The Cape Verde International Days on Mathematics 2013View this Special Issue
Conference Paper | Open Access
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.
Let be a bounded domain in with a sufficiently smooth boundary and let . In this work, we propose a finite difference scheme for the following nonlocal time-fractional thermistor problem: where denotes the Caputo fractional derivative of order , 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 . 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) 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 is the temperature outside . 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 . For more description about the history of thermistors and more detailed accounts of their advantages and applications in industry, refer to [1–4].
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 [5–8]. In , 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 for solving the time-fractional nonlocal thermistor system. Stability and error analysis for this scheme are presented showing that the temporal accuracy is of order.
2. 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  and given by subject to the initial and homogenous boundary conditions and where 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 in (1). While the case corresponds to the steady state thermistor problem, in the case , 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 .
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 ). In the sequel, we will assume the following assumptions: (H1) is a positive Lipshitzian continuous function;(H2) there exist positive constants and such that for all we have (H3).It can be shown (e.g., see [12, 13]) that the quantity where is given next, defines a norm on which is equivalent to the norm.
3. 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 . . We use the following formulation: for all , where is the truncation error. It can be seen from  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 , and define the discrete fractional differential operator by Then (6) becomes Using this approximation, it yields the following finite difference scheme to (1): for , where are approximations to . Scheme (11) can be reformulated in the form To complete the semidiscrete problem, we consider the boundary conditions and the initial condition , noting that If we set then (12) can be rewritten into for all . When , scheme (12) reads When , scheme (12) becomes We define the error term by Then we get from (7) that
3.1. Existence of the Semidiscrete Scheme
Definition 1. We say that 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
3.2. 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 , we have, for , Notice that . Taking , we have or Then Hence, since the standard -norm and the norm defined by (5) are equivalent, we have Suppose now that we have and prove that . Multiplying (16) by and using the fact that , we obtain Following the same as for the case with respect to the nonlocal term , we then have Hence,
4. Stability and Error Analysis
4.1. Stability Result
The weak formulation of (16) is for all and :We have the following unconditional stability result.
Theorem 4. The semidiscretized problem is stable in the sense that for all it holds
Proof. We prove this result by recurrence. First, when , we have, for , On other terms Taking in (36), we have In a similar way, we have We also have We then obtain by (5) and (36) that Dividing both sides of the previous inequality (40) by , we get Suppose now that we have and prove that . Choosing in (33), we obtain Then using the recurrence hypothesis (42), we obtain since . Similarly to the case , we have 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 . Then one has the following error estimates: (a) where and ; is a constant depending on . (b) when ,
Proof. Let the difference between the exact solution of (1) and the solution of the time-discrete problem. Obviously .
(a) We will prove the result by induction. We begin with the first case when . For , by gathering equations corresponding to exact and discrete solutions, the error equation reads Choosing in the previous equation, it yields that To continue the proof, we will need the following lemma which is used in the sequel.
Lemma 6. Let , be two weak solutions of (1). Assume that (H1)–(H3) hold. Then one has where and , and are positive constants.
Proof. We have
If we multiply by and integrate over , 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 , we obtain Then point (a) is verified for . Suppose now that we have proven (a) for all , and prove it also for . We have Taking in (58) and using Lemma 6, we then have Using the induction assumption and the fact that for a positive integer , we have We then have since . Then By using Young’s inequality, we get Hence, For a suitable choice of and dividing both sides by , we get One can show easily that Hence, we have, for all , such that , (b) We are now interested in the case . We will derive again the following estimation by induction: The previous inequality is obvious for . Suppose now that (68) holds for all , and we need to prove that it holds also for . Similarly to the previous case, by combining the corresponding equations of the exact and discrete solutions and taking as a test function, it yields that Notice that Then, similar to the earlier development, we have It follows, for an well chosen such that , that Then the estimate (b) is proved. This completes the proof of the theorem.
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Copyright © 2013 M. R. Sidi Ammi and A. El Hachimi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.