Computational challenge of fractional differential equations and the potential solutions: a survey
type: Item
instance of: scholarly article
title: Computational challenge of fractional differential equations and the potential solutions: a survey
zbMATH Open document ID: 1394.65072
DOI: 10.1155/2015/258265
author: Chunye Gong
author: Yuewen Jiang
author: Jie Liu
author: Wei-min Bao
author: Guo-Jian Tang
published in: Mathematical Problems in Engineering
publication date: 2018-08-27T00:00:00Z
review text: Summary: We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations are \(O(N^2M)\), \(O(NM^2)\), and \(O(NM(M+N)\)) compared with \(O(MN)\) for the classical partial differential equations with finite difference methods, where \(M\), \(N\) are the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator), short memory principle, fast Fourier transform (FFT) based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 6925878
Wikidata QID: Q59117749
describes a project that uses: mlrnd
describes a project that uses: JASMIN
describes a project that uses: FODE
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2015/258265
OpenAlex ID: W2083378485
cites work: On fractional filtering versus conventional filtering in economics
cites work: Q4233638
cites work: Combining differential evolution and particle swarm optimization to tune and realize fractional-order controllers
cites work: Pseudo-state feedback stabilization of commensurate fractional order systems
cites work: Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation
cites work: High-Speed Digital Realizations of Fractional Operators in the Delta Domain
cites work: An efficient iteration method for Toeplitz-plus-band triangular systems generated from fractional ordinary differential equation
cites work: A Lyapunov approach to the stability of fractional differential equations
cites work: LMI stability conditions for fractional order systems
cites work: On the Laguerre Rational Approximation to Fractional Discrete Derivative and Integral Operators
cites work: Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain
cites work: Approximate analytical solution for nonlinear system of fractional differential equations by BPS operational matrices
cites work: Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method
cites work: Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations
cites work: A novel implicit finite difference method for the one-dimensional fractional percolation equation
cites work: Parallel fully implicit two‐grid methods for distributed control of unsteady incompressible flows
cites work: A parallel version of QMRCGSTAB method for large linear systems in distributed parallel environments
cites work: Recent history of fractional calculus
cites work: Q3087388
cites work: Q4000098
cites work: Compact alternating direction implicit method for two-dimensional time fractional diffusion equation
cites work: Time-fractional heat equations and negative absolute temperatures
cites work: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives
cites work: A finite element method for the multiterm time-space Riesz fractional advection-diffusion equations in finite domain
cites work: Finite difference/spectral approximations for the time-fractional diffusion equation
cites work: A second-order accurate numerical method for the two-dimensional fractional diffusion equation
cites work: Stability of a numerical method for a space-time-fractional telegraph equation
cites work: Boundary integral operator for the fractional Laplace equation in a bounded Lipschitz domain
cites work: Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton's optimal portfolio
cites work: Fractional wave equations with attenuation
cites work: Finite Element Method for the Space and Time Fractional Fokker–Planck Equation
cites work: Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method
cites work: Fractional Euler-Lagrange equations revisited
cites work: Fractional kinetic equations: solutions and applications
cites work: A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates
cites work: The fractional Boltzmann transport equation
cites work: An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives
cites work: A predictor-corrector approach for the numerical solution of fractional differential equations
cites work: Compact finite difference method for the fractional diffusion equation
cites work: A finite element method for time fractional partial differential equations
cites work: Analysis for one-dimensional time-fractional Tricomi-type equations by LDG methods
cites work: Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation
cites work: Novel Numerical Methods for Solving the Time-Space Fractional Diffusion Equation in Two Dimensions
cites work: Stability and convergence of a finite volume method for the space fractional advection-dispersion equation
cites work: An explicit and numerical solutions of the fractional KdV equation
cites work: Meshless simulations of the two-dimensional fractional-time convection-diffusion-reaction equations
cites work: A RBF meshless approach for modeling a fractal mobile/immobile transport model
cites work: An implicit RBF meshless approach for time fractional diffusion equations
cites work: A family of Adams exponential integrators for fractional linear systems
cites work: Generalized exponential time differencing methods for fractional order problems
cites work: A second-order accurate numerical approximation for the fractional diffusion equation
cites work: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics
cites work: Q3141902
cites work: The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation
cites work: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative
cites work: Q3413225
cites work: A parallel algorithm for the Riesz fractional reaction-diffusion equation with explicit finite difference method

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A second order finite difference-spectral method for space fractional diffusion equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A GPU parallelized spectral method for elliptic equations in rectangular domains

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Collaborating CPU and GPU for large-scale high-order CFD simulations with complex grids on the TianHe-1A supercomputer

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A parallel domain decomposition method for 3D unsteady incompressible flows at high Reynolds number

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Parallel domain decomposition methods with mixed order discretization for fully implicit solution of tracer transport problems on the cubed-sphere

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A Fully Implicit Domain Decomposition Algorithm for Shallow Water Equations on the Cubed-Sphere

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: An efficient parallel algorithm for the numerical solution of fractional differential equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Performance modeling of serial and parallel implementations of the fractional Adams-Bashforth-Moulton method

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Solving the Caputo fractional reaction-diffusion equation on GPU

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: The short memory principle for solving Abel differential equation of fractional order

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Short memory principle and a predictor-corrector approach for fractional differential equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: The numerical solution of fractional differential equations: speed versus accuracy

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A direct \(O(N \log ^{2} N)\) finite difference method for fractional diffusion equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: An \(O(N \log ^{2}N)\) alternating-direction finite difference method for two-dimensional fractional diffusion equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Multigrid method for fractional diffusion equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Finite element multigrid method for the boundary value problem of fractional advection dispersion equation

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A circulant preconditioner for fractional diffusion equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Preconditioned iterative methods for fractional diffusion equation

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: High order finite difference WENO schemes for fractional differential equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Quasi-compact finite difference schemes for space fractional diffusion equations

Computational challenge of fractional differential equations and the potential solutions: a survey (cites work)
cites work: Discretized Fractional Calculus

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: Numerical solution of fractional differential equations: a survey and a software tutorial
Similarity Score: 0.9115422
Recommender Run: Recommender Run 3

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: Computational solution of a fractional integro-differential equation
Similarity Score: 0.8959948
Recommender Run: Recommender Run 3

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: Fractional Partial Differential Equations and Their Numerical Solutions
Similarity Score: 0.89460737
Recommender Run: Recommender Run 3

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: Computational study on the dynamics of fractional order differential equations with applications
Similarity Score: 0.89163285
Recommender Run: Recommender Run 3

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: On the numerical solution of fractional partial differential equations
Similarity Score: 0.89096385
Recommender Run: Recommender Run 3

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: An efficient computational approach for linear and nonlinear fractional differential equations
Similarity Score: 0.889647
Recommender Run: Recommender Run 3

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: Q4428616
Similarity Score: 0.88745284
Recommender Run: Recommender Run 3

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: Q2991117
Similarity Score: 0.8834689
Recommender Run: Recommender Run 3

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: Q3013388
Similarity Score: 0.88307756
Recommender Run: Recommender Run 3

Computational challenge of fractional differential equations and the potential solutions: a survey (Recommended article)
Recommended article: Numerical comparison of methods for solving fractional differential-algebraic equations (FDAEs)
Similarity Score: 0.88154876
Recommender Run: Recommender Run 3