Analysis of a Time-Fractional Substantial Diffusion Equation of Variable Order

Abstract

A time-fractional substantial diffusion equation of variable order is investigated, in which the variable-order fractional substantial derivative accommodates the memory effects and the structure change of the surroundings of the physical processes with respect to time. The existence and uniqueness of the solutions to the proposed model are proved, based on which the weighted high-order regularity of the solutions, in which the weight function characterizes the singularity of the solutions, are analyzed.

Circuit Implementation of Variable-Order Scaling Fractal-Ladder Fractor with High Resolution

“…Variable-order fractional calculus is expressed through mathematical functions that can be used to model natural phenomena and processes. The studies conducted on variable-order fractional calculus were primarily focused on the mathematical description of variable-order phenomena and processes and on the theory of variableorder fractional calculus application system design [4][5][6][8][9][10][11][12]. However, studies on the implementation of flexible and controllable physical entities of variable-order fractional calculus are scarce.…”