Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions

DOI10.1137/16M1059278zbMath1353.65086MaRDI QIDQ2834570

Yujian Jiao, Zhimin Zhang, Can Huang, Li-Lian Wang

Publication date: 23 November 2016

Published in: SIAM Journal on Numerical Analysis (Search for Journal in Brave)

zbMATH Keywords

fractional differential equationsRiemann-Liouville fractional derivativemixed Volterra-Fredholm integral equationwell-conditioned collocation methodsfractional Birkhoff interpolationfractional integration preconditioners

Mathematics Subject Classification ID

Numerical methods for integral equations (65R20) Numerical methods for initial value problems involving ordinary differential equations (65L05) Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations (65L60) Error bounds for numerical methods for ordinary differential equations (65L70) Fredholm integral equations (45B05) Volterra integral equations (45D05) Fractional ordinary differential equations (34A08) Preconditioners for iterative methods (65F08)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Fractional Sturm-Liouville eigen-problems: theory and numerical approximation
• Exponentially accurate spectral and spectral element methods for fractional ODEs
• A multi-domain spectral method for time-fractional differential equations
• Piecewise polynomial collocation for linear boundary value problems of fractional differential equations
• A spectral collocation method for a weakly singular Volterra integral equation of the second kind
• A second-order accurate numerical method for the two-dimensional fractional diffusion equation
• Remarks on fractional derivatives
• An efficient collocation method for a Caputo two-point boundary value problem
• Generalized Jacobi polynomials/functions and their applications
• Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
• Numerical solution of the space fractional Fokker-Planck equation.
• Finite difference approximations for fractional advection-dispersion flow equations
• Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials
• Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem
• Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis
• Orthogonality of Jacobi polynomials with general parameters
• A fully discrete difference scheme for a diffusion-wave system
• Finite difference methods for two-dimensional fractional dispersion equation
• Generalized Jacobi functions and their applications to fractional differential equations
• Wellposedness of Variable-Coefficient Conservative Fractional Elliptic Differential Equations
• A Well-Conditioned Collocation Method Using a Pseudospectral Integration Matrix
• Spectral Methods
• Variational solution of fractional advection dispersion equations on bounded domains in ℝd
• Integration Preconditioning Matrix for Ultraspherical Pseudospectral Operators
• Hypergeometric Orthogonal Polynomials and Their q-Analogues
• A Space-Time Spectral Method for the Time Fractional Diffusion Equation
• Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel
• Integration Preconditioning of Pseudospectral Operators. I. Basic Linear Operators
• The Numerical Solution of Integral Equations of the Second Kind
• A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels
• Collocation Methods for Volterra Integral and Related Functional Differential Equations
• Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints
• Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations
• Inhomogeneous Dirichlet Boundary-Value Problems of Space-Fractional Diffusion Equations and their Finite Element Approximations
• Numerical Approximation of a Time Dependent, Nonlinear, Space‐Fractional Diffusion Equation
• Polynomial spectral collocation method for space fractional advection–diffusion equation
• Fractional Spectral Collocation Method
• Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey
• The random walk's guide to anomalous diffusion: A fractional dynamics approach

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