Fractional Differential Equations Of Variable Order: Existence Results, Numerical Method And Asymptotic Stability Conditions

Abstract

The concept of variable and distributed order fractional derivative firstly appeared in [9] since many physical processes exhibited memory effects that may vary with time or space variables. Some new variable order fractional derivatives and applications were suggested, for example, Hamilton’s principle [2], variable–order mechanics [3], constitutive relation for vis-coelasiticity [12], fractional diffusion equations [8, 13, 14]. Although the variable-order fractional derivative provides more freedom degrees and new ways to understand the complicated dynamics, the main difficulty is to consider the qualitative theories. Hence, it is a challenging work to define a variable-order function not only can be efficient in explanation of physical phenomena but also for convenience of mathematical analysis