Temperature in a non-homogeneous bar and the multivariable Aleph-function

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International Journal of Advances in Applied Mathematics and Mechanics

Temperature in a non-homogeneous bar and the multivariable Aleph-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 [2] has used the Fox’s H-function and the multivariable H-function defined by Srivastava et al. [8,9] and has obtained a solution to a problem of heat conduction. The object of this paper is to employ Aleph-function and the multivariable Aleph-function in obtaining a solution of the partial differential equation

∂θ

∂t = b ∂

∂x

·

(1 − x²)∂θ

∂x

¸

related to a problem of heat conduction. The result yields a number of particular cases on specialising the parameters and may prove to be useful in several interesting situations appearing in the literature on physics and mechanics. We shall see the particular cases concerning the Aleph-function of two variables, the I-function of two variables and the multivariable H-function.

MSC: 33C99 • 33C60 • 44A20

Keywords: Multivariable Aleph-function • Aleph-function of two variables • I-function of two variables • Heat conduction • Aleph-function • Multivariable H-function

© 2017 The Author(s). This is an open access article under the CC BY-NC-ND license(https://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction and preliminaries

As an exemple of the application of this multivariable Aleph-function in applied mathematics we shall consider the problem of determining a functionθ(x,t) representing the temperature in a non-homogeneous bar with ends at X = ±1 in which the thermal diffusivity is proportional to (1 − x²) and if the lateral surface on the bar is insulated, it satisfies the partial differential equation of heat conduction, see Churchill [4].

∂θ

∂t = b ∂

∂x

·

(1 − x²)∂θ

∂x

¸

(1) where b is a constant, provided thermal coefficient is constant. The boundary conditions of the problem are that both ends of a bar at u = ±1 are also insulated because the conductivity vanishes there ; and the initial conditions

θ(x,0) = f (x),−1 < x < 1 (2)

Here we shall consider

f (x) = (1 − x)^σℵ^{M ,N}_Pi,Qi,ci;r0

Ã

z(1 − x)^h

¯

¯

¯

¯

¯

¡aj, Aj¢

1,n,£ci¡aj i, Aj i¢¤

n_+1,pi;r⁰

¡b_j, B_j¢

1,m,£c_i¡b_{j i}, B_{j i}¢¤

m+1,q_i ;r⁰

! ℵ^0,_{U :W}^n:V









z₁(1 − x)^k1 ... z_r(1 − x)^kr









(3)

∗ Tel.: 06 831 24 968

E-mail address: [email protected]

The multivariable Aleph-function is an extension of the multivariable I-function recently defined by C.K. Sharma and Ahmad [6], itself is a generalization of the multivariable H-function defined by Srivastava et al. [8]. The multivari- able Aleph-function is defined by means of the multiple contour integral. We have

ℵ(z1, ··· , zr) = ℵ^0,_p^{n:m1,n1,···,mr,nr}

i,qi,τi;R:p_{i (1)},q_{i (1)},τ_{i (1)};R⁽¹⁾;···;p_{i (r )},q_{i (r )};τ_{i (r )};R^{(r )}





 z₁ ... z_r

¯

¯

¯

¯

¯

[(aj;α⁽¹⁾_j , ··· ,α^{(r )}_j )1,n], [τi(aj i;α⁽¹⁾_{j i}, ··· ,α^{(r )}_{j i})n_+1,pi] :

· · · , [τi(bj i;β⁽¹⁾_{j i}, ··· ,β^{(r )}_{j i})_m+1,qi] : [(c⁽¹⁾_j ),γ⁽¹⁾_j )1,n₁], [τi⁽¹⁾(c⁽¹⁾

j i⁽¹⁾},γ⁽¹⁾_{j i}(1))_n

1+1,p⁽¹⁾_i ]; ··· ;[(c^{(r )}_j ),γ^{(r )}_j )1,nr], [τi^{(r )}(c^{(r )}

j i^{(r )},γ^{(r )}_{j i}(r ))_n

r+1,p^{(r )}_i ] [(d⁽¹⁾_j ),δ⁽¹⁾_j )1,m₁], [τi⁽¹⁾(d⁽¹⁾

j i⁽¹⁾,δ⁽¹⁾_{j i}(1))_m

1+1,q⁽¹⁾_i ]; ··· ;[(d^{(r )}_j ),δ^{(r )}_j )1,mr], [τi^{(r )}(d^{(r )}

j i^{(r )},δ^{(r )}_{j i}(r ))_m

r+1,q^{(r )}_i ]

!

