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research-article

Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions

Authors:

Zhongqiang Zhang,

George Em KarniadakisAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 38, Issue 5

Pages A3070 - A3093

https://doi.org/10.1137/16M1070323

Published: 01 January 2016 Publication History

Abstract

We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order $0<\beta<1$. From the known structure of the nonsmooth solution and by introducing corresponding correction terms, we can obtain uniformly second-order accuracy from these schemes. We prove the convergence and linear stability of the proposed schemes. Numerical examples illustrate the flexibility and efficiency of the IMEX schemes and show that they are effective for nonlinear and multirate fractional differential systems as well as multiterm fractional differential systems with nonsmooth solutions.

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Cited By

Wang YXie B(2023)A fractional Adams–Simpson-type method for nonlinear fractional ordinary differential equations with non-smooth dataBIT10.1007/s10543-023-00952-463:1Online publication date: 30-Jan-2023
https://dl.acm.org/doi/10.1007/s10543-023-00952-4
Sarumi IFurati KMustapha KKhaliq A(2022)Efficient high-order exponential time differencing methods for nonlinear fractional differential modelsNumerical Algorithms10.1007/s11075-022-01339-292:2(1261-1288)Online publication date: 27-Jul-2022
https://dl.acm.org/doi/10.1007/s11075-022-01339-2
Fu TDu CXu Y(2022)An Effective Finite Element Method with Shifted Fractional Powers Bases for Fractional Boundary Value ProblemsJournal of Scientific Computing10.1007/s10915-022-01854-392:1Online publication date: 1-Jul-2022
https://dl.acm.org/doi/10.1007/s10915-022-01854-3

Index Terms

Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
      2. Partial differential equations
    2. Numerical analysis
      1. Numerical differentiation
2. Theory of computation
  1. Design and analysis of algorithms

Index terms have been assigned to the content through auto-classification.

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Published In

cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 38, Issue 5

^† Special Section on Two Themes: CSE Software and Big Data in CSE

2016

1789 pages

ISSN:1064-8275

DOI:10.1137/sjoce3.38.5

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© 2016, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2016

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Cited By

Wang YXie B(2023)A fractional Adams–Simpson-type method for nonlinear fractional ordinary differential equations with non-smooth dataBIT10.1007/s10543-023-00952-463:1Online publication date: 30-Jan-2023
https://dl.acm.org/doi/10.1007/s10543-023-00952-4
Sarumi IFurati KMustapha KKhaliq A(2022)Efficient high-order exponential time differencing methods for nonlinear fractional differential modelsNumerical Algorithms10.1007/s11075-022-01339-292:2(1261-1288)Online publication date: 27-Jul-2022
https://dl.acm.org/doi/10.1007/s11075-022-01339-2
Fu TDu CXu Y(2022)An Effective Finite Element Method with Shifted Fractional Powers Bases for Fractional Boundary Value ProblemsJournal of Scientific Computing10.1007/s10915-022-01854-392:1Online publication date: 1-Jul-2022
https://dl.acm.org/doi/10.1007/s10915-022-01854-3

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