A new approach of modeling crossover behaviors in heat conduction is a piecewise heat equation with variable coefficients and constant fractional order

The equation showing heat conduction within homogeneous media is very versatile and it has reasonable success being applied within a large number of fields; however, there has also been some success with problems when illustrating heat diffusion within different non-homogeneous media. There are differential operators derived from the generalized Mittag-Leffler kernel to describe some properly defined crossover characteristics; operators based on the power law kernel have been used to describe long-range characteristics; stochastic characteristics are incorporated into mathematical equations in terms of random functions, and so on. However, there seems to be no effective model for complicated crossover behaviors due to the limitations of the existing theories. Yet, piecewise calculus has only recently been introduced and applied in some complex real-life problems in this approach. This research will involve numerical analysis of a heat equation with constant and variable coefficients with the use of a piecewise method. All possible scenarios are considered, and the process of their quantitative modeling is demonstrated.