= 1

(2πω)^r Z

L1

· · · Z

Lr

ψ(s1, ··· , sr)

r

Y

k=1

θk(s_k)z_k^skd s₁· · · d sr (4)

withω =p

−1.

ψ(s1, ··· , sr) =

n Q

j =1Γ µ

1 − aj+

r

P

k=1α^(k)_j s_k

¶

R

P

i =1

"

τi p_i

Q

j =n+1Γ µ

a_{j i}−

r

P

k=1α^(k)_{j i}s_k

¶ _qi Q

j =1Γ µ

1 − bj i+

r

P

k=1β^(k)_{j i} s_k

¶# (5)

and

θk(sk) =

m_k

Q

j =1Γ³

d^(k)_j − δ^(k)_j s_k´ⁿk

Q

j =1Γ³

1 − c^(k)_j + γ^(k)_j s_k´

R^(k)

P

i^(k)=1

"

τ_i^(k)

q_{i (k)}

Q

j =mk+1Γ³

1 − d^(k)_{j i}(k)+ δ^(k)_{j i}(k)s_k´ ^pi (k)

Q

j =nk+1Γ³ c^(k)

j i^(k)− γ^(k)_{j i}(k)s_k´

# (6)

For more details, see Ayant [1]. The condition for absolute convergence of multiple Mellin-Barnes type contour can be obtained by extension of the corresponding conditions for multivariable H-function given by as,

¯¯ar g z_k¯

¯<1

2A^(k)_i π,where

A^(k)_i = n X

j =1

α^(k)_j − τi pi

X

j =n+1

α^(k)_{j i} − τi qi

X

j =1

β^(k)_{j i} +

n_k

X

j =1

γ^(k)_j − τi^(k) pi (k)

X

j =nk+1

γ^(k)_{j i}(k)

+

m_k

X

j =1

δ^(k)_j − τi^(k) q_{i (k)}

X

j =mk+1

δ^(k)_{j i}(k)> 0, k = 1, · · · , r, i = 1, · · · , R, i^(k)= 1, · · · , R^(k) (7)

The complex numbers z_iare not zero.Throughout this document , we assume the existence and absolute convergence conditions of the multivariable Aleph-function.

We may establish the the asymptotic expansion in the following convenient form:

ℵ (z1, ··· , zr) = 0(|z1|^α1, ··· ,|zr|^αr) , max (|z1,|··· ,|zr|) → 0 ℵ (z1, ··· , zr) = 0¡|z1|^β1, ··· ,|zr|^βr¢ ,max(|z1,|··· ,|zr|) → ∞ where

k = 1,··· ,r : αk= min[Re(d^(k)_j /δ^(k)_j )], j = 1,··· ,mk

and

βk= max[Re((c^(k)_j − 1)/γ^(k)_j )], j = 1,··· ,nk

For convenience, we will use the following notations in this paper.

V = m1, n₁; ··· ;mr, n_r (8)

W = pi⁽¹⁾, q_i(1),τi⁽¹⁾; R⁽¹⁾, ··· , pi^{(r )}, q_i(r ),τi^{(r )}; R^{(r )} (9)

A = {(aj;α⁽¹⁾_j , ··· ,α^{(r )}_j )_1,n}, {τi(a_{j i};α⁽¹⁾_{j i}, ··· ,α^{(r )}_{j i})_n+1,pi}, {(c⁽¹⁾_j ;γ⁽¹⁾_j )_1,n1} τi⁽¹⁾(c⁽¹⁾

j i⁽¹⁾;γ⁽¹⁾_{j i}(1))_n1+1,p

i (1), ··· ,{(c^{(r )}_j ;γ^{(r )}_j )_1,nr},τi^{(r )}(c^{(r )}

j i^{(r )};γ^{(r )}_{j i}(r ))_nr+1,p

i (r ) (10)

B = {τi(b_{j i};β⁽¹⁾_{j i}, ··· ,β^{(r )}_{j i})_m+1,qi}, {(d⁽¹⁾_j ;δ⁽¹⁾_j )_1,m1},τi⁽¹⁾(d⁽¹⁾

j i⁽¹⁾;δ⁽¹⁾_{j i}(1))_m1+1,q

i (1), ··· ,

{(d^{(r )}_j ;δ^{(r )}_j )_1,mr},τi^{(r )}(d^{(r )}

j i^{(r )};δ^{(r )}_{j i}(r ))_mr+1,q

i (r ) (11)

The contracted form concerning the multivariable Aleph-function writes:

ℵ(z1, ··· , zr) =





 z1

... zr

¯

¯

¯

¯

¯

¯

¯ A

... B





 (12)

The Aleph- function , introduced by Südland et al. [10], however the notation and complete definition is presented here in the following manner in terms of the Mellin-Barnes type integral.

