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In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moveover, we also present a fractional method of lines, a matrix transfer technique, and an(More)
In this paper, we consider numerical analysis of the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. The explicit and the implicit finite difference method are presented. Their stability and convergence are discussed via new technique of Fourier analysis. The solvability of the implicit finite difference(More)
A physical-mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order. In this paper, an anomalous sub-diffusion equation (ASub-DE) is considered. A new implicit numerical method (INM) and two solution techniques for improving the order of convergence of the INM for solving the(More)
Various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations (FPDE), for example, systems biology, chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments. However, effective numerical methods and numerical analysis for FPDE are still(More)
In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method. The two methods are unconditionally stable and(More)
In this paper, a fractional partial differential equation (FPDE) describing subdiffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally,(More)