Discretized Fractional Calculus

DOI10.1137/0517050zbMath0624.65015OpenAlexW1991982189MaRDI QIDQ3026156

Christian Lubich

Publication date: 1986

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

Full work available at URL: https://doi.org/10.1137/0517050

zbMATH Keywords

special functionsAbel integral equationsfractional integralsEuler's gamma functiondiffusion problemsdiscrete convolutionfractional linear multistep methodsconvolution quadraturesRiemann-Liouville integralsAbel-Liouville integrals

Mathematics Subject Classification ID

Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane (30E20) Numerical methods for integral equations (65R20) Fractional derivatives and integrals (26A33) Gamma, beta and polygamma functions (33B15) Computation of special functions and constants, construction of tables (65D20) Approximate quadratures (41A55) Numerical quadrature and cubature formulas (65D32) Numerical differentiation (65D25)

Related Items

Numerical algorithms for Caputo fractional-order differential equations ⋮ Efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential Equations ⋮ Impulse response of a generalized fractional second order filter ⋮ Two explicit and implicit finite difference schemes for time fractional Riesz space diffusion equation ⋮ Tumor evolution models of phase-field type with nonlocal effects and angiogenesis ⋮ Analytical and numerical investigation on the tempered time-fractional operator with application to the Bloch equation and the two-layered problem ⋮ On Discrete Energy Dissipation of Maxwell'S Equations In a Cole-Cole Dispersive Medium ⋮ Two families of second-order fractional numerical formulas and applications to fractional differential equations ⋮ Optimal Long-Time Decay Rate of Numerical Solutions for Nonlinear Time-Fractional Evolutionary Equations ⋮ A new approach to shooting methods for terminal value problems of fractional differential equations ⋮ Analyses of the contour integral method for time fractional normal-subdiffusion transport equation ⋮ Dynamical analysis of a fractional discrete-time vocal system ⋮ Preconditioned fourth-order exponential integrator for two-dimensional nonlinear fractional Ginzburg-Landau equation ⋮ A formally second‐order <scp>backward differentiation formula</scp> Sinc‐collocation method for the Volterra integro‐differential equation with a weakly singular kernel based on the double exponential transformation ⋮ A compact finite difference scheme for the fourth‐order time‐fractional integro‐differential equation with a weakly singular kernel ⋮ Numerical algorithms for the time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations ⋮ A linearized finite difference scheme for time-space fractional nonlinear diffusion-wave equations with initial singularity ⋮ ADI finite element Galerkin methods for two-dimensional tempered fractional integro-differential equations ⋮ A superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity ⋮ A fast numerical solution of the <scp>3D</scp> nonlinear tempered fractional integrodifferential equation ⋮ Numerical approximations for the nonlinear time fractional reaction–diffusion equation ⋮ Properties of vector-valued \(\tau \)-discrete fractional calculus and its connection with Caputo fractional derivatives ⋮ A corrected \(\mathrm{L}1\) method for a time-fractional subdiffusion equation ⋮ Fractional second linear multistep methods: the explicit forms for solving fractional differential equations and stability analysis ⋮ Fractional exponential fitting backward differential formulas for solving differential equations of fractional order ⋮ Numerical method for solving the fractional evolutionary model of bi-flux diffusion processes ⋮ Modelling discrete time fractional Rucklidge system with complex state variables and its synchronization ⋮ A fourth-order fractional Adams-type implicit-explicit method for nonlinear fractional ordinary differential equations with weakly singular solutions ⋮ A high-order discrete energy decay and maximum-principle preserving scheme for time fractional Allen-Cahn equation ⋮ \(\alpha\)-robust analysis of fast and novel two-grid FEM with nonuniform \(\mathrm{L}1\) scheme for semilinear time-fractional variable coefficient diffusion equations ⋮ The fast method and convergence analysis of the fractional magnetohydrodynamic coupled flow and heat transfer model for the generalized second-grade fluid ⋮ A high-efficiency second-order numerical scheme for time-fractional phase field models by using extended SAV method ⋮ Numerical recovery of a time-dependent potential in subdiffusion ^* ⋮ An efficient Sinc‐collocation method by the single exponential transformation for the nonlinear fourth‐order partial integro‐differential equation with multiterm kernels ⋮ The compact difference scheme for the fourth‐order nonlocal evolution equation with a weakly singular kernel ⋮ Superconvergence analysis of anisotropic finite element method for the time fractional substantial diffusion equation with smooth and nonsmooth solutions ⋮ A novel shifted Jacobi operational matrix method for nonlinear multi‐terms delay differential equations of fractional variable‐order with periodic and anti‐periodic conditions ⋮ A Sharp $\alpha$-Robust $L1$ Scheme on Graded Meshes for Two-Dimensional Time Tempered Fractional Fokker-Planck Equation ⋮ An efficient compact difference-proper orthogonal decomposition