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generalized jacobi functions AND THEIR APPLICATIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS

MATHEMATICS OF COMPUTATION 2016年第300期85卷 1603-1638页

作者： Chen, Sheng Shen, Jie Wang, Li-Lian Xiamen Univ Sch Math Sci Xiamen 361005 Fujian Peoples R China Xiamen Univ Fujian Prov Key Lab Math Modeling & High Performa Xiamen 361005 Fujian Peoples R China Purdue Univ Dept Math W Lafayette IN 47907 USA Nanyang Technol Univ Sch Phys & Math Sci Div Math Sci Singapore 637371 Singapore

In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized jacobi functions (GJFs), which is intrinsically related to fractional calculus and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value problems (FBVPs) of general order, and we show that with an appropriate choice of the parameters in GJFs, the resulting linear systems are sparse and well-conditioned. Moreover, we derive error estimates with convergence rates only depending on the smoothness of data, so true spectral accuracy can be attained if the data are smooth enough. The ideas and results presented in this paper will be useful in dealing with more general FDEs involving Riemann-Liouville or Caputo fractional derivatives.

关键词： Fractional differential equations singularity jacobi polynomials with real parameters generalized jacobi functions weighted Sobolev spaces approximation results spectral accuracy

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generalized jacobi spectral Galerkin method for fractional pantograph differential equation

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2021年第1期44卷 153-165页

作者： Yang, Changqing Lv, Xiaoguang Jiangsu Ocean Univ Dept Sci Lianyungang 222005 Jiangsu Peoples R China

This work is concerned with the extension of the jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.

关键词： Caputo derivative convergence analysis fractional derivatives and integrals Galerkin method generalized jacobi functions

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Estimates for coefficients in jacobi series for functions with limited regularity by fractional calculus

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2024年第4期50卷 68-68页

作者： Liu, Guidong Liu, Wenjie Duan, Beiping Nanjing Audit Univ Sch Math Nanjing 211815 Peoples R China Harbin Inst Technol Sch Math Harbin 150001 Peoples R China Shenzhen MSU BIT Univ Fac Computat Math & Cybernet Shenzhen 518172 Peoples R China

In this paper, optimal estimates on the decaying rates of jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the range of the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>p$$\end{document}-space under the Riemann-Liouville fractional integral. The connection between these new spaces and classical fractional-order Sobolev spaces is then elucidated. Under this framework, the optimal decaying rate of jacobi expansion coefficients is obtained, based on which the projection errors under different norms are given. This work is expected to introduce fractional calculus into traditional fields in approximation theory and to explore the possibility in solving classical problems by this 'new' tool.

关键词： jacobi polynomials generalized jacobi functions jacobi expansion coefficients Fractional calculus Projection error

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Error estimates of generalized spectral iterative methods with accurate convergence rates for solving systems of fractional two - point boundary value problems

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APPLIED MATHEMATICS AND COMPUTATION 2020年 364卷 124638-124638页

作者： Erfani, S. Babolian, E. Javadi, S. Kharazmi Univ Fac Math Sci & Comp 50 Taleghani Ave Tehran *** Iran

The main purpose of this study is to provide an efficient spectral iterative method based on fractional interpolants for solving a class of linear and nonlinear fractional two - point boundary value problems involving left and right fractional derivatives. In order to achieve this goal, by composition of iterative method and spectral method, in each step, the unknown solution of system is expanded by using two classes of generalized jacobi functions to obtain numerically coefficients. For this system of fractional differential equations on the interval [ -1,1 ], the singularities of the solution are considered of types (1 + t)(beta) and (1 - x)(alpha), where 0 < beta, alpha < 1 are left and right singularities indexes. Then, these types of singularities can be well resolved with the mentioned functions. A rigours convergence analysis of the proposed method with accurate spectral rate of convergence is extensively discussed in L-2-norm. Finally, some numerical results are given to demonstrate the effectiveness and applicability of the proposed method and accuracy of the presented convergence rates. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Convergence rate generalized jacobi functions Singularity index Spectral method System of fractional differential equations

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Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives

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JOURNAL OF SCIENTIFIC COMPUTING 2019年第3期81卷 1577-1601页

作者： Deng, Beichuan Zhang, Zhimin Zhao, Xuan Wayne State Univ Dept Math Detroit MI 48202 USA Beijing Computat Sci Res Ctr Beijing 100193 Peoples R China Southeast Univ Sch Math Nanjing 210096 Jiangsu Peoples R China

In this paper, superconvergence points are located for the approximation of the Riesz derivative of order a using classical Lobatto-type polynomials when a. (0, 1) and generalized jacobi functions (GJF) for arbitrary a > 0, respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre-Lobatto expansion. It is observed that the convergence rate for different a at the superconvergence points is at least O( N -2) better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of ordera > 1 with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be O(N -(a+3)/2) for a. (0, 1) and O(N -2) for a > 1. Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points.

