Singular behavior near the initial time, induced by weakly singular kernels in Volterra integro-differential equations, often degrades the accuracy of classical spectral collocation methods. To mitigate this effect, considerable attention has been devoted to the development of piecewise collocation methods employing fractional-order polynomials as basis functions, which effectively capture weakly singular or irregular solution behavior. In this paper, we consider the double weakly singular Volterra integro-differential equations and apply a piecewise collocation method based on a fractional-order polynomial basis. In addition, we establish a rigorous regularity analysis of the solution to this problem. We employ a graded mesh to overcome the low convergence orders usually given in collocation methods based on uniform meshes. Based on the presented analysis, we derive error estimates for the proposed method and investigate the selection of the fractional-order and the mesh grading parameter to achieve an optimal convergence rate of O(M−N), where M denotes the number of subintervals and N denotes the number of collocation points in each subinterval. Numerical results demonstrate the validity and accuracy of the theoretical results and confirm the effectiveness of the numerical approach.

The convergence analysis of the spectral methods of the fractional differential equations is generally carried out on the assumption that the underlying solution is sufficiently smooth. Due to the limited smoothing property of the solution operator, these methods fail to achieve spectral accuracy. This work aims to study a spectral collocation approach for the fractional nonlinear pantograph delay differential equations with nonsmooth solutions. The fractional-order derivative is considered in the Caputo sense. An auxiliary transformation is adapted to match the singularity in the corresponding solution and to maximize the convergence rate of the proposed scheme. Therefore, the solution of the resulting equation will possess better regularity and then the numerical method can achieve the spectral accuracy, which serves as an improvement compared with the existing results in the literature. The spectral convergence rate for the proposed approach is discussed in the weighted L2-norm and the L∞-norm. Finally, numerical results are given to confirm our theoretical analysis.

Fractional delay differential equations (FDDEs) and time-fractional delay partial differential equations (TFDPDEs) are the focus of the present research. The FDDEs is converted into a system of algebraic equations utilizing a novel numerical approach based on the spectral Galerkin (SG) technique. The suggested numerical technique is likewise utilized for TFDPDEs. In terms of shifted Jacobi polynomials, suitable trial functions are developed to fulfill the initial-boundary conditions of the main problems. According to the authors, this is the first time utilizing the SG technique to solve TFDPDEs. The approximate solution of five numerical examples is provided and compared with those of other approaches and with the analytic solutions to test the superiority of the proposed method.

The current manuscript introduces a novel numerical treatment for multi-term fractional differential equations with variable coefficients. The spectral Galerkin approach is developed with the operational matrix of fractional derivatives to get a system of algebraic equations that can be solved using a suitable technique. The operational matrix is introduced based on a combination of the shifted Jacobi polynomials. The proposed approach is applied also for 1+1 and 2+1 multi-term time-fractional diffusion and diffusion-wave equations. In addition, the convergence analysis for the presented approaches is investigated. Some specific numerical examples are given to ascertain the wide applicability and the good efficiency of the suggested algorithms, and to confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left- and right-sided fractional derivatives. The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations. Then, a Legendre-based spectral collocation method is developed for solving the transformed system. Therefore, we can make good use of the advantages of the Gauss quadrature rule. We present the construction and analysis of the collocation method. These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler-Lagrange equations. Two numerical examples are given to confirm the convergence analysis and robustness of the scheme.

This paper presents a spectral collocation technique to solve fractional stochastic Volterra integro-differential equations (FSV-IDEs). The algorithm is based on shifted fractional order Legendre orthogonal functions generated by Legendre polynomials. The shifted fractional order Legendre–Gauss–Radau collocation (SFL-GR-C) method is developed for approximating the FSV-IDEs, with the objective of obtaining a system of algebraic equations. For computational purposes, the Brownian motion function W(x) is discretized by Lagrange interpolation, while the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Numerical examples demonstrate the accuracy and applicability of the proposed technique, even when dealing with non-smooth solutions.

In this paper, we consider an important kind of fractional partial differential equations, namely multi-term time-fractional mixed sub-diffusion and diffusion-wave equation. The crucial importance of the considered equation is due to the fact that it generalizes some substantial types of fractional differential equations that can be widely used in describing many real-life phenomena, some of these equations are the time-fractional sub-diffusion, time-fractional diffusion-wave and time-fractional diffusion equations. In this study, the 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation is trans formed to its integrated form with respect to time. An extension of the operational matrix of second-order derivative to the 2D case is used in combination with the operational matrix of fractional-order integrals and the time-space spectral collocation method to reduce such equations to systems of algebraic equations, which are solved using any suitable solver. As far as the authors know, this is the first attempt to deal with 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation via a spectral approach. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. Our results confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account. (C) 2020 Elsevier B.V. All rights reserved.

In this paper, we study and present a spectral numerical technique for solving a general class of multi-order fractional pantograph equations with varying coefficients and systems of pantograph equations. In this study, the spectral Galerkin approach in combination with the properties of shifted Legendre polynomials is used to reduce such equations to systems of algebraic equations, which are solved using any suitable solver. As far as the authors know, this is the first attempt to deal with fractional pantograph equations via spectral Galerkin approach. The errors and convergence of the adopted approach are rigorously analyzed. The efficiency and accuracy of the technique are tested by considering five different examples, to ensure that the suggested approach is more accurate than the existing other techniques. The obtained results in this paper are comparing favorably with those published by other researchers and with the existing exact solutions, whenever possible.

In this paper, we analyze and implement a new efficient spectral Galerkin algorithm for handling linear one-dimensional telegraph type equation. The principle idea behind this algorithm is to choose appropriate basis functions satisfying the underlying boundary conditions. This choice leads to systems with specially structured matrices which can be efficiently inverted. The proposed numerical algorithm is supported by a careful investigation for the convergence and error analysis of the suggested approximate double expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithm.

