A fractional B-spline collocation method for the numerical solution of fractional predator-prey models

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost

Numerical Solution of Fractional Partial Differential Equations with Normalized Bernstein Wavelet Method

In this paper, normalized Bernstein wavelets are presented. Next, the fractional order integration and Bernstein wavelets operational matrices of integration are derived and finally are used for solving fractional partial differential equations. The operational matrices merged with the collocation method are used in order to convert fractional problems to a number of algebraic equations. In the suggested method the boundary conditions are automatically taken into consideration. An assessment of the error of function approximation based on the normalized Bernstein wavelet is also presented. Some numerical instances are given to manifest the versatility and applicability of the suggested method. Founded numerical results are correlated with the best reported results in the literature and the analytical solutions in order to prove the accuracy and applicability of the suggested method

DOUBLE EXPONENTIAL EULER–SINC COLLOCATION METHOD FOR A TIME–FRACTIONAL CONVECTION–DIFFUSION EQUATION

In this research, a new version of Sinc-collocation method incorporated with a Double Exponential (DE) transformation is implemented for a class of convectiondiffusion equations that involve time fractional derivative in the Caputo sense. Our approach uses the DE Sinc functions in space and the Euler polynomials in time, respectively. The problem is reduced to the solution of a system of linear algebraic equations. A comparison between the proposed approximated solution and numerical/exact/available solution reveals the reliability and significant advantages of our newly proposed method

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

A Shannon-Runge-Kutta-Gill Method for Convection-Diffusion Equations

A Shannon-Rugge-Kutta-Gill method for solving convection-diffusion equations is discussed. This approach transforms convection-diffusion equations into one-dimensional equations at collocations points, which we solve by Runge-Kutta-Gill method. A concrete example solved is used to examine the method's feasibility

Numerical solution for the time-Fractional Diffusion-wave Equations by using Sinc-Legendre Collocation Method

In this paper the numerical solution of fractional diffusion wave equation is proposed. The fractional derivative will be in the Caputo sense. The proposed method will be based on shifted Legendre collocation scheme and sinc function approximation for time and space respectively. The problem is reduced to the problem into a system of algebraic equations after implementing this method. For demonstrating the validity and applicability of the proposed numerical scheme some examples are presented. Keywords: Fractional diffusion equation, Sinc functions, shifted Legendre  polynomials, Collocation method

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