关键词： Superconvergence Riesz fractional derivative Spectral interpolation generalized jacobi functions

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An Efficient Space-Time Method for Time Fractional Diffusion Equation

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JOURNAL OF SCIENTIFIC COMPUTING 2019年第2期81卷 1088-1110页

作者： Shen, Jie Sheng, Chang-Tao Purdue Univ Dept Math W Lafayette IN 47907 USA Nanyang Technol Univ Sch Phys & Math Sci Div Math Sci Singapore 637371 Singapore

A space-time Petrov-Galerkin spectral method for time fractional diffusion equations is developed in this paper. The Petrov-Galerkin method is used to simplify the computation of stiffness matrix but leads to full non-symmetric mass matrix. However, the matrix decomposition method based on eigen-decomposition is numerically unstable for non-symmetric linear systems. A QZ decomposition is adopted instead of eigen-decomposition. The QZ decomposition has essentially the same computational complexity as the eigen-decomposition but is numerically stable. Moreover, the enriched Petrov-Galerkin method is developed to resolve the weak singularity at the initial time. We also carry out the error analysis for the proposed methods and present ample numerical results to validate the accuracy and robustness of our numerical schemes.

关键词： Fractional derivative Spectral method QZ decomposotion generalized jacobi functions Error analysis

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Fractional spectral and pseudo-spectral methods in unbounded domains: Theory and applications

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JOURNAL OF COMPUTATIONAL PHYSICS 2017年 338卷 527-566页

作者： Khosravian-Arab, Hassan Dehghan, Mehdi Eslahchi, M. R. Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math 424 Hafez Ave Tehran Iran Tarbiat Modares Univ Fac Math Sci Dept Appl Math POB 14115-134 Tehran Iran

This paper is intended to provide exponentially accurate Galerkin, Petrov-Galerkin and pseudo-spectral methods for fractional differential equations on a semi-infinite interval. We start our discussion by introducing two new non-classical Lagrange basis functions: NLBFs-1 and NLBFs-2 which are based on the two new families of the associated Laguerre polynomials: GALFs-1 and GALFs-2 obtained recently by the authors in [28]. With respect to the NLBFs-1 and NLBFs-2, two new non-classical interpolants based on the associated- Laguerre-Gauss and Laguerre-Gauss-Radau points are introduced and then fractional (pseudo-spectral) differentiation (and integration) matrices are derived. Convergence and stability of the new interpolants are proved in detail. Several numerical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional derivatives (and integrals) of some functions. Moreover, the pseudo-spectral, Galerkin and Petrov-Galerkin methods are successfully applied to solve some physical ordinary differential equations of either fractional orders or integer ones. Some useful comments from the numerical point of view on Galerkin and Petrov-Galerkin methods are listed at the end. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Fractional calculus Associated Laguerre polynomials generalized associated Iaguerre functions of the first- and second-kind Non-classical Lagrange basis functions Fractional differentiation and integration matrices Convergence Stability Spectral methods Pseudo-spectral methods Galerkin and Petrov-Galerkin methods Fractional Basset force equation Lane-Emden equation Fractional relaxation-oscillation equation Fractional ordinary differential equations jacobi polyfractonomials generalized jacobi functions Preconditioner matrix

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A Hybrid Spectral Element Method for Fractional Two-Point Boundary Value Problems

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高等学校计算数学学报（英文版） 2017年第2期10卷 437-464页

作者： Changtao Sheng Jie Shen Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing and School of Mathematical Sciences Xiamen University Xiamen Fujian 361005 China Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing and School of Mathematical Sciences Xiamen University Xiamen Fujian 361005 China Department of Mathematics Purdue University West LafayetteIN 47907-1957 USA

We propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL)fractional *** first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel,following a similar procedure in [16].We then design a hybrid spectral element method with generalized jacobi functions and Legendre polynomials as basis *** use of generalized jacobi functions allow us to deal with the usual singularity of solutions at t =*** establish the existence and uniqueness of the numerical solution,and derive a hptype error estimates under L2 (I)-norm for the transformed *** results are provided to show the effectiveness of the proposed methods.

关键词： Boundary value problems generalized jacobi functions spectral element method fractional differential equations error bounds

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SUPERCONVERGENCE POINTS OF FRACTIONAL SPECTRAL INTERPOLATION

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2016年第1期38卷 A598-A613页

作者： Zhao, Xuan Zhang, Zhimin Southeast Univ Dept Math Nanjing 210096 Jiangsu Peoples R China Beijing Computat Sci Res Ctr Beijing 100193 Peoples R China Wayne State Univ Dept Math Detroit MI 48202 USA

We investigate superconvergence properties of the spectral interpolation involving fractional derivatives. Our interest in this superconvergence problem is, in fact, twofold: when interpolating function values, we identify the points at which fractional derivatives of the interpolant superconverge;when interpolating fractional derivatives, we locate those points where function values of the interpolant superconverge. For the former case, we apply various Legendre polynomials as basis functions and obtain the superconvergence points, which naturally unify the superconvergence points for the first order derivative presented in [Z. Zhang, SIAM J. Numer. Anal., 50 (2012), pp. 2966-2985], depending on orders of the fractional derivatives. While for the latter case, we utilize the Petrov-Galerkin method based on generalized jacobi functions [S. Chen, J. Shen, and L.-L. Wang, Math. Comp., to appear] and locate the superconvergence points both for function values and fractional derivatives. Numerical examples are provided to verify the analysis of superconvergence points for each case.

关键词： superconvergence fractional derivative spectral collocation Petrov-Galerkin generalized jacobi functions

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