An approach for solving an inverse spherical two-phase Stefan problem arising in modeling of electric contact phenomena

In this paper, a model problem that can be used for mathematical modeling and investigation of arc phenomena in electrical contacts is considered. An analytical approach for the solution of a two-phase inverse spherical Stefan problem where along with unknown temperature functions heat flux function has to be determined is presented. The suggested solution method is obtained from a new form of integral error function and its properties that are represented in the form of series whose coefficients have to be determined. Using integral error function and collocation method, the solution of a test problem is obtained in exact form and approximately.


INTRODUCTION
Partial differential equations play an important role for the development of models in heat conduction and investigated in various aspects (see, for example, literature [1][2][3] and the references therein). To realize the physical changes, some models need to be expressed as free or moving boundary problems. The theory of free boundaries has seen great progress in the last half century. For the general literature up to 2015, we refer to Friedman 4 and Chen et al. 5 Also, a long list of studies and literature therein are devoted to Stefan-type problems and their analytical and numerical solutions. [6][7][8][9][10][11][12][13] Arcing processes are very rapid and include phase transformations. Thus, it is reasonable to use Stefan-type problems for mathematical modeling of this phenomena. Worth to say that exact solution of the problem allows to elucidate and enhance understanding of arcing processes and contribute to the development of the arc theory. Present study is devoted to theoretical investigation and mathematical modeling of arc phenomena in electrical contacts and appears as a continuation of recent studies where mathematical modeling of short arcing is considered. 14,15 While in 2 studies, 14,15 the analytical solutions of the one-and two-phase (direct) Stefan problems are found, in this paper, we consider an inverse Stefan problem for which along with unknown temperature functions, heat flux function has to be determined. In the considered model, heat flux depends on time variable; however, it is well known that besides time variable, heat flux In this study, we consider spherical model that agrees with Holm's so-called ideal sphere usually applied for electric contacts with small contact surface (radius of b < 10 −4 m) and low electric current. 16

Problem statement
In a spherical model, the contact spot is given by the spherical surface of radius b. The heat flux P(t) entering this surface melts the electric contact material (liquid zone b < r < (t)) and then passes further through the solid zone (t) < r < ∞ (for the illustration of the model, see Figure 1).
The heat equations for each zone are with initial condition subjected to boundary condition at r = b and to free boundary the Stefan's condition as well as the condition at infinity 2 (∞, t) = 0.
Here, 1 and 2 are unknown heat functions, P(t) is an unknown heat flux coming from electric arc, T m is a melting temperature of electric contact material, f(r) is a given function, and a 1 , a 2 , 1 , 2 , L, and are given constants. In the equation, power balance is described by Stefan's condition (9). The function (t) describing the interphase location is given in the inverse problem under consideration.

PROBLEM SOLUTION
Suppose that the initial and free boundary conditions are analytic functions and they can be expanded in Taylor series as We represent the solution of (1) to (10) in the following form where coefficients A n , B n , C n , and D n have to be found. Here, i n erfcx is integral error function determined by the following recurrent formulas:

Lemma 1.
The integral error function holds the following properties: The proof of the lemma can be given by the L'Hopital's rule and properties of i n erfcx function.

Theorem 2. Let f be n times differentiable analytic function. Then
Proof. Using Lemma 1, it is easy to see that

Calculation of coefficients
By the theorem and equations (4) and (11), we can write Using Theorem 2 and letting F(r) = rf(r), we have From (8), when we put r = (t), then b will be canceled and there left only Let us take √ t = , and we obtain from Equation 8 1 where ( ) = ( ) To calculate coefficient C n , we apply Leibniz and Faa Di Bruno formulas and Bell polynomials. Using Leibniz formula, we have Using Faa Di Bruno formula and Bell polynomials for a derivative of a composite function, we have and j 1 , j 2 , · · · satisfy the following equations by taking kth derivatives of both sides of (13) at = 0, we have (14) for k = 0, 1, 2, · · ·.
From expression (14), we express coefficients C n . From (7), we have In the same manner, we get recurrent formula from (15) where we express A n in terms of B n as We can express coefficients A n from this expression. In Stefan's condition (9), we take first derivative of and we take √ t = . So where ( ) = ∑ ∞ n=1 n 2 n n−2 . If we multiply both sides of (17) by a( ) and use (7) and (8), we get the following expression where Taking k-times derivative of both sides of (18) at = 0, we get recurrent formula for B n coefficients. By using all these expression on condition (6), we express coefficient of heat flux. From (6), we get Remark 3. For the convergence of temperature functions Θ 1 , Θ 2 , it is possible to follow the idea proposed in Holm. 15

APPROXIMATE SOLUTION OF A TEST PROBLEM
In this section, the collocation method that is practical for engineers is applied using 3 points t 0 = 0, t 1 = 0.5, and t 2 = 1.
To show the effectiveness of the method, we proposed the following problem. Solution is found both exactly and approximately.
Let us consider We represent solution of the problem in the following form .
It is clear that from (20), we get D n and from (21), we get C n . Also, using (21), we can express A n in terms of B n . Thus, we can find B n by (22). Let (t) = b + √ t and i = 1 r Φ(r, t) where i = 1, 2. Taking derivative, we get Let f(r) = r. Using (21), we have Let us transform (22) to obtain Multiply both sides on , we get and from (21), we get For the first derivative at = 0, we get From (21) and (23), we get We find coefficients A n , B n from (12) and (21) when n ≤ 3.
In Figure 2, the graphs of both reconstructed exact (exact_Flux_P(0,t)) and approximate (approx_Flux_P(0,t)) flux functions are shown.  In Figures 3 and 4, we illustrate the graph of relative error function calculated by following formula that is less than 0.0032% at point x = 0, 0 ≤ t ≤ 0.025 Error rel = |exact flux-approximate flux| · 100 exact flux .

CONCLUSION
A mathematical model describing heat propagation in electric contacts is constructed on the base of two-phase spherical inverse Stefan problem. The heat source P(t) which is occurred by arcing, bridging, etc. can be determined from Equation 19. Temperature functions Θ 1 , Θ 2 that are given in the form of series are determined whose coefficients A n , B n , C n , and D n are also determined from Equations 14, 16, and 17. In the test problem, we used maximum principle for error estimate; the deviation does not exceed 3.2 × 10 −5 for 3 points. For better precision, more points has to be taken and better computer characteristics are required.