ℵ(z) = ℵ^{M ,N}_Pi,Qi,ci;r0

µ z

¯

¯

¯

¯

(a_j, A_j)_1,n, [c_i(a_{j i}, A_{j i})]_n+1,pi;r0

(b_j, B_j)_1,m, [c_i(b_{j i}, B_{j i})]_m+1,qi;r0

¶

= 1

2πω Z

LΩ_P^{M ,N}_i,Qi,ci;r0(s)z^sd s (13) for all z different to 0 and

Ω^{M ,N}_P

i,Q_i,c_i;r⁰(s) =

M

Q

j =1Γ(bj− Bjs)Q^N

j =1Γ(1 − aj+ Ajs)

r⁰

P

i =1

c_i

P_i

Q

j =N +1Γ(aj i− Aj is)

Q_i

Q

j =M+1Γ(1 − bj i+ Bj is)

(14)

With

¯¯arg z¯

¯<1

2πΩ wherΩ =X^M

j =1

βj+

N

X

j =1

αj− ci

à _Q

i

X

j =M+1

βj i+

Pi

X

j =N +1

αj i

!

> 0, i = 1, · · · , r

For convergence conditions and other details of Aleph-function, see Südland et al. [10]. The serie representation of Aleph-function is given by Chaurasia et al. [3].

ℵ^{M ,N}_Pi,Qi,ci;r0(z) =

M

X

G=1

X∞ g =0

(−)^gΩ_P^{M ,N}_i,Qi,ci,r0(s)

B_Gg ! z^s (15)

with s = ηG,g=bG+ g

B_G , Pi< Qi, |z| < 1andΩ^{M ,N}_P

i,Q_i,c_i;r⁰(s) is given by (14).

2. Main integral

The integral to be establish here is

1

Z

−1

(1 − x)^σ(1 + x)^υP_w^(u,v)(x)ℵ^{M ,N}_P

i,Q_i,c_i;r⁰









z₁(1 − x)^k1 ... z_r(1 − x)^kr

¯

¯

¯

¯

¯

¯

¯ A

... B







d x =

M

X

G=1

X∞ g =0

(−)^gΩ^{M ,N}_P

i,Q_i,c_i,r⁰(ηG,g)(−)^w2^{υ+σ+1+hηG,g}Γ(1 + υ + w)

w !B_Gg ! z^ηG,gℵ^0,_piⁿ_+2,q^+2:V_i+2,τi;R:W







 2^k1z₁

... 2^krz_r

¯

¯

¯

¯

¯

¯

¯

¯ (−σ − hηG,g: k₁, ··· ,kr), (µ − σ − hηG,g: k₁, ··· ,kr), A

· · ·

(−1 − υ − σ − w − hηG,g: k₁, ··· ,kr), (µ + w − σ − hηG,g: k₁, ··· ,kr), B



 (16) where

Re(υ) > −1; Re



σ + h min

1≤j ≤mi

b_j B_j +

r

X

i =1

ki min

1≤j ≤mi

d^{(i )}_j δ^{(i )}_j



> −1; h > 0,ki> 0, i = 1, · · · , r

| arg zk| <1

2A^(k)_i π, k = 1,··· ,r where A^(k)_i is given (7).

| arg z| <1

2πΩ where Ω =X^M

j =1

βj+

N

X

j =1

αj− ci

à _Q

i

X

j =M+1

βj i+

Pi

X

j =N +1

αj i

!

> 0

Proof. We first replace the multivariable Aleph-function on the L.H.S of (17) by its Mellin-barnes contour integral (4), the Aleph-function in series using (15), Now we interchange the order of summation and integration (which is permissible under the conditions stated). Evaluate the inner integral with the help of result [[9], page 175). Interpreting the resulting Mellin-Barnes contour integral as an Aleph-function of r variables, we arrive at the desired result.