algorithm for fractional viscoelastic plate vibration model ⋮ A second-order finite difference scheme for nonlinear tempered fractional integrodifferential equations in three dimensions ⋮ 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 ⋮ Crank-Nicolson ADI finite difference/compact difference schemes for the 3D tempered integrodifferential equation associated with Brownian motion ⋮ Local discontinuous Galerkin schemes for an ultrasonic propagation equation with fractional attenuation ⋮ An implicit difference scheme for the fourth-order nonlinear partial integro-differential equations ⋮ Discrete approximation of solutions of the Cauchy problem for a linear homogeneous differential-operator equation with a Caputo fractional derivative in a Banach space ⋮ A Compact Difference Scheme for Time-Space Fractional Nonlinear Diffusion-Wave Equations with Initial Singularity ⋮ Crank-Nicolson schemes for sub-diffusion equations with nonsingular and singular source terms in time ⋮ Mittag-Leffler stability of numerical solutions to linear homogeneous time fractional parabolic equations ⋮ Numerical approach for time-fractional Burgers' equation via a combination of Adams-Moulton and linearized technique ⋮ Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions ⋮ Subdiffusion with a time-dependent coefficient: Analysis and numerical solution ⋮ Normalized finite fractional differences: Computational and accuracy breakthroughs ⋮ A high-order difference scheme for the fractional sub-diffusion equation ⋮ Fractional Multistep Methods for Weakly Singular Volterra Integral Equations of the First Kind with Perturbed Data ⋮ Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation ⋮ Stability and numerical analysis of backward problem for subdiffusion with time-dependent coefficients ⋮ A new computational method based on fractional Lagrange functions to solve multi-term fractional differential equations ⋮ On a nonlinear abstract Volterra equation ⋮ A high-order numerical algorithm for two-dimensional time-space tempered fractional diffusion-wave equation ⋮ Numerical solutions for a class of differential equations in linear viscoelasticity ⋮ High order finite difference methods on non-uniform meshes for space fractional operators ⋮ Block backward differentiation formulas for fractional differential equations ⋮ Effects of HIV infection on CD4\(^{+}\) T-cell population based on a fractional-order model ⋮ Numerical solution of fractional diffusion-wave equation based on fractional multistep method ⋮ Higher order finite difference method for the reaction and anomalous-diffusion equation ⋮ Discretization of fractional order differentiator over Paley-Wiener space ⋮ Second-order explicit difference schemes for the space fractional advection diffusion equation ⋮ Stability and convergence analysis of a class of continuous piecewise polynomial approximations for time-fractional differential equations ⋮ 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 ⋮ FFT techniques in the numerical solution of convolution equations ⋮ Numerical methods for singular nonlinear integrodifferential equations ⋮ Numerical simulations for a variable order fractional cable equation ⋮ A family of Adams exponential integrators for fractional linear systems ⋮ Tempered stable Lévy motion and transient super-diffusion ⋮ A survey of singularly perturbed Volterra equations ⋮ The lumped mass FEM for a time-fractional cable equation ⋮ Numerical treatment for solving the perturbed fractional PDEs using hybrid techniques ⋮ Fast numerical solution of weakly singular Volterra integral equations ⋮ A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation ⋮ A second-order difference scheme for the time fractional substantial diffusion equation ⋮ A second-order accurate numerical method for a fractional wave equation ⋮ Numerical solution of multiterm fractional differential equations using the matrix Mittag-Leffler functions ⋮ Numerical solution of fractional differential equations: a survey and a software tutorial ⋮ Exponentially accurate spectral and spectral element methods for fractional ODEs ⋮ Finite difference methods for the time fractional diffusion equation on non-uniform meshes ⋮ Microlocal approach towards construction of nonreflecting boundary conditions ⋮ Wave simulation in 2D heterogeneous transversely isotropic porous media with fractional attenuation: a Cartesian grid approach ⋮ Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation ⋮ Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions ⋮ Efficient computation of the Grünwald-Letnikov fractional diffusion derivative using adaptive time step memory ⋮ Fractional high order methods for the nonlinear fractional ordinary differential equation ⋮ Computational challenge of fractional differential equations and the potential solutions: a survey ⋮ A unified Petrov-Galerkin spectral method for fractional PDEs ⋮ Moving finite element methods for time fractional partial differential equations ⋮ Efficient implementation of rational approximations to fractional differential operators ⋮ High order algorithm for the time-tempered fractional Feynman-Kac equation ⋮ High-order compact difference methods for Caputo-type variable coefficient fractional sub-diffusion equations in conservative form ⋮ On FBDF5 method for delay differential equations of fractional order with periodic and anti-periodic conditions ⋮ Quasi wavelet