Differential quadrature method for space-fractional diffusion equations on 2D irregular domains

In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by L\'{e}vy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table

Spectral tau-Jacobi algorithm for space fractional advection-dispersion problem

In this paper, we use the shifted Jacobi polynomials to approximate the solution of the space fractional advection-dispersion. The method is based on the Jacobi operational matrices of fractional derivative and integration. A double shifted Jacobi expansion is used as an approximating polynomial. We apply this method to solve linear and nonlinear term FDEs by using initial and boundary conditions

Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain

In this article, we first introduce a singular fractional Sturm-Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results

Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface of fractional calculus and numerical methods. Recent studies suggest that fractional differential and integral operators are well suited to model physical phenomena with intrinsic memory retention and anomalous behaviour. The global property of fractional operators presents difficulties in fnding either closed-form solutions or accurate numerical solutions to fractional differential equations. In rare cases, when analytical solutions are available, they often exist only in terms of complex integrals and special functions, or as infinite series. Similarly, obtaining an accurate numerical solution to arbitrary order differential equation is often computationally demanding. Fractional operators are non-local, and so it is practicable that when approximating fractional operators, non-local methods should be preferred. One such non-local method is the spectral method. In this thesis, we solve problems that arise in the ow of non-Newtonian fluids modelled with fractional differential operators. The recurrent theme in this thesis is the development, testing and presentation of tractable, accurate and computationally efficient numerical schemes for various classes of fractional differential equations. The numerical schemes are built around the pseudo{spectral collocation method and shifted Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral methods converge geometrically, are accurate and computationally efficient. The objective of this thesis is to show, among other results, that these features are true when the method is applied to a variety of fractional differential equations. A survey of the literature shows that many studies in which pseudo-spectral methods are used to numerically approximate the solutions of fractional differential equations often to do this by expanding the solution in terms of certain orthogonal polynomials and then simultaneously solving for the coefficients of expansion. In this study, however, the orthogonality condition of the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature are used to numerically find the coefficients of the series expansions. This approach is then applied to solve various fractional differential equations, which include, but are not limited to time{space fractional differential equations, two{sided fractional differential equations and distributed order differential equations. A theoretical framework is provided for the convergence of the numerical schemes of each of the aforementioned classes of fractional differential equations. The overall results, which include theoretical analysis and numerical simulations, demonstrate that the numerical method performs well in comparison to existing studies and is appropriate for any class of arbitrary order differential equations. The schemes are easy to implement and computationally efficient

Numerical solution of fractional partial differential equations by spectral methods

Fractional partial differential equations (FPDEs) have become essential tool for the modeling of physical models by using spectral methods. In the last few decades, spectral methods have been developed for the solution of time and space dimensional FPDEs. There are different types of spectral methods such as collocation methods, Tau methods and Galerkin methods. This research work focuses on the collocation and Tau methods to propose an efficient operational matrix methods via Genocchi polynomials and Legendre polynomials for the solution of two and three dimensional FPDEs. Moreover, in this study, Genocchi wavelet-like basis method and Genocchi polynomials based Ritz- Galerkin method have been derived to deal with FPDEs and variable- order FPDEs. The reason behind using the Genocchi polynomials is that, it helps to generate functional expansions with less degree and small coefficients values to derive the operational matrix of derivative with less computational complexity as compared to Chebyshev and Legendre Polynomials. The results have been compared with the existing methods such as Chebyshev wavelets method, Legendre wavelets method, Adomian decomposition method, Variational iteration method, Finite difference method and Finite element method. The numerical results have revealed that the proposed methods have provided the better results as compared to existing methods due to minimum computational complexity of derived operational matrices via Genocchi polynomials. Additionally, the significance of the proposed methods has been verified by finding the error bound, which shows that the proposed methods have provided better approximation values for under consideration FPDEs