J. Comp. & Math. Sci. Vol.4 (6), 437-440 (2013)
Temperature in the Prism Involving H-function of Several Variables ARBENDRA KUMAR UPADHYAY1, NEELAM PANDEY1, VIPIN DUBEY2 and BINIT SOHGAURA1 1
Govt. Model Sc. College, Rewa, M. P., INDIA. 2 Govt. T. R. S. College, Rewa, M. P., INDIA (Received on: November 21, 2013) ABSTRACT
The aim of this paper is to obtain the temperature in the prism involving H-function of several variables. Keywords: H-function of several variables, temperature and prism.
1. INTRODUCTION The multivariable H–function given in3 is defined as follows:
(c´j; γ′j)1, p1; …….;(cj(r); γ(r)j)1, pr (d´j; δ′j)1, q1; …….;(dj(r); δ(r)j)1, qr ]
= [(1/2πω)r] ∫ ......∫ φ1(ξ1) …….. L 1
H [z1, …, zr] ≡ H 0, n: m1, n1;….;m r, n r
L r
p, q: p1, q1;….;p r, q
z1
[
.
zr
|
φr(ξr) ψ(ξ1, ... ξr) z1ξ1 …. zrξ r dξ1 .. dξr
(aj; α′j, ….αj(r))1, p:
where ω = √(– 1),
(bj; β′j, ….βj(r))1, q:
ψ(ξ1, ... ξr) =
n Π
j=1
(1)
r Γ ( 1 − aj + Σ αj(i) ξi)
p
i=1
r
,
(2)
r
q
Π Γ (aj − Σ αj(i)ξi) Π Γ (1 − bj + Σ βj(i)ξi) j=n+1
φi (ξi) =
i=1
mi Π Π Γ (dj(i) − δj(i)ξi)
j=1
qi
j=1
i=1
ni Π Γ (1 − cj(i) + γj(i)ξi)
j=1
, pi
Π Γ (1 − dj(i) + δj(i)ξi) Π Γ (cj(i) − γj(i)ξi) j = mi + 1
j = ni + 1
Journal of Computer and Mathematical Sciences Vol. 4, Issue 6, 31 December, 2013 Pages (403-459)
(3)
438
Arbendra Kumar Upadhyay, et al., J. Comp. & Math. Sci. Vol.4 (6), 437-440 (2013)
In (1), i in the superscript (i) stands for the number of primes, e.g. ., b(1) = b´, b(2) = b´´, and so on; and an empty product is interpreted as unity. Suppose, as usual, that the parameters aj, j = 1, ‌., p; cj(i), j = 1,‌.,pi; bj, j = 1, ‌., q; dj(i), j = 1,‌.,qi; ∀i ∈{1,‌..,r} are complex numbers and the associated coeficents Îąj(i), j = 1, ‌., p; Îłj(i), j = 1,‌.,pi; βj(i), j = 1, ‌., q; δj(i), j = 1,‌.,qi; ∀i ∈{1,‌..,r} positive real numbers such that the left of the contour. Also
0,
(4)
â„Ś 0,
where the integral n, p, q, mi, ni, pi and qi are constrained by the inequalities p ≼ n ≼ 0, q ≼ 0, qi ≼ mi ≼ 1 and pi ≼ ni ≼ 1 ∀ i ∈ {1, 2, ‌, r) and the inequalities in (4) hold for suitably restricted values of the complex variables z1, ‌., zr. The sequence of parameters in (1) are such that none of the poles of the integrand coincide, that is, the
(5)
poles of the integrand in (1) are simple. The contour Li in the complex Ξi−plane is of the Mellin-Barnes type which runs from – Ď‰âˆž to +Ď‰âˆž with indentations, if necessary, to ensure that all the poles of Γ (dj(i) − δj(i)Ξi), j = 1,‌, mi are separated from those of Γ (1 − cj(i) + Îłj(i)Ξi), i = 1, ‌., ni.
