Heat conduction and the multivariable I-function

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International Journal of Advances in Applied Mathematics and Mechanics

Heat conduction and the multivariable I-function

Research Article

F. Y. AYANT∗

Teacher in High School, 411 Avenue Joseph Raynaud, Le parc Fleuri, Bat B, 83140 Six-Fours les plages, Department : VAR, France

Received 09 March 2017; accepted (in revised version) 06 April 2017

Abstract: Chaurasia et al. [3] have studied the Fox’s H-function and the multivariable H-function defined by Srivastava et al.

[9,10] and has obtained a solution to a problem of heat conduction. The object of this paper is to employ the I- function defined by Rathie [7] and the multivariable I-function defined by Prathima et al. [6] in obtaining a solution of the partial differential equation related to a problem of heat conduction. We shall see the particular cases concerning the I-function of two variables defined by Rathie et al. [8] and the multivariable H-function.

MSC: 33C45 • 33C60 • 26D20

Keywords: Multivariable I-function • ¯I-function • Heat conduction • Partial differential equation • I-function of two variables

• Multivariable H-function

1. Introduction and preliminaries

As an example of the application of the multivariable I-function defined by Prathima et al. [6] in applied mathe- matics, we shall consider the problem of obtaining a solution of a problem of heat conduction. Consider the partial differential equation:

∂φ

∂t = ξ∂²φ

∂t²− ξφx² (1)

whereφ(x,t) → 0 for large of t and when x → ∞ ; this equation is related to the problem of heat conduction given by Churchill [4].

∂φ

∂t = ξ∂²φ

∂t²− η¡ φ − φ0¢

(2) provided thatφ0= 0 and η = ξx².

In this paper we shall assume that,

f (x) = x²^σe^−x² µ

y x^2h

¯

(a_j,αi; A⁰_j)_n,n+1, (a_j,αi; A⁰_j)_p (b_j,βj; B⁰_j)_m,m+1, (b_j,βj; B⁰_j)_q

¶ I





 z₁x^2k¹

... z_rx^2k^r







(3)

The ¯I -function, introduced by Rathie [7], however the notation and complete definition is presented here in the fol- lowing manner in terms of the Mellin-Barnes type integral:

I (z) = ¯I¯ p,q^m,n

µ z

¯

(a_j,αi; A⁰_j)_n,n+1, (a_j,αi; A⁰_j)_p (b_j,βj; B⁰_j)_m,m+1, (b_j,βj; B⁰_j)_q

¶

= 1

2πω Z

LΩ^m,np,q(s)z^sd s (4)

∗ Tel.: 06 831 24 968

E-mail address: [email protected]

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for all z different to 0 and

Ω^m,np,q(s) =

m

Q

j =1Γ^B⁰ⁱ(b_j− βjs)Qⁿ

j =1Γ^A⁰^j(1 − aj+ αjs)

p

Q

j =n+1Γ^A⁰ⁱ(aj− αjs)

q

Q

j =m+1Γ^B⁰^j(1 − bj+ βjs)

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The integral (3) converges when |ar g z| <1

2∆ ,if ∆ > 0 where

∆ =X^m

j =1

B⁰_jβj−

q

X

j =m+1

B⁰_jβj+

n

X

j =1

A⁰_jαj−

p

X

j =n+1

A⁰_jαj (6)

When the poles ofΓ(bj− βjs), j = 1,··· ,m, are simples the integral (8) can be evaluate with the help of the Residue Theorem. We obtain

I (z) =¯

m

X

G=1

X∞ g =0

(−)^gΩ^m,np,q(s)

B⁰_Gg ! z^s (7)

with s = ηG,g=bG+ g

B_G⁰ , p < q, |z| < 1 and Ω^m,np,q(s) is given in (8). For more detail, see Rathie [7].

