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Volume 14, Issue 1
Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy

Junying Cao & Zhenning Cai

DOI: 10.4208/nmtma.OA-2020-0039

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 71-112.

Published online: 2020-10

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  • Abstract

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on each subinterval. The method is shown to be unconditionally stable, and for general nonlinear equations, the uniform sharp numerical order 3 − $ν$ can be rigorously proven for sufficiently smooth solutions at all time steps. The proof provides a general guide for proving the sharp order for higher-order schemes in the nonlinear case. Some numerical examples are given to validate our theoretical results.

  • Keywords

Caputo derivative, fractional ordinary differential equations, high-order numerical scheme, stability and convergence analysis.

  • AMS Subject Headings

65M06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-71, author = {Cao , Junying and Cai , Zhenning}, title = {Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {14}, number = {1}, pages = {71--112}, abstract = {

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on each subinterval. The method is shown to be unconditionally stable, and for general nonlinear equations, the uniform sharp numerical order 3 − $ν$ can be rigorously proven for sufficiently smooth solutions at all time steps. The proof provides a general guide for proving the sharp order for higher-order schemes in the nonlinear case. Some numerical examples are given to validate our theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0039}, url = {http://global-sci.org/intro/article_detail/nmtma/18328.html} }
TY - JOUR T1 - Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy AU - Cao , Junying AU - Cai , Zhenning JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 71 EP - 112 PY - 2020 DA - 2020/10 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0039 UR - https://global-sci.org/intro/article_detail/nmtma/18328.html KW - Caputo derivative, fractional ordinary differential equations, high-order numerical scheme, stability and convergence analysis. AB -

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on each subinterval. The method is shown to be unconditionally stable, and for general nonlinear equations, the uniform sharp numerical order 3 − $ν$ can be rigorously proven for sufficiently smooth solutions at all time steps. The proof provides a general guide for proving the sharp order for higher-order schemes in the nonlinear case. Some numerical examples are given to validate our theoretical results.

Cao , Junying and Cai , Zhenning. (2020). Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy. Numerical Mathematics: Theory, Methods and Applications. 14 (1). 71-112. doi:10.4208/nmtma.OA-2020-0039
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