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. 1. Introduction 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 [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 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