6.5 Temperatures in the prism involving A- function of several variables

The aim of this section is to obtain the temperatures in the prism involving A- function of several variables.

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Introduction

Gautam and Goyal (1981, 82) characterized the multivariable A-function, which is a generalization of multivariable H - function of Shrivastava and Panda (1976).The definition of multivariable A-function runs as follows: A [𝑍1, … . . , 𝑍𝑟] = 𝐴 𝑝,𝑞;𝑝1,𝑞1;…;𝑝𝑟,𝑞𝑟 𝑚,𝑛;𝑚1,𝑛1;…;𝑚𝑟,𝑛𝑟 𝑧1 (𝑎𝑗; 𝐴𝑗 ׳, … , 𝐴𝑗 (𝑟))1,𝑝 ; (𝑐𝑗 ׳; 𝐶𝑗 (𝑟), 𝐶𝑗 ")1,𝑝1; … ; (𝑐𝑗 [ | ] ⋮ 𝑧𝑟 (𝑟))1,𝑝𝑟 (𝑏𝑗 ; 𝐵𝑗 ׳, … , 𝐵𝑗 (𝑟))1,𝑞; (𝑑𝑗 ׳; 𝐷𝑗 (𝑟), 𝐷𝑗 ")1,𝑞1; … ; (𝑑𝑗 (𝑟))1,𝑞𝑟 1 = 𝑠1 … 𝑧𝑟 𝑠𝑟 𝑑𝑠1 … 𝑑𝑠𝑟, (1.1) (2𝜋𝜔)𝑟∫𝐿1 … ∫𝐿𝑟 ∅1(𝑠1) … ∅𝑟(𝑠𝑟)∅(𝑠1, … , 𝑠𝑟)𝑧1 Where ω = √ −1; ∏ ⎾(𝑑𝑗 (𝑖)−𝐷𝑗(𝑖)𝑠𝑖) 𝑚𝑖 𝑗=1 ∏ ⎾(1−𝑐𝑗(𝑖)+𝐶𝑗 (𝑖)𝑠𝑖) 𝑛𝑖 𝑗=1 𝜃𝑖(𝑠𝑖)= ∏ ⎾(1−𝑑𝑗 (𝑖)+𝐷𝑗(𝑖)𝑠𝑖) 𝑞𝑖 𝑗=𝑚𝑖+1 ∏ ⎾(𝑐𝑗(𝑖)−𝐶𝑗 (𝑖)𝑠𝑖) 𝑝𝑖 𝑗=𝑛𝑖+1 ;∀ 𝑖{1, … . , 𝑟} (1.2) Φ (𝑠1, … . , 𝑠𝑟)= ∏ ⎾(1−𝑎𝑗+∑ 𝐴𝑗(𝑖)𝑠𝑖 𝑟 𝑖=1 ) 𝑛 𝑗=1 ∏ ⎾(𝑏𝑗−∑ 𝐵𝑗(𝑖)𝑠𝑖 𝑟 𝑖=1 ) 𝑚 𝑗=1 ∏ ⎾(𝑎𝑗−∑ 𝐴𝑗(𝑖)𝑠𝑖 𝑟 𝑖=1 ) 𝑝 𝑗=𝑛+1 ∏ ⎾(1−𝑏𝑗+∑ 𝐵𝑗(𝑖)𝑠𝑖 𝑟 𝑖=1 ) 𝑞 𝑗=1 . (1.3) (𝑖)s, Here m , n , p , q ,mi , ni , pi , and qi (i=1,….,r) are non-negative integers and all a'js, b'js, 𝑑𝑗 (𝑖)s, 𝐴𝑗 (𝑖),s, 𝐵𝑗 𝑐𝑗 𝜋 (𝑖),s are complex numbers.The multiple integral defining the A-function of r - variables converges absolutely if |arg (𝛾𝑖) 𝑧𝑘| < 2 𝜂𝑖, 𝜉𝑖 ∗= 0 , 𝜂𝑖> 0 (1.4) 𝛾𝑖= ∑ {𝐴𝑗 (𝑖) 𝑝 𝑗=1 }𝐴𝑗(𝑖) ∑ {𝐵𝑗 (𝑖)}−𝐵𝑗(𝑖) 𝑞 𝑗=1 ∑ {𝐷𝑗 (𝑖)}𝐷𝑗(𝑖) 𝑞𝑖 𝑗=1 ∑ {𝐶𝑗 (𝑖)}−𝐶𝑗(𝑖) 𝑝𝑖 𝑗=1 , ∀𝑖∈ (i = 1,…,r}; (1.5) 𝛿𝑖 ∗= 𝐼𝑚[∑ 𝐴𝑗 (𝑖) −∑ 𝐵𝑗 (𝑖) + 𝑞 𝑗=1 ∑ 𝐷𝑗 (𝑖) − 𝑝𝑖 𝑗=1 ∑ 𝐶𝑗 (𝑖) 𝑞𝑖 𝑗=1 𝑝 𝑗=1 ], ∀𝑖∈ (i = 1,…,r}; (1.6) Maths 