A new shifted Jacobi-Gauss-collocation (SJ-G-C) algorithm is presented for solving numerically several classes of fractional integro-differential equations (FI-DEs), namely Volterra, Fredholm and systems of Volterra FI-DEs, subject to initial and nonlocal boundary conditions. The new SJ-G-C method is also extended for calculating the solution of mixed Volterra-Fredholm FI-DEs. The shifted Jacobi-Gauss points are adopted for collocation nodes and the FI-DEs are reduced to systems of algebraic equations. Error analysis is performed and several numerical examples are given for illustrating the advantages of the new algorithm. (C) 2019 Elsevier B.V. All rights reserved.

Herein, three important theorems were stated and proved. The first relates the modified generalized Laguerre expansion coefficients of the derivatives of a function in terms of its original expansion coefficients; and an explicit expression for the derivatives of modified generalized Laguerre polynomials of any degree and for any order as a linear combination of modified generalized Laguerre polynomials themselves is also deduced. The second theorem gives new modified generalized Laguerre coefficients of the moments of one single modified generalized Laguerre polynomials of any degree. Finally, the third theorem expresses explicitly the modified generalized Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its modified generalized Laguerre coefficients. Some spectral applications of these theorems for solving ordinary differential equations with varying coefficients and some specific applied differential problems, by reducing them to recurrence relations in their expansion coefficients of the solution are considered.

Herein, we propose a numerical scheme to solve spectrally hyperbolic partial differential equations (HPDEs) using Galerkin method and approximate the solutions using double shifted Jacobi Polynomials. The main characteristic behind this approach is that it reduces such problems to those of solving systems of algebraic equations which greatly simplifies the problem. The validity and efficiency of the proposed method are investigated and verified through several examples.

This paper focuses on studying a general form of pantograph type Volterra integro-differential equations (PVIDEs). We apply a new collocation spectral approach, based on shifted Chebyshev polynomials, for converting such PVIDEs into systems of algebraic equations. In addition, we apply the new spectral approach for systems of pantograph type Volterra integro-differential equations (SPVIDEs). We investigate the error analysis of the proposed numerical approach. Also, we present some comparisons with other spectral approaches for clarifying the superiority of the new spectral approach.

Spectral methods for solving differential/integral equations are characterized by the representation of the solution by a truncated series of smooth functions. The unknowns to be determined are the expansion coefficients in such a representation. The goal of this article is to give an overview of numerical problems encountered when determining these coefficients and the rich variety of techniques proposed to solve these problems. Therefore, a series of explicit formulae expressing the derivatives, integrals and moments of a class of orthogonal polynomials of any degree and for any order in terms of the same polynomials are addressed. We restrict the current study to the orthogonal polynomials of the Hermite, generalized Laguerre, Bessel, and Jacobi (including Legendre, Chebyshev, and ultraspherical) families. Moreover, formulae expressing the coefficients of an expansion of these polynomials which have been differentiated or integrated an arbitrary number of times in terms of the coefficients of the original expansion are given. In addition, formulae for the polynomial coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its original expanded coefficients are also presented. A simple approach to build and solve recursively for the connection coefficients between different orthogonal polynomials is established. The essential results are summarized in tables which could serve as a useful reference to numerical analysts and practitioners. Finally, applications of these results in solving differential and integral equations with varying polynomial coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, are implemented.

We consider a Galerkin method based on Legendre and Laguerre polynomials and apply it to the Euler-Bernoulli beam equation. The matrices of the method are well structured, which results in substantial reduction of computational cost. Numerical examples demonstrate the efficiency and a high accuracy of the algorithm proposed.

In this paper, we propose an efficient spectral numerical method for solving sine and Klein–Gordon nonlinear variable-order fractional differential equations with the initial and Dirichlet boundary conditions. The approach is based on the shifted Legendre–Gauss and Chebyshev–Gauss collocation methods. The Caputo fractional derivative of variable order is adopted, and the original problems are reduced to systems of algebraic equations. The validity and effectiveness of the method is demonstrated by means of several numerical examples.

In this paper, we construct and analyze a Legendre spectral-collocation method for the numerical solution of distributed-order fractional initial value problems. We first introduce three-term recurrence relations for the fractional integrals of the Legendre polynomial. We then use the properties of the Caputo fractional derivative to reduce the problem into a distributed-order fractional integral equation. We apply the Legendre–Gauss quadrature formula to compute the distributed-order fractional integral and construct the collocation scheme. The convergence of the proposed method is discussed. Numerical results are provided to give insights into the convergence behavior of our method.

This article, presented a shifted Legendre Gauss‐Lobatto collocation (SL‐GL‐C) method which is introduced for solving variable‐order fractional Volterra integro‐differential equation (VO‐FVIDEs) subject to initial or nonlocal conditions. Based on shifted Legendre Gauss‐Lobatto (SL‐GL) quadrature, we treat with integral term in the aforementioned problems. Via the current approach, we convert such problem into a system of algebraic equations. After that we obtain the spectral solution directly for the proposed problem. The high accuracy of the method was proved by several illustrative examples.

In this paper, we propose a new accurate and robust numerical technique to approximate the solutions of fractional variational problems (FVPs) depending on indefinite integrals with a type of fixed Riemann-Liouville fractional integral. The proposed technique is based on the shifted Chebyshev polynomials as basis functions for the fractional integral operational matrix (FIOM). Together with the Lagrange multiplier method, these problems are then reduced to a system of algebraic equations, which greatly simplifies the solution process. Numerical examples are carried out to confirm the accuracy, efficiency and applicability of the proposed algorithm (C) 2017 Elsevier B.V. All rights reserved.