based numerical method for a class of partial integro-differential equation ⋮ Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion ⋮ On the fractional signals and systems ⋮ Analytical impulse response of a fractional second order filter and its impulse response invariant discretization ⋮ Finite difference approximations for the fractional advection-diffusion equation ⋮ Second-order stable finite difference schemes for the time-fractional diffusion-wave equation ⋮ A convergent algorithm for solving higher-order nonlinear fractional boundary value problems ⋮ Compact alternating direction implicit method for two-dimensional time fractional diffusion equation ⋮ Numerical solution of two-sided space-fractional wave equation using finite difference method ⋮ Convergence of the variational iteration method for solving multi-order fractional differential equations ⋮ On the use of matrix functions for fractional partial differential equations ⋮ Fractional variational integrators for fractional variational problems ⋮ Two direct Tustin discretization methods for fractional-order differentiator/integrator. ⋮ Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel. ⋮ A new proof for the convergence of the trigonometric method for the Abel integral equations of the first kind ⋮ Collocation methods for fractional integro-differential equations with weakly singular kernels ⋮ Jacobian-predictor-corrector approach for fractional differential equations ⋮ Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials ⋮ An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives ⋮ Two computational algorithms for the numerical solution for system of fractional differential equations ⋮ A modified memory-based mathematical model describing fluid flow in porous media ⋮ Energy estimates for two-dimensional space-Riesz fractional wave equation ⋮ Block implicit Adams methods for fractional differential equations ⋮ On the naturality of the iteration and recursive specifications ⋮ A stable numerical method for multidimensional time fractional Schrödinger equations ⋮ Solving mixed classical and fractional partial differential equations using short-memory principle and approximate inverses ⋮ An effective numerical method and its utilization to solution of fractional models used in bioengineering applications ⋮ Artificial boundary conditions for certain evolution PDEs with cubic nonlinearity for non-compactly supported initial data ⋮ Numerical approaches to fractional calculus and fractional ordinary differential equation ⋮ A characteristic difference method for the transient fractional convection-diffusion equations ⋮ On distributed order integrator/differentiator ⋮ Double-delay fractional and integer-order tanlock loops ⋮ On the numerical solutions for the fractional diffusion equation ⋮ A collocation-shooting method for solving fractional boundary value problems ⋮ Stability of numerical solutions for Abel-Volterra integral equations of the second kind ⋮ Chaotic behavior of a class of discontinuous dynamical systems of fractional-order ⋮ Spectral approximation methods and error estimates for Caputo fractional derivative with applications to initial-value problems ⋮ A fourth-order approximation of fractional derivatives with its applications ⋮ Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations ⋮ Comparative study of three numerical schemes for fractional integro-differential equations ⋮ Compact finite difference method for the fractional diffusion equation ⋮ Fractional Adams-Moulton methods ⋮ Computational algorithms for computing the fractional derivatives of functions ⋮ Stability properties of two-term fractional differential equations ⋮ Fractional-order relaxation laws in nonlinear viscoelasticity ⋮ Explicit methods for fractional differential equations and their stability properties ⋮ On some explicit Adams multistep methods for fractional differential equations ⋮ An explicit numerical method for the fractional cable equation ⋮ Weighted Paley-Wiener theorem, with applications to stability of the linear multi-step methods for Volterra equations in Hilbert spaces ⋮ A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation ⋮ Incomplete iterative solution of subdiffusion ⋮ A variable step size numerical method based on fractional type quadratures for linear integro-differential equations ⋮ Finite element methods based on two families of second-order numerical formulas for the fractional cable model with smooth solutions ⋮ Fast finite volume methods for space-fractional diffusion equations ⋮ High-order algorithms for Riesz derivative and their applications. III ⋮ Wave propagation in media with singular memory ⋮ On fractional backward differential formulas for fractional delay differential equations with periodic and anti-periodic conditions ⋮ An improved spectral meshless radial point interpolation for a class of time-dependent fractional integral equations: 2D fractional evolution equation ⋮ Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations ⋮ Finite difference methods with non-uniform meshes for nonlinear fractional differential equations ⋮ Fractional Adams-Bashforth/Moulton methods: an application to the fractional Keller-Segel chemotaxis system ⋮ A high-accuracy MOC/FD method for solving fractional advection-diffusion equations ⋮ An analysis of the Grünwald-Letnikov scheme for initial-value problems with weakly singular solutions ⋮ Two finite difference schemes for time fractional diffusion-wave equation ⋮ An adaptation of the Gear scheme for fractional derivatives ⋮ Smoothness and stability of the solutions for nonlinear fractional differential equations ⋮ Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems ⋮ A piecewise memory principle for fractional derivatives ⋮ A note on a stable algorithm for computing the fractional integrals of orthogonal polynomials ⋮ Discrete waveform relaxation method for linear fractional delay differential-algebraic equations ⋮ Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation ⋮ An exponential spline technique for solving fractional boundary value problem ⋮ Superconvergence of finite element approximations for the fractional diffusion-wave equation ⋮ Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations ⋮ Preconditioned quasi-compact boundary value methods for space-fractional diffusion equations ⋮ Galerkin type methods for semilinear time-fractional diffusion problems ⋮ An H2N2 interpolation for Caputo derivative with order in \((1,2)\) and its application to time-fractional wave equations in more than one space dimension ⋮ A Crank-Nicolson-type compact difference method and its extrapolation for time fractional Cattaneo convection-diffusion equations with smooth solutions ⋮ Implicit-explicit time integration of nonlinear fractional differential equations ⋮ Subdiffusion with time-dependent coefficients: improved regularity and second-order time stepping ⋮ Time discretization of fractional subdiffusion equations via fractional resolvent operators ⋮ Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis ⋮ Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models ⋮ A hybrid algorithm for Caputo fractional differential equations ⋮ Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations ⋮ On fractional-Legendre spectral Galerkin method for fractional Sturm-Liouville problems ⋮ Spatial synchrony in fractional order metapopulation cholera transmission ⋮ Numerical algorithm for two dimensional fractional Stokes' first problem for a heated generalized second grade fluid with smooth and non-smooth solution ⋮ Preconditioned GMRES method for a class of Toeplitz linear systems in fractional eigenvalue problems ⋮ Numerical solution of Caputo fractional differential equations with infinity memory effect at initial condition ⋮ A gWSGL numerical scheme for generalized fractional sub-diffusion problems ⋮ A switching fractional calculus-based controller for normal non-linear dynamical systems ⋮ Correction of high-order BDF convolution quadrature for fractional Feynman-Kac equation with Lévy flight ⋮ An investigation of nonlinear time-fractional anomalous diffusion models for simulating transport processes in heterogeneous binary media ⋮ Analysis of a temporal discretization for a semilinear fractional diffusion equation ⋮ Stability and chaos in the fractional Chen system ⋮ A fourth-order dissipation-preserving algorithm with fast implementation for space fractional nonlinear damped wave equations ⋮ A unified spectral method for FPDEs with two-sided derivatives. I: A fast solver ⋮ A unified spectral method for FPDEs with two-sided derivatives. II: Stability, and error analysis ⋮ Weighted estimates for boundary value problems with fractional derivatives ⋮ Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation ⋮ A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations ⋮ Rational Krylov methods for functions of matrices with applications to fractional partial differential equations ⋮ A numerical method for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations ⋮ Legendre spectral methods based on two families of novel second-order numerical formulas for the fractional activator-inhibitor system ⋮ Trapezoidal methods for fractional differential equations: theoretical and computational aspects ⋮ Discontinuous dynamical systems for fault detection. A unified approach including fractional and integer order dynamics ⋮ On some generalizations of the implicit Euler method for discontinuous fractional differential equations ⋮ An \(H^1\) convergence of the spectral method for the time-fractional non-linear diffusion equations ⋮ Discrete subdiffusion equations with memory ⋮ Numerical algorithm for the time fractional Fokker-Planck equation ⋮ The study of fractional order controller with SLAM in the humanoid robot ⋮ A weighted average finite difference method for the fractional convection-diffusion equation ⋮ Exact and discretized stability of the Bagley-Torvik equation ⋮ Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation ⋮ Weighted average finite difference methods for fractional diffusion equations ⋮ Continued fraction expansion approaches to discretizing fractional order derivatives-an expository review ⋮ General linear and spectral Galerkin methods for the Riesz space fractional diffusion equation ⋮ The global analysis on the spectral collocation method for time fractional Schrödinger equation ⋮ Local discontinuous Galerkin methods for the time tempered fractional diffusion equation ⋮ Numerical approximation to the fractional derivative operator ⋮ A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations ⋮ A new family of predictor-corrector methods for solving fractional differential equations ⋮ Approximate solutions for the fractional advection-dispersion equation using Legendre pseudo-spectral