2016; 1(5): 14-16 Anamika Dubey Department of Mathematical Sciences, A.P.S .University, Rewa, Madhya Pradesh, India Neelam Pandey Department of Mathematics and Computer Sciences, Govt. Model Science College Rewa, Madhya Pradesh, India Anamika Dubey Department of Mathematical Sciences, A.P.S .University, Rewa, Madhya Pradesh, India http://www.mathsjournal.com 𝜂𝑖= 𝑅𝑒[∑ 𝐴𝑗 (𝑖) −∑ 𝐴𝑗 (𝑖) −∑ 𝐷𝑗 (𝑖) − 𝑚 𝑗=1 ∑ 𝐵𝑗 (𝑖) 𝑞 𝑗=𝑚+1 𝑛 𝑗=1 + ∑ 𝐷𝑗 (𝑖) + 𝑝 𝑗=𝑛+1 ∑ 𝐵𝑗 (𝑖) − 𝑛𝑖 𝑗=1 ∑ 𝐶𝑗 (𝑖) + 𝑞𝑖 𝑗=𝑚𝑖+1 ∑ 𝐶𝑗 (𝑖) 𝑝𝑖 𝑗=𝑛𝑖+1 𝑚𝑖 𝑗=1 ] , ∀𝑖∈ (i = 1,…,r}; (1.7) If we take 𝐴𝑗 ′s, 𝐵𝑗 ′s, 𝐶𝑗 ′s and 𝐷𝑗 ′s as real and positive and m = 0, the A-function reduces to multivariable H -function of Shrivastava and Panda (1976),We are utilizing the multivariable A-function characterized in the accompanying brief structure all through the content. 1 𝑃:𝑃𝑟(𝑟) 𝑧1 𝑚,𝑛;𝑀𝑟[ | 𝑠1 … 𝑧𝑟 𝑠𝑟 𝑑𝑠1 … 𝑑𝑠𝑟 (1.8) A[𝑧1, … . . , 𝑧𝑟]=𝐴 𝑝,𝑞;𝑁𝑟 (2𝜋𝜔)𝑟∫𝐿1 … ∫𝐿𝑟 ∅1(𝑠1) … ∅𝑟(𝑠𝑟)∅(𝑠1, … , 𝑠𝑟)𝑧1 𝑄:𝑄𝑟(𝑟)]= ⋮ 𝑧𝑟 𝐿 𝑆𝑖𝑛 1 2 ⁄ 𝑛𝜋 𝛤(𝜔) Where ω = √ −1; 𝐹𝑟𝑜𝑚 Gradshteyn ∫(𝑠𝑖𝑛𝜋𝑥𝐿) ⁄ 𝜔−1 𝐿 𝑂 𝑠𝑖𝑛𝜋𝑥𝐿 ⁄ dx = 2𝜔−1𝛤{1 2 ⁄ (𝜔 ±𝑛+1)} , (1.9) 𝑟𝜋𝑥 𝑠𝜋𝑦 Where n is any integer and 𝜔> 0. 2. Formulation of the problem All four faces of an infinitely long rectangular prism, formed by the planes x = 0, x =𝜌, y = 0 and y =𝜎, are kept at temperature zero. Let the initial temperature distribution be p(x, y), and derive this expression for the temperature c (x, y, t) in the prism is given by [1] as follows: C(x, y, t) = ∑ ∑ 𝐴𝑟𝑠 ∞ 𝑠=1 ∞ 𝑟=1 exp[-𝜋2 kt (𝑠2 𝑝2 +𝑟2 𝜎2 ) ] sin 𝜌 sin 𝜎 , (2.1) 4 𝑟𝜋𝑥 𝑠𝜋𝑦 Where 𝐴𝑟,𝑠 = 𝜌𝜎 ∫∫𝑝(𝑥, 𝑦) 𝜌 0 𝜎 0 sin 𝜌 sin 𝜎 dx dy. (2.2) 3.

