A piecewise heat equation with constant and variable order coefficients: A new approach to capture crossover behaviors in heat diffusion

Abstract

While the equation depicting the heat flow within homogeneous has been applied in several fields with some success, it has also faced several difficulties to depict heat diffusion in some non-homogeneous media. For particular behaviours adequate differential operators have been applied, for instance, a long-range behaviour has been depicted using operators based on power law kernel; stochastic behaviours have been included into mathematical equation using random function, some well-defined crossover behaviours have been depicted using the differential operators based on the generalized Mittag-Leffler kernel. Nevertheless, complex crossovers behaviours have not been modelled efficiently due to limitation of existing theories. Nevertheless, very recently piecewise calculus was proposed and applied in some complex world problems with great success. In this paper, heat equation with constant and variable coefficients will be subjected to piecewise numerical analysis. Several cases are considered, and their numerical simulation depicted.

Keywords

• ecewise calculus
• crossover behaviours
• heat flow
• complex media
• nonlocality
• randomness