method ⋮ Numerical methods for solving space fractional partial differential equations using Hadamard finite-part integral approach ⋮ Modeling and computing of fractional convection equation ⋮ Numerical analysis of linear and nonlinear time-fractional subdiffusion equations ⋮ Asymptotic stability of compact and linear \(\theta \)-methods for space fractional delay generalized diffusion equation ⋮ Analysis of a time-stepping discontinuous Galerkin method for fractional diffusion-wave equations with nonsmooth data ⋮ Multigrid methods for time-fractional evolution equations: a numerical study ⋮ Direct and integrated radial functions based quasilinearization schemes for nonlinear fractional differential equations ⋮ Efficient radial basis functions approaches for solving a class of fractional optimal control problems ⋮ A second-order accurate scheme for two-dimensional space fractional diffusion equations with time Caputo-Fabrizio fractional derivative ⋮ Time two-grid algorithm based on finite difference method for two-dimensional nonlinear fractional evolution equations ⋮ A local mesh-less collocation method for solving a class of time-dependent fractional integral equations: 2D fractional evolution equation ⋮ Weak Galerkin finite-element method for time-fractional nonlinear integro-differential equations ⋮ Fractional-order uniaxial visco-elasto-plastic models for structural analysis ⋮ High-order central difference scheme for Caputo fractional derivative ⋮ A tunable finite difference method for fractional differential equations with non-smooth solutions ⋮ Construction of new generating function based on linear barycentric rational interpolation for numerical solution of fractional differential equations ⋮ A Petrov-Galerkin spectral element method for fractional elliptic problems ⋮ Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions ⋮ A high-order scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equation ⋮ A fourth-order scheme for space fractional diffusion equations ⋮ An explicit form for higher order approximations of fractional derivatives ⋮ An exponentially convergent scheme in time for time fractional diffusion equations with non-smooth initial data ⋮ Spectral approximations for nonlinear fractional delay diffusion equations with smooth and nonsmooth solutions ⋮ Finite difference method for two-dimensional nonlinear time-fractional subdiffusion equation ⋮ Fractional backward differential formulas for the distributed-order differential equation with time delay ⋮ Fully discrete spectral methods for solving time fractional nonlinear sine-Gordon equation with smooth and non-smooth solutions ⋮ Adaptive discretization of fractional order viscoelasticity using sparse time history ⋮ Pitfalls in fast numerical solvers for fractional differential equations ⋮ An implicit MLS meshless method for 2-D time dependent fractional diffusion-wave equation ⋮ On explicit stability conditions for a linear fractional difference system ⋮ High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations. II ⋮ Discrete-space time-fractional processes ⋮ A formally second-order BDF finite difference scheme for the integro-differential equations with the multi-term kernels ⋮ High order WSGL difference operators combined with Sinc-Galerkin method for time fractional Schrödinger equation ⋮ A self-singularity-capturing scheme for fractional differential equations ⋮ Finite difference schemes for time-fractional Schrödinger equations via fractional linear multistep method ⋮ Gauss-Lobatto-Legendre-Birkhoff pseudospectral scheme for the time fractional reaction–diffusion equation with Neumann boundary conditions ⋮ Analysis of Legendre pseudospectral approximations for nonlinear time fractional diffusion-wave equations ⋮ Quasi-wavelet based numerical method for fourth-order partial integro-differential equations with a weakly singular kernel ⋮ Identification of Potential in Diffusion Equations from Terminal Observation: Analysis and Discrete Approximation ⋮ Backward Diffusion-Wave Problem: Stability, Regularization, and Approximation ⋮ Finite difference approximations and dynamics simulations for the Lévy Fractional Klein-Kramers equation ⋮ Stability Analysis of Linear Time‐Invariant Distributed‐Order Systems ⋮ A Generalized Spectral Collocation Method with Tunable Accuracy for Variable-Order Fractional Differential Equations ⋮ A generalized quasi-boundary value method for recovering a source in a fractional diffusion-wave equation ⋮ NUMERICAL SIMULATIONS FOR THE VARIABLE ORDER TWO-DIMENSIONAL REACTION SUB-DIFFUSION EQUATION: LINEAR AND NONLINEAR ⋮ Unnamed Item ⋮ An extrapolated finite difference method for two-dimensional fractional boundary value problems with non-smooth solution ⋮ Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions ⋮ Julia Sets and Their Control in a Three-Dimensional Discrete Fractional-Order Financial Model ⋮ The Product Midpoint Rule for Abel-Type Integral Equations of the First Kind with Perturbed Data ⋮ Two Fully Discrete Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data ⋮ Time Discretization of a Tempered Fractional Feynman--Kac Equation with Measure Data ⋮ Approximate solution of Bagley-Torvik equations with variable coefficients and three-point boundary-value conditions ⋮ A posteriori error analysis for approximations of time-fractional subdiffusion problems ⋮ Two-grid methods for nonlinear