Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions

DOI10.1016/j.cma.2017.08.029zbMath1439.65081arXiv1701.00996OpenAlexW2589697132MaRDI QIDQ2310114

Fanhai Zeng, Zhongqiang Zhang, George Em. Karniadakis

Publication date: 6 April 2020

Published in: Computer Methods in Applied Mechanics and Engineering (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1701.00996

zbMATH Keywords

low regularitytime-fractional diffusion-wave equationshifted Grünwald-Letnikov formulaFODEs

Mathematics Subject Classification ID

Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Fractional ordinary differential equations (34A08) Fractional partial differential equations (35R11)

Related Items (84)

A self-singularity-capturing scheme for fractional differential equations ⋮ Analysis of Legendre pseudospectral approximations for nonlinear time fractional diffusion-wave equations ⋮ Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model ⋮ An alternating direction implicit Legendre spectral method for simulating a 2D multi-term time-fractional Oldroyd-B fluid type diffusion equation ⋮ Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative ⋮ Fast Crank-Nicolson compact difference scheme for the two-dimensional time-fractional mobile/immobile transport equation ⋮ A stable fast time-stepping method for fractional integral and derivative operators ⋮ An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations ⋮ Reproducing kernel particle method for two-dimensional time-space fractional diffusion equations in irregular domains ⋮ An analysis of the Grünwald-Letnikov scheme for initial-value problems with weakly singular solutions ⋮ Local discontinuous Galerkin method based on a family of second-order time approximation schemes for fractional mobile/immobile convection-diffusion equations ⋮ Efficient numerical algorithm with the second-order time accuracy for a two-dimensional nonlinear fourth-order fractional wave equation ⋮ Numerical schemes for the time-fractional mobile/immobile transport equation based on convolution quadrature ⋮ A novel discrete fractional Grönwall-type inequality and its application in pointwise-in-time error estimates ⋮ Fast method and convergence analysis for the magnetohydrodynamic flow and heat transfer of fractional Maxwell fluid ⋮ An extrapolated finite difference method for two-dimensional fractional boundary value problems with non-smooth solution ⋮ A vertex-centred finite volume method for the 3D multi-term time and space fractional Bloch-Torrey equation with fractional Laplacian ⋮ The convergence rate for difference approximations to fractional boundary value problems ⋮ Two high-order time discretization schemes for subdiffusion problems with nonsmooth data ⋮ A Legendre spectral method on graded meshes for the two-dimensional multi-term time-fractional diffusion equation with non-smooth solutions ⋮ Stochastic P-Bifurcation Analysis of Fractional Smooth and Discontinuous Oscillator with an Extended Fast Method ⋮ An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data ⋮ Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations ⋮ An alternating direction implicit compact finite difference scheme for the multi-term time-fractional mixed diffusion and diffusion-wave equation ⋮ Analytical and numerical investigation on the tempered time-fractional operator with application to the Bloch equation and the two-layered problem ⋮ Two families of second-order fractional numerical formulas and applications to fractional differential equations ⋮ Analysis of a finite difference method based on L1 discretization for solving multi‐term fractional differential equation involving weak singularity ⋮ Implicit-explicit time integration of nonlinear fractional differential equations ⋮ Novel superconvergence analysis of anisotropic triangular <scp>FEM</scp> for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients ⋮ Convergence analysis of a fast second‐order time‐stepping numerical method for two‐dimensional nonlinear time–space fractional Schrödinger equation ⋮ An optimal error estimate for the two-dimensional nonlinear time fractional advection-diffusion equation with smooth and non-smooth solutions ⋮ An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated space-time white noise ⋮ A fast time stepping Legendre spectral method for solving fractional cable equation with smooth and non-smooth solutions ⋮ The fast method and convergence analysis of the fractional magnetohydrodynamic coupled flow and heat transfer model for the generalized second-grade fluid ⋮ A novel finite element method for the distributed-order time fractional Cable equation in two dimensions ⋮ A distributed‐order fractional diffusion equation with a singular density function: Analysis and approximation ⋮ A Time-Fractional Mean-Field Control Modeling Subdiffusive Advective Transport ⋮ Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation ⋮ Fast compact finite difference schemes on graded meshes for fourth-order multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions ⋮ Finite volume element methods for two-dimensional time fractional reaction–diffusion equations on triangular grids ⋮ An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data ⋮ The use of a time-fractional transport model for performing computational homogenisation of 2D heterogeneous media exhibiting memory effects ⋮ A higher order numerical method for time fractional partial differential equations with nonsmooth data ⋮ A two-grid ADI finite element approximation for a nonlinear distributed-order fractional sub-diffusion equation ⋮ Numerical algorithm for two dimensional fractional Stokes' first problem for a heated generalized second grade fluid with smooth and non-smooth solution ⋮ Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains ⋮ An investigation of nonlinear time-fractional anomalous diffusion models for simulating transport processes in heterogeneous binary media ⋮ A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations ⋮ An \(H^1\) convergence of the spectral method for the time-fractional non-linear diffusion equations ⋮ Fourth-order compact difference schemes for the two-dimensional nonlinear fractional mobile/immobile transport models ⋮ The unified theory of shifted convolution quadrature for fractional calculus ⋮ Müntz Spectral Method for Two-Dimensional Space-Fractional Convection-Diffusion Equation ⋮ Analytical and numerical solutions of a multi-term time-fractional Burgers' fluid model ⋮ Error analysis of nonlinear time fractional mobile/immobile advection-diffusion equation with weakly singular solutions ⋮ A New Class of Semi-Implicit Methods with Linear Complexity for Nonlinear Fractional Differential Equations ⋮ An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains ⋮ A class of efficient time-stepping methods for multi-term time-fractional reaction-diffusion-wave equations ⋮ Approximation methods for the distributed order calculus using the convolution quadrature ⋮ Fast high-order difference scheme for the modified anomalous subdiffusion equation based on fast discrete sine transform ⋮ A Fast High Order Method for the Time-Fractional Diffusion Equation ⋮ A preconditioning technique for an all-at-once system from Volterra subdiffusion equations with graded time steps ⋮ Modeling and computing of fractional convection equation ⋮ Numerical analysis of linear and nonlinear time-fractional subdiffusion equations ⋮ Semi-implicit Galerkin-Legendre spectral schemes for nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions ⋮ Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains ⋮ A finite volume method for the two-dimensional time and space variable-order fractional Bloch-Torrey equation with variable coefficients on irregular domains ⋮ Fourier spectral approximation for time fractional Burgers equation with nonsmooth solutions ⋮ Error analysis of a finite element method with GMMP temporal discretisation for a time-fractional diffusion equation ⋮ A transformed \(L 1\) method for solving the multi-term time-fractional diffusion problem ⋮ Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions ⋮ Efficient Multistep Methods for Tempered Fractional Calculus: Algorithms and Simulations ⋮ Spectral approximations for nonlinear fractional delay diffusion equations with smooth and nonsmooth solutions ⋮ Finite difference method for two-dimensional nonlinear time-fractional subdiffusion equation ⋮ On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems ⋮ Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media ⋮ Finite element methods based on two families of second-order numerical formulas for the fractional cable model with smooth solutions ⋮ Error estimates for a robust finite element method of two-term time-fractional diffusion-wave equation with nonsmooth data ⋮ Fully discrete spectral methods for solving time fractional nonlinear sine-Gordon equation with smooth and non-smooth solutions ⋮ Numerical investigation of ordinary and partial differential equations with variable fractional order by Bernstein operational matrix ⋮ The local discontinuous Galerkin finite element method for a multiterm time-fractional initial-boundary value problem ⋮ A weak Galerkin finite element method on temporal graded meshes for the multi-term time fractional diffusion equations ⋮ Sharp error estimate of Grünwald-Letnikov scheme for a multi-term time fractional diffusion equation ⋮ Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations ⋮ Superconvergence analysis of FEM for 2D multi-term time fractional diffusion-wave equations with variable coefficient