3. Solution of the problem

The solution of the problem to be obtained is

θ(x,t) = 2^u X^∞

N =0 M

X

G=1

X∞ g =0

ANe−λN (N +1)tP_N^(u,v)(x)z^ηG,gℵ^0,_piⁿ_+2,q^+2:V_i+2,τi;R:W







 2^k1z₁

... 2^krz_r

¯

¯

¯

¯

¯

¯

¯

¯

(−σ − hηG,g: k₁, ··· ,kr), (1 − µ − σ − hηG,g: k₁, ··· ,kr), A

· · ·

(−1 − υ − σ − u − N − hηG,g: k₁, ··· ,kr), (N − σ − hηG,g: k₁, ··· ,kr), B



 (17)

where

A_N=(−)^gΩ^{M ,N}_P

i,Q_i,c_i,r⁰(ηG,g)(−)^N2^u+ηG,gΓ(2N + u + v + 1)Γ(N + u + v + 1) Γ(N + u + 1)BGg !

provided that

Re(υ) > −1; Re



σ + h min

1≤j ≤M

bj

B_j+

r

X

i =1

k_i min

1≤j ≤M

d^{(i )}_j δ^{(i )}_j



> −1; h > 0, k_i> 0, i = 1, · · · , r

| arg zk| <1

2A^(k)_i πwhereA^(k)_i are given in (7)

|ar g z| <1

2πΩ where, Ω =X^M

j =1

βj+

N

X

j =1

αj− ci

à _Q

i

X

j =M+1

βj i+

Pi

X

j =N +1

αj i

!

> 0

Proof. The solution of the problem can be written as Churchill [4].

θ(x,t) = X^∞

N =0

RNe^{−λN (N +1)}P_N^(u,v)(x) (18)

If t = 0 in (18), then by vertu of (3)

f (x) = (1 − x)^σℵ^{M ,N}_P

i,Q_i,c_i;r⁰

Ã

z(1 − x)^h

¯

¯

¯

¯

¯

(aj, Aj)_1,n,£ci(aj i, Aj i)¤ n+1,pi;r⁰

(b_j, B_j)_1,m,£c_i(b_{j i}, B_{j i})¤

m+1,qi;r⁰

! ℵ^0,_{U :W}^n:V









z₁(1 − x)^k1 ...

z_r(1 − x)^kr









= X∞ N =0

R_NP_N^(u,v)(x)with − 1 ≤ x ≤ 1 (19)

The Eq. (3) is valid since f (x) is continuous in the closed interval −16^x61 and has a piece wise continuous derivative there, then u > −1, v > −1, the Jacobi series (19) associated with f (x) converges uniformly to f (x) in −1 + ²6^x6 1 − ²,0 < ² < 1.

Now multiply both sides of (19) by (1 − x)^u(1 + x)^vP_w^(u,v)(x), u > −1, v > −1 and integrating from −1 to 1

(1 − x)^σ+u(1 + x)^υP^(u,v)_w (x)ℵ^{M ,N}_P

i,Q_i,c_i;r⁰

³

z(1 − x)^h´ ℵ^0,_{U :W}^n:V









z₁(1 − x)^k1 ... z_r(1 − x)^kr

¯

¯

¯

¯

¯

¯

¯ A

... B







 d x

= X∞ N =0

R_N

1

Z

−1

(1 − x)^v(1 + x)^vP_N^(u,v)(x)P_w^(u,v)(x)d x (20)

Now using (16) and the orthogonality property of Jacobi polynomials, we have

R_N=(−)^N2^uΓ(2N + u + v + 1)Γ(N + u + v + 1) Γ(N + u + 1)

M

X

G=1

X∞ g =0

(−)^gΩ^{M ,N}_P

i,Qi,ci,r⁰(ηG,g)2^ηG,g w !B_Gg ! z^ηG,g

ℵ^0,_piⁿ_+2,q^+2:V_i+2,τi;R:W







 2^k1z₁

... 2^krz_r

¯

¯

¯

¯

¯

¯

(−σ − hηG,g: k₁, ··· ,kr), (1 − µ − σ − hηG,g: k₁, ··· ,kr), A

· · · ·

(−1 − υ − σ − u − N − hηG,g: k₁, ··· ,kr), (N − σ − hηG,g: k₁, ··· ,kr), B







 (21)

with the help of (18) and (21), the solution is obtained.

