This study presents a generalized multiscale nonlocal elasticity theory that leverages distributed order fractional calculus to accurately capture coexisting and effects within macroscopic continuum. The behavior is captured via constitutive relations derived from thermodynamic formulation. governing equations of the inhomogeneous continuum are obtained Hamilton principle. As generalization con...

The Riccati differential equation with a fractional derivative of variable order is considered. A in the original implies hereditary property medium, i.e., dependence current state dynamic system on its previous states. software called Numerical Solution Fractional-Differential Equation (briefly NSFDRE) created; it allows one to compute numerical solution Cauchy problem for order. algorithm imp...

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Fell...

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

In this paper we propose a finite difference scheme for temporal discretization of the time-fractional thermistor problem, which is obtained from the so-called thermistor problem by replacing the first-order time derivative with a fractional derivative of order α (0 ≤ α ≤ 1). An existence result is established for the semidiscrete problem. Stability and error analysis are then provided, showing...