_{1}

We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face . The partial differential equation and one of the conditions on the free boundary include a time Caputo’s fractional derivative of order . Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in: 1) Roscani-Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802 - 815; 2) Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237 - 249 and 3) Voller, Int. J. Heat Mass Transfer, 74 (2014), 269 - 277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lamé-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74 - 82, which is recovered by taking the limit when the order .

Heat transfer problems with a phase-change such as melting and freezing have been studied in the last century due to their wide scientific and technological applications, see [

A review of a long bibliography on moving and free boundary problems for phase-change materials (PCM) for the heat equation is given in [

We consider a semi-infinite material, with constant thermal coefficients, which is initially solid at its melting temperature T_{m}. At time

We consider that one of the four thermal coefficients is unknown and that it will be determined by a fractional phase-change problem by imposing an over-specified heat flux condition of the type described in [

Fractional differential equations have been developed in the last decades, see for example the books [

In this paper, the differential equation and a governing condition for the free boundary include a fractional time derivative of order

where

We also define two very important functions, which will be useful in the next section:

1) Wright Function [

2) Mainardi Function [

We note that the Mainardi Function is a particular case of the Wright Function.

Some basic properties for the Caputo fractional derivative and for the Wright Function are the following:

where the classical error and the complementary error functions are defined by:

The method for the determination of unknown thermal coefficients through a one-phase fractional Lamé- Clapeyron-Stefan problem with an over-specified boundary condition at the fixed face

where

latent heat of fusion by unit of mass,

the fixed face

The unknown thermal coefficient can be chosen among the four following ones:

The goal of the present work is to obtain in Section II:

1) The solution of the one-phase time fractional Lamé-Clapeyron-Stefan of order

2) The restrictions on the data of the corresponding problem for the four different cases in order to have a unique explicit solution (see

We remark that the results and explicit formulae obtained in [

First, we obtain a preliminary property in order to have a solution to problem (8)-(14).

Lemma 1. The solution of the problem (8)-(14) with

where the dimensionaless coefficient

Proof. Following [

Now, we will study the four following cases:

Case 1: Determination of

Case 2: Determination of

Case 3: Determination of

Case 4: Determination of

whose results are summarized in

Remark 1.

In a analogous manner, we can compute the explicit formulae for the four thermal coefficients of the solid phase of the semi-infinite material by using a solidification process instead of a fusion process.

Theorem 2 (Case 1: Determination of the thermal coefficient c).

If data verify the condition:

then the solution of the Case 1 (problem (8)-(14) with

where the coefficient

Moreover, the temperature

the dimensionless coefficient

Proof. From condition (18) we obtain expression (20), taking into account the definition of the diffusion coefficient and the expression (20) from condition (17), we obtain the Equation (21) for the dimensionless coefficient

have the following properties [

and the real function

is a positive strictly decreasing function because

and

owning to the fact

Then, we get the expression (24) for the diffusion coefficient.

Theorem 3. If the parameter

where the dimensionless coefficient

In particular, the inequality (19) is transformed in the following one:

Proof. It follows from (6) and properties of functions

Theorem 4 (Case 2: Determination of the thermal coefficient l).

If data verify the condition:

then the solution of the Case 2 (problem (8)-(14) with

where the coefficient

and the real function

Moreover, the temperature

the dimensionless coefficient

Proof. From (17), we obtain the Equation (40) for the coefficient

Theorem 5. If the parameter

where the dimensionless coefficient

In particular, the inequality (38) is transformed in the following one:

Proof. It follows from (6) and properties of functions

Theorem 6 (Case 3: Determination of the thermal coefficient k).

For any data, the solution of the Case 3 (problem (8)-(14) with

where the coefficient

Moreover, the temperature

the dimensionless coefficient

Proof. From (18) we have that the coefficient

Therefore, from (17) we obtain the expressions (50) and (54) for the conductivity

Theorem 7. For any data, if the parameter

where the dimensionless coefficient

Proof. It follows from (6) and properties of functions

Theorem 8 (Case 4: Determination of the thermal coefficient r).

For any data, the solution of the Case 4 (problem (8)-(14) with

where the coefficient

the dimensionless coefficient

Proof. It is similar to the proof of the Case 3 (see Theorem 6).

Theorem 9. For any data, if the parameter

where the dimensionless coefficient

Proof. It is similar to the proof of the Case 3 (see Theorem 7).

Now, in order to summarize our results on the determination of one unknown thermal coefficient through a fractional Lamé-Clapeyron-Stefan problem with an over-specified heat flux boundary condition on the fixed face, we show the formula and restrictions for data for the four cases for the fractional Lamé-Clapeyron-Stefan problem with

Case # | Explicit formulae for the unknown thermal coefficient | Equation that must satisfy the parameter | Restrictions on data |
---|---|---|---|

1 | |||

2 | |||

3 | ----------- | ||

4 | ----------- |

Case # | Explicit formulae for the unknown thermal coefficient | Equation that must satisfy the parameter | Restrictions on data |
---|---|---|---|

1 | |||

2 | |||

3 | ----------- | ||

4 | ----------- |

The present work has been sponsored by the Projects PIP N˚ 0534 from CONICET―Univ. Austral, and by AFOSR-SOARD Grant FA9550-14-1-0122.

Domingo AlbertoTarzia, (2015) Determination of One Unknown Thermal Coefficient through the One-Phase Fractional Lamé-Clapeyron-Stefan Problem. Applied Mathematics,06,2182-2191. doi: 10.4236/am.2015.613191