New fractional pseudospectral methods with accurate convergence rates for fractional differential equations. ETNA - Electronic Transactions on Numerical Analysis

The main purpose of this paper is to introduce generalized fractional pseudospectral integration and differentiation matrices using a family of fractional interpolants, called fractional Lagrange interpolants. We develop novel approaches to the numerical solution of fractional differential equations with a singular behavior at an end-point. To achieve this goal, we present efficient and stable methods based on three-term recurrence relations, generalized barycentric representations, and Jacobi-Gauss quadrature rules to evaluate the corresponding matrices. In a special case, we prove the equivalence of the proposed fractional pseudospectral methods using a suitable fractional Birkhoff interpolation problem. In fact, the fractional integration matrix yields the stable inverse of the fractional differentiation matrix, and the resulting system is well-conditioned. We develop efficient implementation procedures for providing optimal error estimates with accurate convergence rates for the interpolation operators and the proposed schemes in the $L^{2}$-norm. Some numerical results are given to illustrate the accuracy and performance of the algorithms and the convergence rates

Traveling wave solutions of a biological reaction-convection-diffusion equation model by using (G′/G)(G'/G) expansion method

In this paper, the $(G'/G)$-expansion method is applied to solve a biological reaction-convection-diffusion model arising in mathematical biology. Exact traveling wave solutions are obtained by this method. This scheme can be applied to a wide class of nonlinear partial differential equations

A localized Newton basis functions meshless method for the numerical solution of the 2D nonlinear coupled Burgers' equations

Purpose - The purpose of this paper is to introduce a local Newton basis functions collocation method for solving the 2D nonlinear coupled Burgers' equations. It needs less computer storage and flops than the usual global radial basis functions collocation method and also stabilizes the numerical solutions of the convectiondominated equations by using the Newton basis functions.Design/methodology/approach - A meshless method based on spatial trial space spanned by the local Newton basis functions in the "native" Hilbert space of the reproducing kernel is presented. With the selected local sub-clusters of domain nodes, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. The method leads to a systemof ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs).Findings - The method is successfully used for solving the 2D nonlinear coupled Burgers' equations for reasonably high values of Reynolds number (Re). It is a well-known issue in the analysis of the convectiondiffusion problems that the solution becomes oscillatory when the problem becomes convection-dominated if the standard methods are followed without special treatments. In the proposed method, the authors do not detect any instability near the front, hence no technique is needed. The numerical results show that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re.Originality/value - The authors used more stable basis functions than the standard basis of translated kernels for representing of kernel-based approximants for the numerical solution of partial differential equations (PDEs). The local character of the method, having a well-structured implementation including enforcing the Dirichlet and Neuman boundary conditions, and producing accurate and stable results for flow with reasonably high values of Re for the numerical solution of the 2D nonlinear coupled Burgers' equations without any special technique are the main values of the paper

Radial basis functions method for solving the fractional diffusion equations

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. The paper presents a meshless method based on spatial trial spaces spanned by the radial basis functions (RBFs) for the numerical solution of a class of initial-boundary value fractional diffusion equations with variable coefficients on a finite domain. The space fractional derivatives are defined by using RiemannLiouville fractional derivative. We first provide Riemann-Liouville fractional derivatives for the five kinds of RBFs, including the Powers, Gaussian, Multiquadric, Matern and Thin-plate splines, in one dimension. The time-dependent fractional diffusion equation is discretized in space with the RBF collocation method and the remaining system of ordinary differential equations (ODEs) is advanced in time with an ODE method using a method of lines approach. Some numerical results are given in order to demonstrate the efficiency and accuracy of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method. The stability of the linear systems arising from discretizing Riemann-Liouville fractional differential operator with RBFs is also analysed. (c) 2018 Elsevier Inc. All rights reserved