physics, Die Grundlehren der Mathematischen Wissenschaften. 52, Springer, New York,NY, USA, 1966.

The Study of Triple Integral Equations with Generalized...

Documents

Transcript of The Study of Triple Integral Equations with Generalized...