time fractional diffusion equations by \(L 1\)-Galerkin FEM ⋮ A novel alternating-direction implicit spectral Galerkin method for a multi-term time-space fractional diffusion equation in three dimensions ⋮ Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the sinc-collocation method based on the double exponential transformation ⋮ Efficient difference schemes for the Caputo-tempered fractional diffusion equations based on polynomial interpolation ⋮ A kind of generalized backward differentiation formulae for solving fractional differential equations ⋮ High-order finite difference method based on linear barycentric rational interpolation for Caputo type sub-diffusion equation ⋮ Collocation approach based on an extended cubic \(B\)-spline for a second-order Volterra partial integrodifferential equation ⋮ Fractional BDF methods for solving fractional differential matrix equations ⋮ Explicit and implicit schemes for fractional–order Hantavirus model ⋮ Correction of high-order \(L_k\) approximation for subdiffusion ⋮ Efficient high-order exponential time differencing methods for nonlinear fractional differential models ⋮ A fractional Adams-Simpson-type method for nonlinear fractional ordinary differential equations with non-smooth data ⋮ Correction of High-Order BDF Convolution Quadrature for Fractional Evolution Equations ⋮ Numerical Analysis of Nonlinear Subdiffusion Equations ⋮ An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data ⋮ A Generalized Spectral Collocation Method with Tunable Accuracy for Fractional Differential Equations with End-Point Singularities ⋮ Second Order Shifted Approximations for the First Derivative ⋮ The efficient ADI Galerkin finite element methods for the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics ⋮ L1 scheme for solving an inverse problem subject to a fractional diffusion equation ⋮ Compact Crank–Nicolson and Du Fort–Frankel Method for the Solution of the Time Fractional Diffusion Equation ⋮ A numerical study by using the Chebyshev collocation method for a problem of biological invasion: Fractional Fisher equation ⋮ Time discretization and convergence to superdiffusion equations via Poisson distribution ⋮ On the nonreflecting boundary operators for the general two dimensional Schrödinger equation ⋮ High-order Time Stepping Schemes for Semilinear Subdiffusion Equations ⋮ On the stability analysis of weighted average finite difference methods for fractional wave equation ⋮ Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise ⋮ Efficient Preconditioner Updates for Semilinear Space–Time Fractional Reaction–Diffusion Equations ⋮ Fast and Parallel Runge--Kutta Approximation of Fractional Evolution Equations ⋮ A Hilbert Space Approach to Fractional Difference Equations ⋮ Sparse Approximations of Fractional Matérn Fields ⋮ Sharp Error Estimate of the Nonuniform L1 Formula for Linear Reaction-Subdiffusion Equations ⋮ A New Family of Difference Schemes for Space Fractional Advection Diffusion Equation ⋮ Numerical Solution of the Coupled System of Nonlinear Fractional Ordinary Differential Equations ⋮ Spectral Collocation Method for a Class of Integro-Differential Equations with Erdelyi-Kober Fractional Operator ⋮ Numerical Schemes for Time-Space Fractional Vibration Equations ⋮ FINITE DIFFERENCE METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS ⋮ HOW TO APPROXIMATE THE FRACTIONAL DERIVATIVE OF ORDER 1 < α ≤ 2 ⋮ The Finite Difference Methods for Fractional Ordinary Differential Equations ⋮ A Parallel-in-Time Algorithm for High-Order BDF Methods for Diffusion and Subdiffusion Equations ⋮ Low Earth-orbit satellite attitude stabilization with fractional regulators ⋮ On linear stability of predictor–corrector algorithms for fractional differential equations ⋮ A New Class of Semi-Implicit Methods with Linear Complexity for Nonlinear Fractional Differential Equations ⋮ Numerical simulation of a point-source initiated flame ball with heat losses ⋮ A High Order Finite Difference Method for Tempered Fractional Diffusion Equations with Applications to the CGMY Model ⋮ A Fast High Order Method for the Time-Fractional Diffusion Equation ⋮ Finite difference schemes for two-dimensional time-space fractional differential equations ⋮ Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data ⋮ Some temporal second order difference schemes for fractional wave equations ⋮ Convolution quadrature time discretization of fractional diffusion-wave equations ⋮ Unnamed Item ⋮ Numerical analysis of backward subdiffusion problems ⋮ Solving spatial-fractional partial differential diffusion equations by spectral method ⋮ Diffusive Approximation of a Time-Fractional Burger's Equation in Nonlinear Acoustics ⋮ A unified difference-spectral method for time–space fractional diffusion equations ⋮ Finite difference method for Riesz space fractional diffusion equations with delay and a nonlinear source term ⋮ Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions ⋮ Finite difference methods and their physical constraints for the fractional klein-kramers equation ⋮ Efficient Multistep Methods for Tempered Fractional Calculus: Algorithms and Simulations ⋮ A Discretization of Caputo Derivatives with Application to Time Fractional SDEs and Gradient Flows ⋮ The G^α -scheme for Approximation of Fractional Derivatives: Application to the Dynamics of Dissipative Systems ⋮ Numerical approximation of stochastic