Numerical solution via Laplace transforms of a fractional order evolution equation

DOI10.1216/JIE-2010-22-1-57zbMath1195.65122OpenAlexW1965557233MaRDI QIDQ973894

William McLean, Vidar Thomée

Publication date: 26 May 2010

Published in: Journal of Integral Equations and Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1216/jie-2010-22-1-57

zbMATH Keywords

convergenceresolventfinite elementsquadrature formulaLaplace transformationquadraturefractional order diffusion equationfractional order wave equationspatially semidiscrete approximation

Mathematics Subject Classification ID

Heat equation (35K05) Wave equation (35L05) Transform methods (e.g., integral transforms) applied to PDEs (35A22) Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs (65M60) Laplace transform (44A10) Method of lines for initial value and initial-boundary value problems involving PDEs (65M20) Fractional partial differential equations (35R11)

Related Items

Numerical approximation of Riccati fractional differential equation in the sense of Caputo-type fractional derivative ⋮ Fast numerical contour integral method for fractional diffusion equations ⋮ Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature ⋮ Numerical solution of multi-term time fractional wave diffusion equation using transform based local meshless method and quadrature ⋮ A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations ⋮ Parallel mixed FEM simulation of a class of single-phase models with non-local operators ⋮ On the numerical solution of Bagley-Torvik equation via the Laplace transform ⋮ Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation ⋮ ITERATIVE SOLUTION OF SHIFTED POSITIVE-DEFINITE LINEAR SYSTEMS ARISING IN A NUMERICAL METHOD FOR THE HEAT EQUATION BASED ON LAPLACE TRANSFORMATION AND QUADRATURE ⋮ On the numerical approximation of Volterra integro-differential equation using Laplace transform ⋮ Convergence analysis of iterative Laplace transform methods for the coupled PDEs from regime-switching option pricing ⋮ The lumped mass FEM for a time-fractional cable equation ⋮ Time-stepping error bounds for fractional diffusion problems with non-smooth initial data ⋮ A localized transform-based meshless method for solving time fractional wave-diffusion equation ⋮ Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel ⋮ Semidiscrete Finite Element Analysis of Time Fractional Parabolic Problems: A Unified Approach ⋮ A posteriori error analysis for a continuous space-time finite element method for a hyperbolic integro-differential equation ⋮ Optimal order finite element approximation for a hyperbolic integro-differential equation ⋮ Laplace inversion for the solution of an abstract heat equation without the forward transform of the source term ⋮ A hybridizable discontinuous Galerkin method for fractional diffusion problems ⋮ Galerkin type methods for semilinear time-fractional diffusion problems ⋮ Analyses of the contour integral method for time fractional normal-subdiffusion transport equation ⋮ Asymptotics and discretization of a weakly singular kernel: application to viscous flows in a network of thin tubes ⋮ Alternating direction implicit Galerkin methods for an evolution equation with a positive-type memory term ⋮ Second-order difference approximations for Volterra equations with the completely monotonic kernels ⋮ \(khp\)-adaptive spectral projection based discontinuous Galerkin method for the numerical solution of wave equations with memory ⋮ Numerical solution of distributed-order time-fractional diffusion-wave equations using Laplace transforms ⋮ Numerical modelling of advection diffusion equation using Chebyshev spectral collocation method and Laplace transform ⋮ Fast Laplace transform methods for free-boundary problems of fractional diffusion equations ⋮ Optimal error analysis of a FEM for fractional diffusion problems by energy arguments ⋮ L1 scheme for solving an inverse problem subject to a fractional diffusion equation ⋮ FINITE ELEMENT APPROXIMATION OF A TIME-FRACTIONAL DIFFUSION PROBLEM FOR A DOMAIN WITH A RE-ENTRANT CORNER ⋮ Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion ⋮ A transform based local RBF method for 2D linear PDE with Caputo-Fabrizio derivative ⋮ On the approximation of time-fractional telegraph equations using localized kernel-based method ⋮ Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions ⋮ Numerical approximation of fractional-order Volterra integrodifferential equation ⋮ Fast and Parallel Runge--Kutta Approximation of Fractional Evolution Equations ⋮ Analysis of a Time-Stepping Scheme for Time Fractional Diffusion Problems with Nonsmooth Data ⋮ An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid ⋮ Numerical solution of fractional order anomalous subdiffusion problems using radial kernels and transform ⋮ Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature ⋮ A finite difference method for an anomalous sub-diffusion equation, theory and applications ⋮ Regularity theory for time-fractional advection-diffusion-reaction equations ⋮ High‐order compact finite difference and laplace transform method for the solution of time‐fractional heat equations with dirchlet and neumann boundary conditions ⋮ Moving collocation methods for time fractional differential equations and simulation of blowup ⋮ Discontinuous Galerkin method for an evolution equation with a memory term of positive type ⋮ Temporally semidiscrete approximation of a Dirichlet boundary control for a fractional/normal evolution equation with a final observation ⋮ Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term ⋮ Analysis of a time-stepping discontinuous Galerkin method for fractional diffusion-wave equations with nonsmooth data ⋮ Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data ⋮ A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law ⋮ A Parallel-in-Time-and-Space HPC Framework for a Class of Fractional Evolution Equations ⋮ Numerical Approximation of Semilinear Subdiffusion Equations with Nonsmooth Initial Data ⋮ A spectral projection based method for the numerical solution of wave equations with memory ⋮ Finite element approximations for fractional evolution problems ⋮ Exponentially convergent method for the approximation of a differential equation with fractional derivative and unbounded operator coefficient in a Banach space ⋮ Approximations of the Mittag-Leffler operator function with exponential accuracy and their applications to solving evolutionary equations with fractional time derivative ⋮ Error estimation of a discontinuous Galerkin method for time fractional subdiffusion problems with nonsmooth data ⋮ Numerical methods for fractional partial differential equations ⋮ Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data

Uses Software

• SPECFUN

Cites Work

• Time discretization via Laplace transformation of an integro-differential equation of parabolic type
• A second-order accurate numerical method for a fractional wave equation
• Semigroups of linear operators and applications to partial differential equations
• The accuracy and stability of an implicit solution method for the fractional diffusion equation
• Time fractional diffusion: A discrete random walk approach
• On the numerical inversion of the Laplace transform of certain holomorphic mappings
• Discretization with variable time steps of an evolution equation with a positive-type memory term
• Fractional diffusion and wave equations
• A Spectral Order Method for Inverting Sectorial Laplace Transforms
• The Stability in L p and W p 1 of the L 2 -Projection onto Finite Element Function Spaces
• A Numerical Method for a Partial Integro-Differential Equation
• On the Quasi-Optimality in $L_\infty$ of the $\overset{\circ}{H}^1$-Projection into Finite Element Spaces*
• Numerical solution of an evolution equation with a positive-type memory term
• Algorithm 715: SPECFUN–a portable FORTRAN package of special function routines and test drivers
• Maximum-norm estimates for resolvents of elliptic finite element operators
• An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations
• A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature
• AnH¹-Galerkin Mixed Finite Element Method for an Evolution Equation with a Positive-Type Memory Term
• A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature
• Time discretization of an evolution equation via Laplace transforms
• Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term
• Optimizing Talbot’s Contours for the Inversion of the Laplace Transform
• Exponentially Convergent Algorithms for the Operator Exponential with Applications to Inhomogeneous Problems in Banach Spaces
• Generation of Analytic Semigroups by Strongly Elliptic Operators
• The random walk's guide to anomalous diffusion: A fractional dynamics approach