Volume 39, Issue 1
Boundary Value Methods for Caputo Fractional Differential Equations

Yongtao Zhou, Chengjian Zhang & Huiru Wang

J. Comp. Math., 39 (2021), pp. 108-129.

Published online: 2020-09

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  • Abstract

This paper deals with the numerical computation and analysis for Caputo fractional differential equations (CFDEs). By combining the $p$-order boundary value methods (BVMs) and the $m$-th Lagrange interpolation, a type of extended BVMs for the CFDEs with $\gamma$-order ($0 <\gamma < 1$) Caputo derivatives are derived. The local stability, unique solvability and convergence of the methods are studied. It is proved under the suitable conditions that the convergence order of the numerical solutions can arrive at $\min\left\{p, m-\gamma+1\right\}$. In the end, by performing several numerical examples, the computational efficiency, accuracy and comparability of the methods are further illustrated.

  • Keywords

Fractional differential equations, Caputo derivatives, Boundary value methods, Local stability, Unique solvability, Convergence.

  • AMS Subject Headings

34A08, 34K28, 65L20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yongtaozh@126.com (Yongtao Zhou)

cjzhang@hust.edu.cn (Chengjian Zhang)

hrwang2013@163.com (Huiru Wang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-39-108, author = {Zhou , Yongtao and Zhang , Chengjian and Wang , Huiru }, title = {Boundary Value Methods for Caputo Fractional Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {39}, number = {1}, pages = {108--129}, abstract = {

This paper deals with the numerical computation and analysis for Caputo fractional differential equations (CFDEs). By combining the $p$-order boundary value methods (BVMs) and the $m$-th Lagrange interpolation, a type of extended BVMs for the CFDEs with $\gamma$-order ($0 <\gamma < 1$) Caputo derivatives are derived. The local stability, unique solvability and convergence of the methods are studied. It is proved under the suitable conditions that the convergence order of the numerical solutions can arrive at $\min\left\{p, m-\gamma+1\right\}$. In the end, by performing several numerical examples, the computational efficiency, accuracy and comparability of the methods are further illustrated.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1907-m2018-0252}, url = {http://global-sci.org/intro/article_detail/jcm/18280.html} }
TY - JOUR T1 - Boundary Value Methods for Caputo Fractional Differential Equations AU - Zhou , Yongtao AU - Zhang , Chengjian AU - Wang , Huiru JO - Journal of Computational Mathematics VL - 1 SP - 108 EP - 129 PY - 2020 DA - 2020/09 SN - 39 DO - http://doi.org/10.4208/jcm.1907-m2018-0252 UR - https://global-sci.org/intro/article_detail/jcm/18280.html KW - Fractional differential equations, Caputo derivatives, Boundary value methods, Local stability, Unique solvability, Convergence. AB -

This paper deals with the numerical computation and analysis for Caputo fractional differential equations (CFDEs). By combining the $p$-order boundary value methods (BVMs) and the $m$-th Lagrange interpolation, a type of extended BVMs for the CFDEs with $\gamma$-order ($0 <\gamma < 1$) Caputo derivatives are derived. The local stability, unique solvability and convergence of the methods are studied. It is proved under the suitable conditions that the convergence order of the numerical solutions can arrive at $\min\left\{p, m-\gamma+1\right\}$. In the end, by performing several numerical examples, the computational efficiency, accuracy and comparability of the methods are further illustrated.

Yongtao Zhou, Chengjian Zhang & Huiru Wang. (2020). Boundary Value Methods for Caputo Fractional Differential Equations. Journal of Computational Mathematics. 39 (1). 108-129. doi:10.4208/jcm.1907-m2018-0252
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