time-fractional diffusion ⋮ FRACTIONAL DUFFING'S EQUATION AND GEOMETRICAL RESONANCE ⋮ Unnamed Item ⋮ Spectral approximations to the fractional integral and derivative ⋮ Discretized fractional substantial calculus ⋮ Efficient Numerical Solution of the Multi-Term Time Fractional Diffusion-Wave Equation ⋮ Design of a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems ⋮ On the Construction and Properties of $m$-step Methods for FDEs ⋮ High Order Algorithms for the Fractional Substantial Diffusion Equation with Truncated Lévy Flights ⋮ Optimal Collocation Nodes for Fractional Derivative Operators ⋮ NUMERICAL SOLUTION OF A MODEL FOR TURBULENT DIFFUSION ⋮ Time-Splitting Schemes for Fractional Differential Equations I: Smooth Solutions ⋮ Uniform $l^{1} $ convergence in the Crank-Nicolson method of a linear integro-differential equation for viscoelastic rods and plates ⋮ Petrov--Galerkin and Spectral Collocation Methods for Distributed Order Differential Equations ⋮ Numerical methods for fractional partial differential equations ⋮ The Poisson distribution, abstract fractional difference equations, and stability ⋮ A backward Euler difference scheme for the integro-differential equations with the multi-term kernels ⋮ A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces ⋮ An alternating direction implicit Legendre spectral method for simulating a 2D multi-term time-fractional Oldroyd-B fluid type diffusion equation ⋮ Constructions of second order approximations of the Caputo fractional derivative ⋮ Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes ⋮ Fast Crank-Nicolson compact difference scheme for the two-dimensional time-fractional mobile/immobile transport equation ⋮ A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations ⋮ Wave-heat coupling in one-dimensional unbounded domains: artificial boundary conditions and an optimized Schwarz method ⋮ Numerical analysis for compact difference scheme of fractional viscoelastic beam vibration models ⋮ A novel discrete fractional Grönwall-type inequality and its application in pointwise-in-time error estimates ⋮ A second-order \(L2\)-\(1_\sigma\) Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients ⋮ Fast method and convergence analysis for the magnetohydrodynamic flow and heat transfer of fractional Maxwell fluid ⋮ A predictor-corrector compact finite difference scheme for a nonlinear partial integro-differential equation ⋮ Global superconvergence analysis of nonconforming finite element method for time fractional reaction-diffusion problem with anisotropic data ⋮ The convergence rate for difference approximations to fractional boundary value problems ⋮ Two time-stepping schemes for sub-diffusion equations with singular source terms ⋮ Solving time-fractional advection-dispersion equation by variable weights particle tracking method ⋮ An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data ⋮ Benchmark problems for Caputo fractional-order ordinary differential equations ⋮ Approximating the Basset force by optimizing the method of van Hinsberg et al. ⋮ The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations ⋮ An improved algorithm based on finite difference schemes for fractional boundary value problems with nonsmooth solution ⋮ Nonstandard finite difference schemes for a fractional-order Brusselator system ⋮ High-order numerical algorithms for Riesz derivatives via constructing new generating functions ⋮ New Laguerre filter approximators to the Grünwald-Letnikov fractional difference ⋮ Optimal error estimate of the Legendre spectral approximation for space-fractional reaction-advection-diffusion equation ⋮ Finite difference approach for variable order reaction-subdiffusion equations ⋮ Efficient high order algorithms for fractional integrals and fractional differential equations ⋮ Superconvergence of FEM for distributed order time fractional variable coefficient diffusion equations ⋮ High-order algorithms for Riesz derivative and their applications. I. ⋮ Reduced spline method based on a proper orthogonal decomposition technique for fractional sub-diffusion equations ⋮ The unified theory of shifted convolution quadrature for fractional calculus ⋮ A Crank-Nicolson-type finite-difference scheme and its algorithm implementation for a nonlinear partial integro-differential equation arising from viscoelasticity ⋮ Numerical solution of the time-fractional Navier-Stokes equations for incompressible flow in a lid-driven cavity ⋮ Analysis of BDF2 finite difference method for fourth-order integro-differential equation ⋮ Necessity of introducing non-integer shifted parameters by constructing high accuracy finite difference algorithms for a two-sided space-fractional advection-diffusion model ⋮ Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview ⋮ Time-local discretization of fractional and related diffusive operators using Gaussian quadrature with applications ⋮ Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term ⋮ Asymptotic expansions and approximations for the Caputo derivative ⋮ Finite element method for fractional parabolic integro-differential equations with smooth and nonsmooth initial data ⋮ Higher order methods for fractional differential equation based on fractional backward differentiation formula of order three ⋮ Hopf bifurcation, chaos control and synchronization of a chaotic fractional-order system with chaos entanglement function ⋮ Generalized