Solution in terms of A- function Consider 𝑟𝜋𝑥 𝑠𝜋𝑦 𝑧1 (sin 𝜌 )𝜔1(sin 𝜎)𝜇1 𝑟𝜋𝑥 𝑠𝜋𝑦 P(x, y) = (sin 𝑂,𝑛:(𝑚1,𝑛1);……..;………………..;(𝑚𝑟,𝑛𝑟) [ ] (3.1) 𝜌 )𝛾−1(sin 𝜎)𝛿−1 A𝑝,𝑞:(𝑝1,𝑞1);……………………..;(𝑝𝑟,𝑞𝑟) 𝑟𝜋𝑥 𝑠𝜋𝑦 ⋮ 𝑧𝑟(sin 𝜌 )𝜔𝑟(sin 𝜎)𝜔𝑟 𝑍1.: Where A [ 𝑍𝑟] is the A-function of several variables. Combining (3.1) and (2.1), making use of the definition of A- function changing the order of integration, after using the Mellin bernes integral,we arrive at 𝑧12−𝜔1−𝜇1 𝑠𝜋 𝑟𝜋 (1−𝛾,𝜔1,……………..,𝜔𝑟),(1−𝛿,𝜇1,…………..,𝜇𝑟),…………………,……….. 𝑂,𝑛+2:(𝑚1,𝑛1);……..;………………..;(𝑚𝑟,𝑛𝑟)[[ | 𝐴𝑟,𝑠=24−𝛾−𝛿 sin 2 sin ……………,(1 2 A𝑝+2,𝑞+2:(𝑝1,𝑞1);……………………..;(𝑝𝑟,𝑞𝑟) 2−𝛾 2±𝑠 2,𝜔 2)(1 2−𝛿 2±𝑟 2,𝜇 2):……………………………………………..,], (3.2) ⋮ 𝑧𝑟2−𝜔𝑟−𝜇𝑟 𝑟𝜋𝑥 𝑠𝜋𝑦 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑡ℎ𝑎𝑡 𝜔≥0, 𝜇≥0, 𝑅𝑒(𝛾) > 0, 𝑅𝑒(𝛿)> 0 Putting the value of 𝐴𝑟,𝑠 from (3.2) in (2.1), we get following required solution of the problem: C(x,y,t) = ∑ ∑ 24−𝛾−𝛿 ∞ 𝑠=1 ∞ 𝑟=1 exp[-𝜋2 kt (𝑠2 𝑝2 +𝑟2 𝜎2 ) ] sin 𝜌 sin 𝜎 , 𝑂,𝑛+2:(𝑚1,𝑛1);……..;………………..;(𝑚𝑟,𝑛𝑟) A𝑝+2,𝑞+2:(𝑝1,𝑞1);……………………..;(𝑝𝑟,𝑞𝑟) 𝑧12−𝜔1−𝜇1 𝑟𝜋𝑥 𝑠𝜋𝑦 (1−𝛾,𝜔1,……………..,𝜔𝑟),(1−𝛿,𝜇1,…………..,𝜇𝑟),…………………,……….. [[ | 𝜌 sin 𝜎 , (3.3) ……………,(1 2−𝛾 2±𝑠 2,𝜔 2)(1 2−𝛿 2±𝑟 2,𝜇 2):……………………………………………..,] sin ⋮ 𝑧𝑟2−𝜔𝑟−𝜇𝑟 provided that ω ≥0, 𝜇≥0, 𝑅𝑒(𝛾) > 0, 𝑅𝑒(𝛿)> 0 4. Special Cases On specializing the parameters in (3.3), we get the following result in terms of A – function, which is a result given by shrivastava [69, p. 71(6)]: http://www.mathsjournal.com 𝑟𝜋𝑥 𝑠𝜋𝑦 𝑚,𝑛+2:(𝑚1,𝑛1);……..;………………..;(𝑚𝑟,𝑛𝑟) C(x,y,t) = ∑ ∑ 24−𝛾−𝛿 ∞ 𝑠=1 ∞ 𝑟=1 exp[-𝜋2 kt (𝑠2 𝑝2 +𝑟2 𝜎2 ) ] sin 𝜌 sin 𝜎 , H𝑝+2,𝑞+4:(𝑝1,𝑞1);……………………..;(𝑝𝑟,𝑞𝑟) 𝑧12−𝜔1−𝜇1 𝑟𝜋𝑥 𝑠𝜋𝑦 (1−𝛾,𝜔1,……………..,𝜔𝑟),(1−𝛿,𝜇1,…………..,𝜇𝑟),(𝑎𝑗,𝛼𝑗)1,𝑝…………………,……….. [[ | 𝜌 sin 𝜎, (4.1) ……………,(𝑏𝑗,𝛽𝑗)1,𝑞 ,(1 2−𝛾 2±𝑠 2,𝜔 2)(1 2−𝛿 2±𝑟 2,𝜇 2):……………………………………………..,]sin ⋮ 𝑧𝑟2−𝜔𝑟−𝜇𝑟 1 Provided that |arg z| < ( 2)𝜋𝐴, 𝜔≥0, 𝜇≥𝑂, 𝑅𝑒(𝛾) > 0, 𝑅𝑒(𝛿) > 0, 𝑤ℎ𝑒𝑟𝑒 A is given as: ∑ 𝐴𝑗 (𝑖) −∑ 𝐵𝑗 (𝑖) + 𝑞 𝑗=𝑛+1 ∑ 𝐷𝑗 (𝑖) − 𝑝𝑖 𝑗=1 ∑ 𝐶𝑗 (𝑖) 𝑞𝑖 𝑗=𝑚+1 𝑝 𝑗=1 = A > 0,

Conclusion

Specializing the parameters of 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 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.

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