Cmjv02i05p0777

J. Comp. & Math. Sci. Vol.2 (5), 777-779 (2011)

Problem Related to Cooling of a Sphere and I-Function S. S. SHRIVASTAVA and HARISH MISHRA Department of Mathematics, Govt. P.G. College, Shahdol, M. P. India ABSTRACT In this paper we present the solution to a classical problem in three dimensions, the cooling of a sphere involving I-function. The assumed symmetries in the problem will permit us to reduce the dimension of the problem to one spatial dimension and time.

Keywords: I-function, H-function, initial temperature, cooling of a sphere.

1. INTRODUCTION

|arg x| < ½ BĎ&#x20AC;, â&#x2C6;&#x20AC; i â&#x2C6;&#x2C6; (1, 2, â&#x20AC;Ś, r).

The I-function of one variable is defined by Saxena1 and we will represent here in the following manner:

Given a sphere whose initial temperature depends only on the distance from the center (e.g., a constant initial temperature) and whose boundary is kept at a constant temperature, predict the temperature at any point inside the sphere at a later time. This is the problem, for example, of determining the temperature of the center of a potato that has been put in a hot oven. The reader might also conjecture that this problem is important for medical examiners who want to determine the time of an individual's death. Early researchers, notably Kelvin, used this problem to determine the age of the earth based on assumptions about its initial temperature and its temperature today.

,

I , x| ,

, :

,

, ,

,

,

, ,

,

θ s x ds

(1)

where Ď&#x2030; = â&#x2C6;&#x161;(â&#x20AC;&#x201C; 1), θ s

â&#x2C6;? â&#x2C6;?

â&#x2C6;&#x2018; â&#x2C6;? â&#x2C6;?

integral is convergent, when (B>0, A â&#x2030;¤ 0), where ŕ­Ž

ŕ­Ż

ŕą&#x; ŕą&#x; B â&#x2C6;&#x2018;ŕ­Źŕ­¨ŕ­&#x20AC;ଵ Îąŕ­¨ â&#x2C6;&#x2018;ŕ­¨ŕ­&#x20AC;୬ାଵ ι୨୧ â&#x2C6;&#x2018;ŕ­Ť ŕ­¨ŕ­&#x20AC;ଵ β୨ â&#x2C6;&#x2018;ŕ­¨ŕ­&#x20AC;୫ାଵ β୨୧

(2)

A â&#x2C6;&#x2018; Îą â&#x2C6;&#x2018; β

(3)

(4)

This cooling problem may remind the reader of Newton's law of cooling, which is encountered in ordinary differential equations texts. Recall that this law states

Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)

S. S. Shrivastava, et al., J. Comp. & Math. Sci. Vol.2 (5), 777-779 (2011)

778

that the rate at which a body cools is proportional to the difference of its temperature and the temperature of the environment. Quantitatively, if T = T(t) is the temperature of a body and Te is the temperature of its environment, then T '(t) = K(Te â&#x2C6;&#x2019; T), where K is the constant of proportionality. But the reader should note that this law applies only in the case that the body has a uniform, homogeneous temperature. In the PDE problem we are considering, the temperature may vary radially throughout the body.

In this paper, we shall make application of following modified form of the integral2:

sin x sin nx dx

! " % "#

!

(5)

where, Re (s) > 0.

2. INTEGRAL The integral, which we need as follows: Ď&#x20AC;

m, l

â&#x2C6;Ť (sin x)Ď&#x2030; â&#x20AC;&#x201C; 1 sin nx I p , q :[zr (sin x)Îť |] dx i i 0 = 21 â&#x2C6;&#x2019; Ď&#x2030; Ď&#x20AC; sin [z 2 |

I

......., (1/2 â&#x2C6;&#x2019; Ď&#x2030;/2 Âą n/2 , Îť/2)

The integral (6) can be obtained easily by making use of (5) and the definition of I-function as given in (1).