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients
• Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation
• Finite difference methods for the time fractional diffusion equation on non-uniform meshes
• Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation
• Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence
• High-order algorithms for Riesz derivative and their applications. II
• Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain
• Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative
• Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation
• Spline collocation methods for linear multi-term fractional differential equations
• Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials
• A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations
• The Galerkin finite element method for a multi-term time-fractional diffusion equation
• Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations
• Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients
• Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain
• A second-order accurate numerical method for a fractional wave equation
• Second-order stable finite difference schemes for the time-fractional diffusion-wave equation
• The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type
• Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations
• High-order algorithms for Riesz derivative and their applications. I.
• Detailed error analysis for a fractional Adams method
• Finite difference approximations for fractional advection-dispersion flow equations
• Quasi-compact finite difference schemes for space fractional diffusion equations
• A high-order spectral method for the multi-term time-fractional diffusion equations
• Numerical methods for solving the multi-term time-fractional wave-diffusion equation
• Nonpolynomial collocation approximation of solutions to fractional differential equations
• Finite difference/spectral approximations for the time-fractional diffusion equation
• Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations
• A Fourier method for the fractional diffusion equation describing sub-diffusion
• Weighted average finite difference methods for fractional diffusion equations
• Pitfalls in fast numerical solvers for fractional differential equations
• Generalized Jacobi functions and their applications to fractional differential equations
• Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations
• Discretized Fractional Calculus
• Convolution quadrature time discretization of fractional diffusion-wave equations
• A Finite Element Method with Singularity Reconstruction for Fractional Boundary Value Problems
• A Generalized Spectral Collocation Method with Tunable Accuracy for Variable-Order Fractional Differential Equations
• Numerical Algorithms for Time-Fractional Subdiffusion Equation with Second-Order Accuracy
• A Stability Analysis of Convolution Quadraturea for Abel-Volterra Integral Equations
• A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels
• HOW TO APPROXIMATE THE FRACTIONAL DERIVATIVE OF ORDER 1 < α ≤ 2
• The Finite Difference Methods for Fractional Ordinary Differential Equations
• Superconvergence of a Discontinuous Galerkin Method for Fractional Diffusion and Wave Equations
• A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation
• A class of second order difference approximations for solving space fractional diffusion equations
• Fourth Order Accurate Scheme for the Space Fractional Diffusion Equations
• The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation
• Fractional Spectral Collocation Method
• Spectral Methods

This page was built for publication: Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions