A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations

A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations

Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping

In this article, a general type of two-dimensional time-fractional telegraph equation explained by the Caputo derivative sense for (1 < α ≤ 2) is considered and analyzed by a method based on the Galerkin weak form and local radial point interpolant (LRPI) approximation subject to given appropriate initial and Dirichlet boundary conditions. In the proposed method, so-called meshless local radial point interpolation (MLRPI) method, a meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. The point interpolation method is proposed to construct shape functions using the radial basis functions. In the MLRPI method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares. Two numerical examples are presented and satisfactory agreements are achieved.

https://imechanica.org/files/i2015-15033-5_036524400461.pdf

Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation

In this review paper, we are mainly concerned with the finite difference methods, the Galerkin finite element methods, and the spectral methods for fractional partial differential equations (FPDEs)...

Numerical methods for fractional partial differential equations

In this paper, we implement the radial basis functions for solving a classical type of time-fractional telegraph equation defined by Caputo sense for ð1oαr2Þ. The presented method which is coupled of the radial basis functions and finite difference scheme achieves the semi-discrete solution. We investigate the stability, convergence and theoretical analysis of the scheme which verify the validity of the proposed method. Numerical results show the simplicity and accuracy of the presented method. & 2013 Elsevier Ltd. All rights reserved.

Numerical solution of fractional telegraph equation by using radial basis functions

The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular boundary method.

In this paper, with the aim of extending an elegant and straightforward numerical approximation to describe one of the most common physical phenomena has been undertaken. In this regard, the generalization of advection–diffusion equation, namely, the time-fractional advection–diffusion equation with understanding sense variable-order fractional derivative, is taken into consideration. An efficient and accurate approach is relying on the Kansa scheme and finite difference method to provide a mathematical framework to treat the spatial discretization and temporal term, respectively. The meshless collocation approach is utilized for interior scattered points and those on the boundary. Thus, the problem under consideration is reduced to a system of linear algebraic equations. The use of the radial basis function as shape function brings many advantages for proposal numerical method in terms of improved accuracy by setting an appropriate shape parameter and applied for solving high-dimensional models without extra cost. The validity and accuracy of the proposed approach is investigated by four various examples involving three benchmark examples and a practical application of pollution transfer phenomena.

Vahid Reza Hosseini

Papers

Numerical solution of fractional telegraph equation by using radial basis functions

Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping

Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation

The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular boundary method.

The meshless approach for solving 2D variable-order time-fractional advection–diffusion equation arising in anomalous transport