Adams method for solving fractional delay differential equations ⋮ Galerkin FEM for a time-fractional Oldroyd-B fluid problem ⋮ A source identification problem in a time-fractional wave equation with a dynamical boundary condition ⋮ Numerical studies for fractional-order logistic differential equation with two different delays ⋮ Solution of singular integral equations via Riemann-Liouville fractional integrals ⋮ Alternating direction implicit difference scheme for the multi-term time-fractional integro-differential equation with a weakly singular kernel ⋮ Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order ⋮ Discrete and continuous fractional persistence problems -- the positivity property and applications ⋮ Convergence analysis of the anisotropic FEM for 2D time fractional variable coefficient diffusion equations on graded meshes ⋮ Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions ⋮ Numerical methods for multi-term fractional (arbitrary) orders differential equations ⋮ Galerkin spectral method for nonlinear time fractional cable equation with smooth and nonsmooth solutions ⋮ Efficient solution of time-fractional differential equations with a new adaptive multi-term discretization of the generalized Caputo-Dzherbashyan derivative ⋮ An explicit fourth-order energy-preserving scheme for Riesz space fractional nonlinear wave equations ⋮ A spectral deferred correction method for fractional differential equations ⋮ Fractional diffusion on bounded domains ⋮ Fractional convolution quadrature based on generalized Adams methods ⋮ Exponential quadrature rules for linear fractional differential equations ⋮ Stability and convergence of finite difference method for two-sided space-fractional diffusion equations ⋮ Error analysis of nonlinear time fractional mobile/immobile advection-diffusion equation with weakly singular solutions ⋮ A thermodynamically consistent fractional visco-elasto-plastic model with memory-dependent damage for anomalous materials ⋮ Fractional damping through restricted calculus of variations ⋮ The accuracy and stability of an implicit solution method for the fractional diffusion equation ⋮ A spectral collocation method for fractional chemical clock reactions ⋮ A class of efficient time-stepping methods for multi-term time-fractional reaction-diffusion-wave equations ⋮ An efficient second order stabilized scheme for the two dimensional time fractional Allen-Cahn equation ⋮ Approximation methods for the distributed order calculus using the convolution quadrature ⋮ Backward difference formulae: the energy technique for subdiffusion equation ⋮ On failed methods of fractional differential equations: the case of multi-step generalized differential transform method ⋮ A novel high order compact ADI scheme for two dimensional fractional integro-differential equations ⋮ Stability and convergence of 3-point WSGD schemes for two-sided space fractional advection-diffusion equations with variable coefficients ⋮ Numerical solutions of fractional wave equations using an efficient class of FDM based on the Hermite formula ⋮ Exact and numerical solutions of time-fractional advection-diffusion equation with a nonlinear source term by means of the Lie symmetries ⋮ Asymptotic behavior of mild solutions for nonlinear fractional difference equations ⋮ 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 ⋮ Numerical simulation of fractional evolution model arising in viscoelastic mechanics ⋮ Two \(L1\) schemes on graded meshes for fractional Feynman-Kac equation ⋮ An effective finite element method with singularity reconstruction for fractional convection-diffusion equation ⋮ Two linearized schemes for time fractional nonlinear wave equations with fourth-order derivative ⋮ The formally second-order BDF ADI difference/compact difference scheme for the nonlocal evolution problem in three-dimensional space ⋮ Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space ⋮ Explicit representation of discrete fractional resolvent families in Banach spaces ⋮ Fast ADI difference/compact difference schemes for the nonlocal evolution equation with weakly singular kernels in three dimensions ⋮ Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media ⋮ Fast BDF2 ADI methods for the multi-dimensional tempered fractional integrodifferential equation of parabolic type ⋮ An implicit difference scheme for the fourth-order nonlinear non-local PIDEs with a weakly singular kernel ⋮ A localized meshless technique for solving 2D nonlinear integro-differential equation with multi-term kernels ⋮ ADI Galerkin finite element scheme for the two-dimensional semilinear partial intergro-differential equation with a weakly singular kernel ⋮ Fractional systems: theoretical foundations ⋮ A computational algorithm for simulating fractional order relaxation-oscillation equation ⋮ The global stability investigation of the mathematical design of a fractional-order HBV infection ⋮ The numerical solution of linear multi-term fractional differential equations: Systems of equations ⋮ A sharp discrete convolution sum estimate ⋮ Two fast numerical methods for a generalized Oldroyd-B fluid model ⋮ Upper and lower estimates for the separation of solutions to fractional differential equations ⋮ Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations ⋮ A fractional-order dependent collocation method with graded mesh for impulsive fractional-order system