We consider the fractional cable equation. For solution of fractional Cable equation involving Caputo fractional derivative, a new difference scheme is constructed based on Crank Nicholson difference scheme. We prove that the proposed method is unconditionally stable by using spectral stability technique.

In this paper, Haar wavelet operational matrix method is proposed to solve a class of fractional partial differential equations. We derive the Haar wavelet operational matrix of fractional order integration. Meanwhile, the Haar wavelet operational matrix of fractional order differentiation is obtained. The operational matrix of fractional order differentiation is utilized to reduce the initial ...

In this paper, we consider the numerical solutions of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two kinds of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection-dispersion equation (RFADE). RFDE is obtained from the standard diffusion equation by replacing the secondo...

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0 < α < 1. The order of convergence of the numerical method is ...

In this paper, using the Lie group analysis method, we study the invariance properties of the general time fractional fifth-order Korteweg-de Vries (KdV) equation. A systematic research to derive Lie point symmetries of the equation is performed. In the sense of point symmetry, all of the geometric vector fields and the symmetry reductions of the equation are obtained, the exact power series so...

In this paper, we present a numerical solution to an ordinary differential equation of a fractional order in one-dimensional space. The solution to this equation can describe a steady state of the process of anomalous diffusion. The process arises from interactions within complex and non-homogeneous background. We present a numerical method which is based on the finite differences method. We co...

This paper presents the formulation of time-fractional Klein-Gordon equation using the Euler-Lagrange variational technique in the Riesz derivative sense and derives an approximate solitary wave solution. Our results witness that He’s variational iteration method was very efficient and powerful technique in finding the solution of the proposed equation. The basic idea described in this paper is...

An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized twolevel schemes are used. The numeri...

We develop an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs). We first introduce a new family of interpolants, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points. We perform such a construction following a spectral theory recently developed in [M. Zayernouri ...