The multivariable I-function is defined in term of multiple Mellin-Barnes type integral:

I (z₁, ··· , zr) = Ip,q:p^0,n:m1¹,q^,n1¹;···;p^;···;mr,q^r^,nr^r





 z₁ ... z_r

¯

(a_j;α⁽¹⁾_j , ··· ,α^{(r )}_j ; A_j)_1,p: (c⁽¹⁾_j ,γ⁽¹⁾_j ;C⁽¹⁾_j )_1,p₁; ··· ;(c^{(r )}_j ,γ^{(r )}_j ;C^{(r )}_j )_1,p_r (b_j;β⁽¹⁾_j , ··· ,β^{(r )}_j ; B_j)_1,q: (d⁽¹⁾_j ,δ⁽¹⁾_j ; D⁽¹⁾_j )_1,q₁; ··· ;(d^{(r )}_j ,δ^{(r )}_j ; D^{(r )}_j )_1,q_r





 (8)

= 1

(2πω)^r Z

L1

· · · Z

Lr

φ(s1, ··· , sr)

r

Y

i =1

θi(s_i)z_i^sⁱd s₁· · ·d sr (9)

where f (s₁, L, s_r),θi(s_i), i = 1,...,r are given by,

φ(s1, ··· , sr) =

n

Q

j =1Γ^A^j µ

1 − a j +P^r

i =1α^{(i )}_j s_j

¶

p

Q

j =n+1Γ^A^j µ

aj−

r

P

i =1α^{(i )}_j sj

¶ _q Q

j =1Γ^B^j µ

1 − b j +

r

P

i =1β^{(i )}_j sj

¶ (10)

θi(s_i) =

n_i

Q

j =1Γ^C^{(i )}^j ³

1 − c^{(i )}_j + γ^{(i )}_j s_i´m_i

Q

j =1Γ^D^{(i )}^j ³

d^{(i )}_j − δ^{(i )}_j s_i´

p_i

Q

j =ni+1Γ^C^{(i )}^j ³

c^{(i )}_j − γ^{(i )}_j s_i´ ^qi

Q

j =mi+1Γ^D^{(i )}^j ³

1 − d^{(i )}_j + δ^{(i )}_j s_i´

(11)

For more details, see Prathima et al. [6].

Following the result of Braaksma [2] the I-function of r variables is analytic if,

U_i=

p

X

j =1

A_jα^{(i )}_j −

q

X

j =1

B_jβ^{(i )}_j +

pi

X

j =1

C^{(i )}_j γ^{(i )}_j −

qi

X

j =1

D^{(i )}_j δ^{(i )}_j ≤ 0, i = 1,··· ,r (12)

The integral (18) converges absolutely if

¯¯arg(z_k)¯

¯<1

2∆kπ,k = 1,··· ,r where

∆k= −

p

X

j =n+1

A_jα^(k)_j −

q

X

j =1

B_jβ^(k)_j +

m_k

X

j =1

D^(k)_j δ^(k)_j −

q_k

X

j =mk+1

D^(k)_j δ^(k)_j +

n_k

X

j =1

C^(k)_j γ^(k)_j −

p_k

X

j =nk+1

C^(k)_j γ^(k)_j > 0 (13)

The parameters mj, nj, pj, qj( j = 1,··· ,r ),n, p, q are non negative integers (for more details, see Prathima et al. [7]) α^{(i )}_j ( j = 1,··· , p;i = 1,··· ,r ),β^{(i )}_j ( j = 1,··· , q;i = 1,··· ,r ),γ^{(i )}_j ( j = 1,··· , pi; i = 1,··· ,r ) and δ^{(i )}_j , ( j = 1,··· , qi; i = 1,··· ,r )

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are assumed to be positive quantities for standardization purpose. a_j( j = 1,··· , p),bj( j = 1,··· , q),c^{(i )}_j ( j = 1,··· , pi, i = 1, ··· ,r ),d^{(i )}_j ( j = 1,··· , qi, i = 1,··· ,r ) are complex numbers.

The exposants A_j( j = 1,··· , p),Bj( j = 1,··· , q),C^{(i )}_j ( j = 1,··· , pi; i = 1,··· ,r ),D^{(i )}_j ( j = 1,··· , qi; i = 1,··· ,r ) of various gamma function involved in (10) and (11) may take non integer values.

The contour L_i in the complex s_i -plane is of Mellin Barnes type which runs from c − i ∞ to c + i ∞ (c real) with indentation, if necessary, in such a manner that all singularities ofΓ^D^{(i )}^j ¡d^{(i )}_j − δ^{(i )}_j s_i¢, j = 1,··· ,mi to the right and Γ^C^{(i )}^j ¡1 − c^{(i )}_j − γ^{(i )}_j s_i¢, j = 1,··· ,niare to the left of L_i.

