in this article we implement an operational matrix of fractional integration for legendre polynomials. we proposed an algorithm to obtain an approximation solution for fractional differential equations, described in riemann-liouville sense, based on shifted legendre polynomials. this method was applied to solve linear multi-order fractional differential equation with initial conditions, and the...

We develop and analyze multilevel methods for nonuniformly elliptic operators whose ellipticity holds in a weighted Sobolev space with an A2–Muckenhoupt weight. Using the so-called Xu–Zikatanov (XZ) identity, we derive a nearly uniform convergence result, under the assumption that the underlying mesh is quasi-uniform. We also consider the so-called α-harmonic extension to localize fractional po...

Obtaining analytical or numerical solution of fractional differential equations is one of the troublesome and challenging issue among mathematicians and engineers, specifically in recent years. The purpose of this paper Lie Symmetry method is developed to solve second-order fractional differential equations, based on conformable fractional derivative. Some numerical examples are presented to il...

Optimization problems with partial differential equation (PDE) constraints arise in many science and engineering applications. Their robust and efficient solution present many mathematical challenges and requires a tight integration of properties and structures of the underlying problem, of fast numerical PDE solvers, and of numerical nonlinear optimization. This workshop brought together exper...

In this article, approximate solutions of some PDE fractional order are investi?gated with the help a new semi-analytical method called optimal auxiliary function method. The proposed was tested upon time-fractional Fisher equation, Fornberg-Whitham and Inviscid Burger equation. beauty is that there no need for discretization assumptions small or large parameters provides an ap?proximate soluti...

We study a linear fractional Fokker–Planck equation that models non-local diffusion in the presence of a potential field. The non-locality is due to the appearance of the 'fractional Laplacian' in the corresponding PDE, in place of the classical Laplacian which distinguishes the case of regular diffusion. We prove existence of weak solutions by combining a splitting technique together with a Wa...

Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in applications such as fluid flow, finance and others. In this paper we examine some practical numerical methods to solve a class of initial- boundary value fractional partial differential equations with variable coefficients on a finite domain. S...

In this paper, we introduce a family of fractional-order Chebyshev functions based on the classical Chebyshev polynomials. We calculate and derive the operational matrix of derivative of fractional order $gamma$ in the Caputo sense using the fractional-order Chebyshev functions. This matrix yields to low computational cost of numerical solution of fractional order differential equations to the ...

This paper presents a method to extend PDE surfaces to high dimensional spaces. We review a common existing analytic solution, and show how it can be used straightforwardly to increase the dimension of the space the surface is embedded within. We then further develop a numerical scheme suitable for increasing the number of variables that parametrise the surface, and investigate some of the prop...