Fractional variable-order model of heat transfer in time-varying fractal media

In recent years technological progress and rapid growth of computational power have made it relatively easy to find solutions for most of the real-world problems connected to partial differential equations. However, even nowadays researchers often struggle to effectively model diffusion processes in heterogeneous, complex media, where the most notable example can be fractal structures. The problem can become even more difficult for time-varying non-homogeneous media. Previous researches have shown that variable-order fractional calculus can be extremely useful in this particular area. Using such mathematical methods allows us to consider investigated medium as the homogeneous non-integer order structure, what can greatly decrease computational time without significant loss of accuracy. In the paper a non-integer variable-order model for heat diffusion in 2-dimensional fractal-like structure is presented. The paper addresses the relations between system order and number of iterations of fractal. Various types of order switching methods are investigated. The model is validated with numerical finite element method simulation. For investigated structure the obtained model is both fast and accurate way to predict heat transfer process results in time-varying environment.