3. FORMULATE THE PROBLEM For simplicity, let us consider a sphere of radius Ď = Ď&#x20AC; whose initial temperature is T0 = constant. We will assume that the boundary is held at zero degrees for all time t > 0. If u is the temperature, then in general u will depend on three spatial coordinates and time. But a little reflection shows that the temperature will depend only on the distance from the center of the sphere and on time. Evidently, the temperature u must satisfy the heat equation ut = kâ&#x2C6;&#x2020;u, where k is the diffusivity and â&#x2C6;&#x2020; is the Laplacian. It should be clear that a spherical coordinate system , , is more appropriate than a rectangular system, and u = u(Ď , t). Because u does not depend on the angles Ď&#x2020; and θ, the Laplacian takes on a particularly simple form: â&#x2C6;&#x2020;

m, l + 1 pi + 1, qi + 2: r

(1 â&#x20AC;&#x201C; Ď&#x2030;, Îť), â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś

Proof:

],

(6)

â&#x2C6;&#x201A; 2 â&#x2C6;&#x201A;

â&#x2C6;&#x201A;Ď Ď â&#x2C6;&#x201A;Ď

Therefore, we may formulate the model as

ut = k(uĎ Ď + uĎ ), 0

provided that Îť > 0, Ď&#x2030; > 0, Re[Ď&#x2030; + 2Îť min 1â&#x2030;¤jâ&#x2030;¤m (bj/βj)] > 0, |arg z| < ½ BĎ&#x20AC;;

u(Ď&#x20AC;, t) = 0, t > 0,

where B is given in (2).

u(Ď , 0) = T0, 0

Ď < Ď&#x20AC;, t > 0.

(7) (8)

Ď < Ď&#x20AC;.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)

(9)

S. S. Shrivastava, et al., J. Comp. & Math. Sci. Vol.2 (5), 777-779 (2011)

Observe that there is an implied boundary condition at Ď = 0, namely that the temperature should remain bounded.

To solve problem (7) â&#x2C6;&#x2019; (9) we assume u(Ď , t) = y(Ď )g(t). Substituting into the PDE and separating variables gives, following solution, which is given in3. u(Ď , t) =

! ! â&#x2C6;&#x2018; c e !

(10)

Now combining (13) and (11) and making the use of the integral (6), we derive cn = 22 â&#x2C6;&#x2019; Ď&#x2030; sin

m, l + 1

I pi + 1, qi + 2: r

(1 â&#x20AC;&#x201C; Ď&#x2030;, Îť), â&#x20AC;Śâ&#x20AC;Ś.. [z 2 |

],(14)

â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś., (1/2 â&#x2C6;&#x2019; Ď&#x2030;/2 Âą n/2 , Îť/2)

Putting the value of Cn from (14) in (10), we get the required result (12).

5. SPECIAL CASES

where "

779

# $

$

% % ! ! "

(11)

4. SOLUTION OF THE PROBLEM The solution of the problem to be obtained is u(Ď , t) = â&#x2C6;&#x2018;

Ă&#x2014; 22 â&#x2C6;&#x2019; Ď&#x2030; sin

#

! &'

m, l + 1

On specializing the parameters, Hfunction may be reduced to G-function, Lauricellaâ&#x20AC;&#x2122;s functions Legendre functions, Bessel functions, hypergeometric functions, Appellâ&#x20AC;&#x2122;s functions, Kampe de Ferietâ&#x20AC;&#x2122;s functions and several other higher transcendental functions. Therefore the result (12) is of general nature and may reduced to be in different forms, which will be useful in the literature on applied Mathematics and other branches.

I pi + 1, qi + 2: r

(1 â&#x20AC;&#x201C; Ď&#x2030;, Îť), â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś. [z 2 | â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś, (1/2 â&#x2C6;&#x2019; Ď&#x2030;/2 Âą n/2 , Îť/2) ],

(12)

where all conditions of convergence are same as in (6).

Proof: Choose m, l

f(Ď ) = u(Ď , 0) = T0 = Ď â&#x2C6;&#x2019; 1 (sin Ď )Ď&#x2030; â&#x2C6;&#x2019; 1 I pi, qi: r [z (sin Ď )Îť ] (13)

REFERENCES 1. Saxena, V. P.: Formal Solution of Certain New Pair of Dual Integral Equations Involving H-function, Proc.

Nat. Acad. Sci. India, 52(A), III (1982).

2. Gradshteys, I. S. & Ryzhik, I. M.: Table of Integrals, Series and Products, Academic Press, New York, 372 (1980).

3. J. David Logan: Partial Differential Equations: Springer, (2004).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)