We shall note,

X = m1, n₁; ··· ;mr, n_r; Y = p1, q₁; ··· ; pr, q_r (14)

A = (aj;α⁽¹⁾_j , ··· ,α^{(r )}_j ; A_j)_1,p; (c⁽¹⁾_j ,γ⁽¹⁾_j ;C⁽¹⁾_j )_1,p₁; ··· ;(c^{(r )}_j ,γ^{(r )}_j ;C^{(r )}_j )_1,p_r (15)

B = (bj;β⁽¹⁾_j , ··· ,β^{(r )}_j ; B_j)_1,q: (d⁽¹⁾_j ,δ⁽¹⁾_j ; D⁽¹⁾_j )_1,q₁; ··· ;(d^{(r )}_j ,δ^{(r )}_j ; D^{(r )}_j )_1,q_r (16) The multivariable I-function write:

I (z₁, ··· , zr) =





 z₁ ... xr

¯

¯ A B





 (17)

2. Main integral

Lemma 2.1.

Z∞

−∞

x^2σe^−x²H₂_υ(x)d x =pπ4^υ−σΓ(2σ + 1)

Γ(σ − υ + 1) (18)

We have the following integral.

Theorem 2.1.

Z∞

−∞

x²^σe^x²H_2υ(x) ¯I_p,q^m,n µ

y x^2h

¯

(aj,αi; A⁰j)_n,n+1, (aj,αi; A⁰j)p

(bj,βj; B⁰j)_m,m+1, (bj,βj; B⁰j)q

¶





 z1x^2k¹

... z_rx^2k^r





 d x

=p

π4^υ−σX^m

G=1

X∞ g =0

4^h^η^G,g(−)^gΩ^m,np,q(s)

B_Gg ! y^η^G,gI_p+1,q+1;Y^0,n+1;X





 z₁4^−k¹ ... z_r4^−k^r

¯

(−2σ − 2hG,g; 2k1, ··· ,2kr; 1), A

· · ·

(−υ − σ − hηG,g; k1, ··· ,kr; 1), B





 (19)

Ω^m,np,q(s) is defined by (4), provided that

min{h, h,σ,ki} > 0,i = 1,··· ,r ;Re



1 + hb_j βj +

r

X

i =1

k_i min

1≤j ≤mi

d^{(i )}_j δ^{(i )}_j



> 0

¯¯ar g (z_k)¯

¯<1

2∆kπ,k = 1,··· ,r ∆kis defined by (12).

|ar g z| <1

2∆,if∆ > 0 where ∆ =X^m

j =1

B⁰_jβj−

q

X

j =m+1

B⁰_jβj+

n

X

j =1

A⁰_jαj−

p

X

j =n+1

A⁰_jαj

Proof. To prove (18), first expressing the I-function in serie with the help of (7), and we interchange the order of summations and x-integral (which is permissible under the conditions stated). Expressing the I-function of r variables defined by Prathima et al. [6] in Mellin-contour integral and interchange the order of integrations which is justifiable due to absolute convergence of the integral involved in the process. Now evaluating the inner x-integral with the help of Lemma. Interpreting the Mellin-Barnes contour integral in multivariable I-function, we obtain the desired result (19).

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3. Solution

The solution of (1) to be establish is, φ(x,t) =X^∞

α=0 m

X

G=1

X∞ g =0

2^{α−2σ−2hη}^G,g⁻¹²(−)^gΩ^m,np,q(s)

α!BGg ! y^η^G,ge^{(1+2α)ξt−}^x2² I_p+1,q+1;Y^0,n+1;X





 z₁4^−k¹ ... z_r4^−k^r

¯

¡−2σ − 2hG,g; 2k₁, ··· ,2kr; 1¢ , A

· · ·

¡_α

2$ − σ − hηG,g; k1, ··· ,kr; 1¢ ,B





 (20)

Provided that,

min{h, h,σ,ki} > 0,i = 1,··· ,r ; Re



1 + hbj

βj +

r

X

i =1

k_i min

1≤j ≤mi

d^{(i )}_j δ^{(i )}_j



> 0

¯¯arg(z_k)¯

¯<1

2∆kπ,k = 1,··· ,r, ∆k} is defined by (12).

| arg z| <1

2∆,if∆ > 0where ∆ =X^m

j =1

B⁰_jβj−

q

X

j =m+1

B⁰_jβj+

n

X

j =1

A⁰_jαj−

p

X

j =n+1

A⁰_jαj

Proof. The solution of (1) can be written as [[1], page 360, Eq. 2.3]