4. Particular cases

a) If r = 2, the multivariable Aleph-function reduces to Aleph-function of two variables defined by Sharma [5] and we have

θ(x,t) = 2^u X^∞

N =0 M

X

G=1

X∞ g =0

A_Ne−λN (N +1)tP_N^(u,v)(x)z^ηG,gℵ^0,_piⁿ_+2,q^+2:V_i+2,τi;R:W







 2^k1z₁

... 2^k2z₂

¯

¯

¯

¯

¯

¯

¯

¯

(−σ − hηG,g: k₁, k₂), (1 − µ − σ − hηG,g: k₁, k₂), A

· · ·

(−1 − υ − σ − u − N − hηG,g: k₁, k₂), (N − σ − hηG,g: k₁, k₂), B



 (22)

where

A_N=(−)^gΩ_P^{M ,N}_i,Qi,ci,r0(ηG,g)(−)^N2^u+ηG,gΓ(2N + u + v + 1)Γ(N + u + v + 1)

Γ(N + u + 1)BGg ! (23)

under the same notations and conditions that (17) with r = 2.

b) If r = 2 and τi,τi⁰,τi⁰⁰→ 1 the multivariable Aleph-function reduces to I-function of two variables defined by Sharma et al. [7] and we have

θ(x,t) = 2^u X^∞

N =0 M

X

G=1

X∞ g =0

A_Ne−λN (N +1)tP_N^(u,v)(x)z^ηG,gI^0,_p^n+2:V

i+2,qi+2,;R:W







 2^k1z1

... 2^k2z₂

¯

¯

¯

¯

¯

¯

¯

¯

(−σ − hηG,g: k1, k2), (1 − µ − σ − hηG,g: k1, k2), A

· · ·

(−1 − υ − σ − u − N − hηG,g: k₁, k₂), (N − σ − hηG,g: k₁, k₂), B



 (24)

Where ANis defined by (23), under the same conditions and notations that (17) with r = 2 and τi,τi⁰,τi⁰⁰→ 1.

c) The multivariable Aleph-function reduces to multivariable H-function and we have the following formula,

θ(x,t) = 2^u X^∞

N =0 M

X

G=1

X∞ g =0

A_Ne−λN (N +1)tP_N^(u,v)(x)z^ηG,gH_p+2,q+2:W^0,n+2:V







 2^k1z₁ ... 2^krz_r

¯

¯

¯

¯

¯

¯

¯

¯

(−σ − hηG,g: k₁, ··· ,kr), (1 − µ − σ − hηG,g: k₁, ··· ,kr), A

· · ·

(−1 − υ − σ − u − N − hηG,g: k₁, ··· ,kr), (N − σ − hηG,g: k₁, ··· ,kr), B



 (25)

where

A_N=(−)^gΩ_P^{M ,N}

i,Q_i,c_i,r⁰(ηG,g)(−)^N2^u+ηG,gΓ(2N + u + v + 1)Γ(N + u + v + 1) Γ(N + u + 1)BGg !

under the same conditions and notations that (17).

5. Conclusion

Specializing the parameters of the Aleph-function and the multivariable Aleph-function, we can obtain a large number of results involving various special functions of one and several variables useful in Mathematics analysis, Applied Mathematics, 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] F.Y. Ayant, An integral associated with the Aleph-functions of several variables. International Journal of Mathe- matics Trends and Technology (IJMTT) 31(3) (2016) 142-154.

[2] V.B.L. Chaurasia, Temperature in a non-homogeneous bar and the H-function of the several variables, The Math- ematics Education 21 (2) 33–37.

[3] V.B.L. Chaurasia, Y. Singh, New generalization of integral equations of fredholm type using Aleph function Int. J.

of Modern Math. Sci. 9(3) (2014) 208–220.

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

[5] k. Sharma, On the integral representation and applications of the generalized function of two variables, Interna- tional Journal of Mathematical Engineering and Sciences 3(1) (2014) 1–13.

[6] C.K. Sharma, S.S. Ahmad, On the multivariable I-function. Acta ciencia Indica Math. 20(2) (1994) 113–116.

[7] C.K. Sharma, P.L. Mishra, On the I-function of two variables and its properties. Acta Ciencia Indica Math. 17 (1991) 667–672

[8] 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.

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

[10] N. Südland, B. Baumann,T.F. Nonnenmacher, Open problem: Who knows about the Aleph functions?, Fract. Calc.

Appl. Anal. 1(4) (1998) 401–402.

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