The Galerkin finite element method for a multi-term time-fractional diffusion equation

DOI10.1016/j.jcp.2014.10.051zbMath1352.65350arXiv1401.8049OpenAlexW2081811871MaRDI QIDQ728813

Zhi Zhou, Yikan Liu, Bangti Jin, Raytcho D. Lazarov

Publication date: 20 December 2016

Published in: Journal of Computational Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1401.8049

zbMATH Keywords

finite element methoderror estimatesemidiscrete schemeCaputo derivativemulti-term time-fractional diffusion equation

Mathematics Subject Classification ID

Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs (65M60) Fractional partial differential equations (35R11)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Exponentially accurate spectral and spectral element methods for fractional ODEs
• Stability and convergence of an effective finite element method for multiterm fractional partial differential equations
• Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems
• Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation
• Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain
• Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations
• Numerical simulations of 2D fractional subdiffusion problems
• An operational method for solving fractional differential equations with the Caputo derivatives
• Boundary particle method for Laplace transformed time fractional diffusion equations
• Numerical methods for solving the multi-term time-fractional wave-diffusion equation
• Finite difference/spectral approximations for the time-fractional diffusion equation
• A fully discrete difference scheme for a diffusion-wave system
• Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H − s , 0 ≤ s ≤ 1
• Numerical identification of a Robin coefficient in parabolic problems
• An inverse problem for a one-dimensional time-fractional diffusion problem
• Alternating Direction Implicit Galerkin Finite Element Method for the Two-Dimensional Time Fractional Evolution Equation
• An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements
• Error Estimates of Crank–Nicolson-Type Difference Schemes for the Subdiffusion Equation
• A Space-Time Spectral Method for the Time Fractional Diffusion Equation
• Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations
• Superconvergence of a Discontinuous Galerkin Method for Fractional Diffusion and Wave Equations
• Parabolic Control Problems in Measure Spaces with Sparse Solutions
• Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation
• Numerical solution of multi‐term fractional differential equations
• Existence and Uniqueness of the Weak Solution of the Space-time Fractional Diffusion Equation and a Spectral Method Approximation
• Galerkin Finite Element Methods for Parabolic Problems

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