From Gradshteyn L
L sin ½ nπ Γ(ω)
0
ω–1
âˆŤ (sin Ď€x/L)ω – 1 sin nĎ€x/L dx = 2
Γ{½ (ω ¹ n + 1)}
,
(6)
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 = a, y = 0 and y = b, are kept at temperature zero. Let the initial temperature distribution be f(x, y), and derive this expression for the temperature u(x, y, t) in the prism is given by1, as follows:
u x, y, t ∑ ∑ B exp Ď€ kt sin
sin ,
Journal of Computer and Mathematical Sciences Vol. 4, Issue 6, 31 December, 2013 Pages (403-459)
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Arbendra Kumar Upadhyay, et al., J. Comp. & Math. Sci. Vol.4 (6), 437-440 (2013)
where
B f x, y sin sin dxdy.
439
(8)
3. SOLUTION IN TERMS OF H-FUNCTION: Consider f(x, y) = sin ,
.H ,
sin
sin sin ¾ , ;‌‌‌‌.; , ] , ;‌‌‌‌.; , [ ¾ sin sin
(9)
where H is the H-function of several variables, defined in (1). Combining (9) and (8), making use of the definition of H-function as given in (1), changing the order of integration, after using the integral (6), we arrive at B 2 sin
sin
, , ;‌‌‌‌.; , .H , , ;‌‌‌‌.; ,
2 Âľ [ ] Âľ 2
, , ,¾ ,‌‌.:‌‌‌..,‌‌.:‌‌‌,‌‌.. ¾ , :‌‌‌.,‌‌..:‌‌‌,‌‌..
‌‌.., / / / , /
,
(10)
provided that Ν # 0, ¾ # 0, Re ω ( 0, )* δ ( 0 and |arg " | 0 1" 2, 34 5 1, ‌ , 8 , where 1" is given in (4).
Putting the value of Bmn from (10) in (7), we get following required solution of the problem:
u x, y, t ∑ exp Ď€ kt sin ∑ 2
, , ;‌‌‌‌.; , .H , , ;‌‌‌‌.; ,
sin
2 Âľ [ ] 2 Âľ
Journal of Computer and Mathematical Sciences Vol. 4, Issue 6, 31 December, 2013 Pages (403-459)
440
Arbendra Kumar Upadhyay, et al., J. Comp. & Math. Sci. Vol.4 (6), 437-440 (2013) , , ,¾ ,‌‌.:‌‌‌..,‌‌.:‌‌‌,‌‌.. ¾ , :‌‌‌.,‌‌..:‌‌‌,‌‌..
‌‌.., / / / , /
sin
sin ,
(11)
provided that Ν # 0, ¾ # 0, Re ω ( 0, )* δ ( 0 and |arg " | 0 1" 2, 34 5 1, ‌ , 8 , where 1" is given in (4).
4. SPECIAL CASE: On specializing the parameters in (11), we get the following result in terms of H– function, which is a result given by Shrivastava [4, p. 71 (6)]:
u x, y, t ∑ exp Ď€ kt sin ∑ 2
sin
, , ,Âľ , , & , , % , , / / / , / / / / ,Âľ/
#,$ z2 Âľ | 9 H ,
9 sin
sin ,
(12)
provided that |arg z| < (1/2)Ď&#x20AC;A, Îť # 0, Âľ # 0, Re Ď&#x2030; ( 0, )* δ ( 0, where A is given as: n
p
m
q
ÎŁ Îąj â&#x20AC;&#x201C; ÎŁ Îąj + ÎŁ βj â&#x20AC;&#x201C; ÎŁ βj â&#x2030;Ą A > 0,
j=1
j=n+1
j=1
j=m+1
REFERENCES 1. Churchill, R.V.: Fourier series and Boundary Value Problems, McGrawâ&#x20AC;&#x201C; Hill, New York (1988). 2. Gradshteyn, I. S. and Ryzhik, I. M.: Tables of Integrals, Series and Products, Academic Press, Inc. New York, (1980).
3. Srivastava, H. M., Gupta, K. C. and Goyal, S. P.: The H-function of one and two variables with applications, South Asian Publishers, New Delhi, (1982). 4. Srivastava, S. S.: Studies on Some Contributions of Special Functions in Various Discipline, Minor Research Project Submitted to MPCOST, Bhopal, (2008).
Journal of Computer and Mathematical Sciences Vol. 4, Issue 6, 31 December, 2013 Pages (403-459)