Analytical solution of two-phase spherical Stefan problem by heat polynomials and integral error functions

On the base of the Holm model, we represent two phase spherical Stefan problem and its analytical solution, which can serve as a mathematical model for diverse thermo-physical phenomena in electrical contacts. Suggested solution is obtained from integral error function and its properties which are represented in the form of series whose coefficients have to be determined. Convergence of solution series is proved.


INTRODUCTION
In most electric contacts with small contact surface (b < 10 −4 ) and low electric current it is sufficient to use Holm's ideal sphere [1] for investigation of diverse thermo-physical phenomena in electric contacts.Such processes like arcing and bridging are so fleeting (nana second range) that their experimental study is very difficult or sometimes impossible and the need of modeling is due not only to the need to optimize the planning experiment, but also due to the impossibility to use a different approach.So called Stefan type problems which take in account phase transformations, agree with experimental data [2] and can serve as a model for afore mentioned processes [3][4][5].From theoretical point of view, these problems are among the most challenging problems in the theory of non-linear parabolic equations, which along with the desired solution an unknown moving boundary has to be found.In some specific cases it is possible to construct Heat potentials for which, boundary value problems can be reduced to integral equations [3,4,6].However, in the case of domains that degenerate at the initial time, there are additional difficulties because of the singularity of integral equations, which belong to the class of pseudo -Volterra equations which are unsolvable in the general case.First attempts to solve Stefan problem by proposed method are given in [7,8].This study is devoted to suggest analytical solution for two phase spherical Stefan problem with one free boundary which is based on the use of integral error functions and their properties.

TWO PHASE SPHERICAL STEFAN PROBLEM Preliminaries
1. Complementary error function is represented as following: Lemma 1 By L'Hopital's rule it is not difficult to show that

Analytical solution
Let's consider two-phase Stephan Problem, which enables to describe heat transfer in electrical contacts.The heat flux P(t) entering in the sphere of the radius b melts contact material (liquid zone b < r < α(t)) and passes further through the solid zone α(t) < r < ∞.The heat equations for each zone are as follows: and the Stefan's condition is as Here we represent α(t) as follows where α 1 , α 2 , α 3 , ... have to be determined.By making substitution θ = u r + T m and r = x + b in (2)-(13) we reduce problem (2)-(12) to the following problem: 020031-2 where We represent solution in the form Here coefficients A 2n , A 2n+1 , B n ,C n have to be found.Moreover, it is necessary to find unknown moving β (t).Using Hermite polynomials like in [7] we represent (24) in the form of heat polynomials: To find A 2n we use multinomial coefficients of Newton's Polynomials.Thus to derive recurrent formula for A 2n , we take both sides of (19) 2k times derivative at τ = 0 and get following expressions: where l = 1, 2, ... and A 0 = 0.
Thus A 2n , coefficients can be explicitly expressed from (28) and (29) where C i, j [4l] or C i, j [4l − 2] multinomial coefficients or sums of coefficients at β i, j .By Lemma 1 and condition (16) we get following theorem.

FIGURE 1 .
FIGURE 1. Temperature distribution in electric contact.Holm hemisphere