Stability and Error Analysis of the Semidiscretized Fractional Nonlocal Thermistor Problem

A finite difference scheme is proposed for temporal discretization of the nonlocal time-fractional thermistor problem. Stability and error analysis of the proposed scheme are provided.


Introduction
Let Ω be a bounded domain in R with a sufficiently smooth boundary Ω and let = Ω × (0, ). In this work, we propose a finite difference scheme for the following nonlocal time-fractional thermistor problem: = 0, on = Ω × (0, ) , where ( , )/ denotes the Caputo fractional derivative of order (0 < < 1), Δ is the Laplacian with respect to the spacial variables, is supposed to be a smooth function prescribed next, and is a fixed positive real. Here ] denotes the outward unit normal and / ] = ] ⋅ ∇ is the normal derivative on Ω. Such problems arise in many applications, for instance, in studying the heat transfer in a resistor device whose electrical conductivity is strongly dependent on the temperature . When = 1, (1) describes the diffusion of the temperature with the presence of a nonlocal term. Constant is a dimensionless parameter, which can be identified with the square of the applied potential difference at the ends of the conductor. Function is the positive thermal transfer coefficient. The given value 0 is the temperature outside Ω. For the sake of simplicity, boundary conditions are chosen of homogeneous Neumann type. Mixed or more general boundary conditions which model the coupling of the thermistor to its surroundings appear naturally. ( ) is the temperature inside the conductor, and ( ) is the temperature dependent electrical conductivity. Recall that (1) is obtained from the so-called nonlocal thermistor problem by replacing the first-order time derivative with a fractional derivative of order (0 < < 1). For more description about the history of thermistors and more detailed accounts of their advantages and applications in industry, refer to [1][2][3][4].
In recent years, it has been turned out that fractional differential equations can be used successfully to model many phenomena in various fields as fluids mechanics, viscoelasticity, chemistry, and engineering [5][6][7][8]. In [4], existence and uniqueness of a positive solution to a generalized nonlocal thermistor problem with fractional-order derivatives were proved. In this work, a finite difference method is proposed 2 Conference Papers in Mathematics for solving the time-fractional nonlocal thermistor system. Stability and error analysis for this scheme are presented showing that the temporal accuracy is of 2 − order.

Formulation and Statement of the Problem
We consider the time-fractional thermistor problem (1), which is obtained from by replacing the first-order time derivative with a fractional derivative on Caputo sense as defined in [9] and given by subject to the initial and homogenous boundary conditions and where (0 < < 1) is the order of the time-fractional derivative.
(1) covers (2) and extends it to general cases. The classical nonlocal thermistor problem (2) with the time derivative of integer order can be obtained by taking the limit → 1 in (1). While the case = 0 corresponds to the steady state thermistor problem, in the case 0 < < 1, the Caputo fractional derivative depends on and uses the information of the solutions at all previous time levels (non-Markovian process). In this case, the physical interpretation of fractional derivative is that it represents a degree of memory in the diffusing material [10].
In the analysis of the numerical method, we will assume that problem (1) has a unique and sufficiently smooth solution which can be established by assuming more hypotheses and regularity on the data (see [11]). In the sequel, we will assume the following assumptions: (H1) : R → R is a positive Lipshitzian continuous function; (H2) there exist positive constants and such that for all ∈ R we have It can be shown (e.g., see [12,13]) that the quantity where 0 is given next, defines a norm on 1 (Ω) which is equivalent to the ‖ ⋅ ‖ 1 (Ω) norm.