φ(x,t) =X^∞

α=0

A_αe^{(1+2α)ξt−}^x2² H_α(x) (21)

where H_α(x) is the Hermite polynomial. If t = 0, then by vertue of (3), we have

x²^σe^x²H_2υ(x) ¯I_p,q^m,n µ

y x^2h

¯

(aj,αi; A⁰j)_n,n+1, (aj,αi; A⁰j)p

(bj,βj; B⁰j)_m,m+1, (bj,βj; B⁰j)q

¶ I





 z1x^2k¹ ... z_rx^2k^r





= X∞ α=0

A_αe⁻^x2² H_α(x) (22)

Multiplying both sides of (22) by H_β(x) and integrating from −∞ to ∞ with respect to x, and using (19) and the or- thogonality property of Hermite polynomials [5], we find.

A_β=

m

X

G=1

X∞ g =0

2^{α−2σ−2hη}^G,g⁻¹²(−)^gΩ^m,np,q(s)





 z₁4^−k¹

... z_r4^−k^r

¯

(−2σ − 2hηG,g; 2k1, ··· ,2kr; 1), A

· · ·

α2− σ − hηG,g; k₁, ··· ,kr; 1), B





 (23) With the help of (21) and (23), the solution (20) is established.

4. Particular case

a) If r = 0 the multivariable I-function reduces to I-function of two variables defined by Rathie et al. [8].

φ(x,t) =X^∞

α=0 m

X

G=1

X∞ g =0

2^{α−2σ−2hη}^G,g⁻¹²(−)^gΩ^m,np,q(s)





 z₁4^−k¹

... z₂4^−k²

¯

(−2σ − 2hηG,g; 2k1, 2k2; 1), A

· · ·

α2− σ − hηG,g; k₁, k₂; 1), B







H_α(x) (24)

under the same conditions and notations that (20) with r = 2.

b) The multivariable I-function reduces to multivariable H-function defined by Srivastava et al. [9,10] and we obtain.

φ(x,t) =X^∞

α=0 m

X

G=1

X∞ g =0

2^{α−2σ−2hη}^G,g⁻¹²(−)^gΩ^m,np,q(s)

α!BGg ! y^η^G,ge^{(1+2α)ξt−}^x2² H_p+1,q+1;Y^0,n+1;X





 z₁4^−k¹

... z_r4^−k^r

¯

(−2σ − 2hηG,g; 2k1, ··· ,2kr), A

· · ·

α2− σ − hηG,g; k₁, ··· ,kr), B







H_α(x) (25)

under the same conditions and notations that (20) with Aj= Bj= C^{(i )}_j = D^{(i )}_j = 1.

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5. Conclusion

Specializing the parameters of the I-function and the multivariable I-function, we can obtain a large number of results involving various special functions of one and several variables useful in Mathematics analysis, Applied Math- ematics, Physics and Mechanics. The result derived in this paper is of general character and may prove to be useful in several interesting situations appearing in the literature of sciences.

References

[1] B.R. Bhonsle, Heat conduction and Hermite polynomials. Proc. Acad. Sci. India, Sect A 36 (1966) 359–360.

[2] B.L.J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes integrals, Compositio Mathematical 15 (1964) 239–341.

[3] V.B.L. Chaurasia, Heat conduction and the H-function of several complex variables, Jnanabha 13 (1983) 39–46.

[4] R.V. Churchill, Fourier series and boundary value problems, McGraw-Hill Book Co., New York, 1942.

[5] A. Erdelyi et al., Tables of integral transforms, II, McGraw-Hill, New York, 1954.

[6] J. Prathima, V. Nambisan, S.K. Kurumujji, A Study of I-function of Several Complex Variables, International Jour- nal of Engineering Mathematics 2014 (2014) 1–12.

[7] A.K. Rathie, A new generalization of generalized hypergeometric function, Le Matematiche 52(2) 297–310.

[8] A.K. Rathie, K.S. Kumari, T.M. Vasudevan Nambisan, A study of I-functions of two variables, Le Matematiche 64(1) 285–305.

[9] H.M. Srivastava, R. Panda, Some expansion theorems and generating relations for the H-function of several com- plex variables, Comment. Math. Univ. St. Paul. 24 (1975) 119–137.

[10] H.M. Srivastava, R. Panda, Some bilateral generating function for a class of generalized hypergeometric polynomials, J. Reine Angew Math 283/284, (1976) 265–274.

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