The operational matrix of fractional integration for shifted Chebyshev polynomials

A new shifted Chebyshev operational matrix (SCOM) of fractional integration of arbitrary order is introduced and applied together with spectral tau method for solving linear fractional differential equations (FDEs). The fractional integration is described in the Riemann–Liouville sense. The numerical approach is based on the shifted Chebyshev tau method. The main characteristic behind the approach using this technique is that only a small number of shifted Chebyshev polynomials is needed to obtain a satisfactory result. Illustrative examples reveal that the present method is very effective and convenient for linear multi-term FDEs.     

A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations

A new explicit formula for the integrals of shifted Chebyshev polynomials of any degree for any fractional-order in terms of shifted Chebyshev polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of linear multi-order fractional differential equations (FDEs) by considering their integrated forms. The shifted Chebyshev spectral tau (SCT) method based on the integrals of shifted Chebyshev polynomials is applied to construct the numerical solution for such problems. The method is then tested on examples. It is shown that the SCT yields better results.     

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

A spectral shifted Legendre Gauss–Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials \({P_{L,n}(x), \ x\in[0,L]}\) is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre–Gauss–Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge–Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.     

The Sinc–Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients

This paper deals with the numerical solution of classes of fractional convection–diffusion equations with variable coefficients. The fractional derivatives are described based on the Caputo sense. Our approach is based on the collocation techniques. The method consists of reducing the problem to the solution of linear algebraic equations by expanding the required approximate solution as the elements of shifted Legendre polynomials in time and the Sinc functions in space with unknown coefficients. The properties of Sinc functions and shifted Legendre polynomials are then utilized to evaluate the unknown coefficients. Several examples are given and the numerical results are shown to demonstrate the efficiency of the newly proposed method.     

Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations

In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of shifted Chebyshev polynomials themselves. We develop also a direct solution technique for solving the linear multi-order fractional differential equations (FDEs) with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials T_L,n(x) with x ∈ (0, L), L > 0 and n is the polynomial degree. We presented a shifted Chebyshev collocation method with shifted Chebyshev–Gauss points used as collocation nodes for solving nonlinear multi-order fractional initial value problems. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.     

The alternative Legendre Tau method for solving nonlinear multi-order fractional differential equations

In this paper, the alternative Legendre polynomials (ALPs) are used to approximate the solution of a class of nonlinear multi-order fractional differential equations (FDEs). First, the operational matrix of fractional integration of an arbitrary order and the product operational matrix are derived for ALPs. These matrices together with the spectral Tau method are then utilized to reduce the solution of the mentioned equations into the one of solving a system of nonlinear algebraic equations with unknown ALP coefficients of the exact solution. The fractional derivatives are considered in the Caputo sense and the fractional integration is described in the Riemann-Liouville sense. Numerical examples illustrate that the present method is very effective for linear and nonlinear multi-order FDEs and high accuracy solutions can be obtained only using a small number of ALPs.     

On shifted Jacobi spectral method for high-order multi-point boundary value problems

This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss–Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved.     

A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.     

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Two‐dimensional shifted Legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices

In this paper, a numerical solution of fractional partial differential equations (FPDEs) for electromagnetic waves in dielectric media will be discussed. For the solution of FPDEs, we developed a numerical collocation method using an algorithm based on two‐dimensional shifted Legendre polynomials approximation, which is proposed for electromagnetic waves in dielectric media. By implementing the partial Riemann–Liouville fractional derivative operators, two‐dimensional shifted Legendre polynomials approximation and its operational matrix along with collocation method are used to convert FPDEs first into weakly singular fractional partial integro‐differential equations and then converted weakly singular fractional partial integro‐differential equations into system of algebraic equation. Some results concerning the convergence analysis and error analysis are obtained. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2017 John Wiley & Sons, Ltd.     

Fractional differential transform method combined with the Adomian polynomials

A modification of the fractional differential transform method (FDTM) for solving nonlinear fractional differential equations (FDEs) is presented. In this technique, the nonlinear term is replaced by its Adomian polynomial of index k. Then the dependent variable components are replaced in the recurrence relation by their corresponding differential transform components of the same index. Thus nonlinear FDEs can be easily solved with less computational work for any analytic nonlinearity due to the properties and available algorithms of the Adomian polynomials. Numerical examples with different types of nonlinearities are solved and good results are obtained.     

A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation

The space-time fractional diffusion-wave equation (FDWE) is a generalization of classical diffusion and wave equations which is used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. This paper reports an accurate spectral tau method for solving the two-sided space and time Caputo FDWE with various types of nonhomogeneous boundary conditions. The proposed method is based on shifted Legendre tau (SLT) procedure in conjunction with the shifted Legendre operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided fractional derivatives. We focus primarily on implementing this algorithm in both temporal and spatial discretizations. In addition, convergence analysis is provided theoretically for the Dirichlet boundary conditions, along with graphical analysis for several special cases using other conditions. These suggest that the Legendre Tau method converges exponentially provided that the data in the given FDWE are smooth. Finally, several numerical examples are given to demonstrate the high accuracy of the proposed method.     

Solving high‐order linear differential equations by a Legendre matrix method based on hybrid Legendre and Taylor polynomials

A numerical method for solving the high‐order linear differential equations with variable coefficients under the mixed conditions is presented. The method is based on the hybrid Legendre and Taylor polynomials. The solution is obtained in terms of Legendre polynomials. Comparison of the present solution is made with the existing solution and excellent agreement is noted. Illustrative examples are included to demonstrate the validity and applicability of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A coupled finite and boundary spectral element method for linear water-wave propagation problems

A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to 1:3. In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by Cerrato et al. [A. Cerrato, J. A. González, L. Rodríguez-Tembleque, Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem. 62 (2016) 22–34.] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld’s radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre–Gauss–Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical p-convergence of spectral methods.     

Numerical solution of nonlinear integral equations using alternative Legendre polynomials

In this paper, the operational matrices of integration and the product for the alternative Legendre polynomials (ALPs) are first derived. Then, using these operational matrices and the collocation method, the nonlinear Volterra–Fredholm–Hammerstein integral equations are reduced to a set of nonlinear algebraic equations with unknown ALP coefficients. Some error estimations are provided and the efficiency and accuracy is verified by applying the method to some examples chosen from other literature.     

Legendre Wavelets Direct Method for the Numerical Solution of Time-Fractional Order Telegraph Equations

In this paper, a Legendre wavelet collocation method for solving a class of time-fractional order telegraph equation defined by Caputo sense is discussed. Fractional integral formula of a single Legendre wavelet in the Riemann–Liouville sense is derived by means of shifted Legendre polynomials. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence analysis and error analysis of the proposed method are investigated. Several examples are presented to show the applicability and accuracy of the proposed method.     

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.     

A New Approach for Solving a Class of Delay Fractional Partial Differential Equations

In this article, a new numerical approach has been proposed for solving a class of delay time-fractional partial differential equations. The approximate solutions of these equations are considered as linear combinations of Müntz–Legendre polynomials with unknown coefficients. Operational matrix of fractional differentiation is provided to accelerate computations of the proposed method. Using Padé approximation and two-sided Laplace transformations, the mentioned delay fractional partial differential equations will be transformed to a sequence of fractional partial differential equations without delay. The localization process is based on the space-time collocation in some appropriate points to reduce the fractional partial differential equations into the associated system of algebraic equations which can be solved by some robust iterative solvers. Some numerical examples are also given to confirm the accuracy of the presented numerical scheme. Our results approved decisive preference of the Müntz–Legendre polynomials with respect to the Legendre polynomials.