Time Discretization: A Finite Difference Scheme
We introduce a finite difference approximation to discretize the time-fractional derivative. Let = / be the length of each time step, for some large . = , = 0, 1, . . . , . We use the following formulation: for all 0 ≤ ≤ − 1, where +1 is the truncation error. It can be seen from [14] that the truncation error verifies where is a constant depending only on . On the other hand, by change of variables, we have Let us denote = ( + 1) 1− − 1− , = 0, 1, . . . , and define the discrete fractional differential operator by Then (6) becomes Using this approximation, it yields the following finite difference scheme to (1): for = 1, . . . , − 1, where +1 ( ) are approximations to ( , +1 ). Scheme (11) can be reformulated in the form To complete the semidiscrete problem, we consider the boundary conditions and the initial condition 0 = 0 , noting that > 0, = 0, 1, . . . , If we set then (12) can be rewritten into for all ≥ 1. When = 0, scheme (12) reads When = 1, scheme (12) becomes We define the error term +1 by Then we get from (7) that

Existence of the Semidiscrete Scheme
Definition 1. We say that +1 is a weak solution of (11) if where At each time step, we solve a discretized fractional thermistor problem.

Theorem 2.
Let hypotheses (H1)-(H3) be satisfied; then there exists at least a weak solution of (12), such that Existence and uniqueness results follow from general results of elliptic problems [3,4,13]. From now on, we denote by a generic constant which may not be the same at different occurrences.

A Priori Estimates.
We search a priori estimates for solutions.

Lemma 3. There exists a positive constant independent of , such that
Proof. We prove this result by recurrence. First, when = 0, we have, for V ∈ 1 0 (Ω), 4

Stability Result.
The weak formulation of (16) is for all ≥ 1 and V ∈ 1 (Ω): We have the following unconditional stability result.

Theorem 4. The semidiscretized problem is stable in the sense that for all > 0 it holds
Proof. We prove this result by recurrence. First, when = 0, we have, for V ∈ 1 (Ω), On other terms Taking V = 1 in (36), we have In a similar way, we have We also have Conference Papers in Mathematics 5 We then obtain by (5) and (36) that Dividing both sides of the previous inequality (40) by Suppose now that we have and prove that ‖ +1 ‖ 1 (Ω) ≤ ‖ 0 ‖ 2 + . Choosing V = +1 in (33), we obtain Then using the recurrence hypothesis (42), we obtain +1 2 Then We have the following error analysis for the solution of the semidiscretized problem.
Theorem 5. Let be the exact solution of (1) and let ( ) be the time-discrete solution with the initial condition 0 ( ) = ( , 0). Then one has the following error estimates: where 0 ≤ < 1 and , = /(1 − ); is a constant depending on .
(a) We will prove the result by induction. We begin with the first case when 0 ≤ < 1. For = 1, by gathering equations corresponding to exact and discrete solutions, the error equation reads Choosing V = 1 in the previous equation, it yields that To continue the proof, we will need the following lemma which is used in the sequel.
Proof. We have If we multiply by and integrate over Ω, we get The proof of Lemma 6 is now completed. Now, we continue the proof of Theorem 5. Using (50), it follows that Then, by (5), we have It follows that For a good choice of and using (20) and 0 = 1, we obtain Then point (a) is verified for = 1. Suppose now that we have proven (a) for all = 1, . . . , , and prove it also for = + 1.
We have Conference Papers in Mathematics 7 Taking V = +1 in (58) and using Lemma 6, we then have Using the induction assumption and the fact that −1 / −1 +1 < 1 for a positive integer , we have +1 2 We then have +1 2 By using Young's inequality, we get Hence, For a suitable choice of and dividing both sides by ‖ +1 ‖ 1 (Ω) , we get One can show easily that Hence, we have, for all , such that ≤ , (b) We are now interested in the case → 1. We will derive again the following estimation by induction: The previous inequality is obvious for = 1. Suppose now that (68) holds for all = 1, 2, . . . , , and we need to prove that it holds also for = + 1. Similarly to the previous case, by combining the corresponding equations of the exact and discrete solutions and taking V = +1 as a test function, it yields that Notice that Then, similar to the earlier development, we have (1 − ( + )) +1 2 It follows, for an well chosen such that 1 − ( + ) > 0, that +1 1 (Ω) ≤ ( + 1) 2 .
Then the estimate (b) is proved. This completes the proof of the theorem.