Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems
-
Joel A. Rosenfeld
and
Warren E. Dixon
Abstract
This manuscript presents a kernelized predictor corrector (KPC) method for fractional order initial value problems, which replaces linear interpolation with interpolation by a radial basis function (RBF) in a predictor-corrector scheme. Specifically, the class of Wendland RBFs is employed as the basis function for interpolation, and a convergence rate estimate is proved based on the smoothness of the particular kernel selected. Use of the Wendland RBFs over Mittag-Leffler kernel functions employed in a previous iteration of the kernelized method removes the problems encountered near the origin in [11]. This manuscript performs several numerical experiments, each with an exact known solution, and compares the results to another frequently used fractional Adams-Bashforth-Moulton method. Ultimately, it is demonstrated that the KPC method is more accurate but requires more computation time than the algorithm in [4].
Acknowledgments
The work of the authors was supported in part by NSF award 1509516 and Office of Naval Research grant N00014-13-1-0151. The first author was further supported by the Air Force Office of Scientific Research (AFOSR) under contract numbers FA9550-15-1-0258, FA9550-16-1-0246, FA9550-18-1-0122, and FA9550-21-1-0134 during the preparation of the manuscript. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsoring agencies.
References
[1] O. P. Agrawal and P. Kumar, Comparison of five numerical schemes for fractional differential equations. In: Advances in Fractional Calculus, Springer, 2007, 43–60.10.1007/978-1-4020-6042-7_4Search in Google Scholar
[2] N. Aronszajn, Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 (1950), 337–404, 10.2307/1990404.Search in Google Scholar
[3] S. De Marchi and R. Schaback, Stability of kernel-based interpolation. Adv. in Computational Math. 32 (2010), 155–161; 10.1007/s10444-008-9093-4.Search in Google Scholar
[4] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, 2010.10.1007/978-3-642-14574-2Search in Google Scholar
[5] K. Diethelm and N. J. Ford, Analysis of fractional differential equations. J. of Math. Anal. and Appl. 265 (2002), 229–248.10.1007/978-3-642-14574-2Search in Google Scholar
[6] K. Diethelm and N. J. Ford, Multi-order fractional differential equations and their numerical solution. Appl. Math. and Computation 154 (2004), 621–640.10.1016/S0096-3003(03)00739-2Search in Google Scholar
[7] K. Diethelm, N. J. Ford, and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlin. Dynamics 29 (2002), 3–22.10.1023/A:1016592219341Search in Google Scholar
[8] L. C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, Amer. Math. Soc., 1998.Search in Google Scholar
[9] G. E. Fasshauer, Meshfree Approximation Methods with Matlab (With CD-ROM), Vol. 6. World Scientific Publishing Company, 2007.10.1142/6437Search in Google Scholar
[10] R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics 6, No 2 (2018), 16 pp.; 10.3390/math6020016.Search in Google Scholar
[11] J. A. Rosenfeld and W. E. Dixon, Approximating the Caputo fractional derivative through the Mittag-Leffler reproducing kernel Hilbert space and the kernelized Adams–Bashforth–Moulton method. SIAM J. on Numer. Anal. 55 (2017), 1201–1217.10.1137/16M1056894Search in Google Scholar
[12] J. A. Rosenfeld, B. Russo, and W. E. Dixon, The Mittag-Leffler reproducing kernel Hilbert spaces of entire and analytic functions. J. of Math. Anal. and Appl. 463 (2018), 576–592.10.1016/j.jmaa.2018.03.036Search in Google Scholar
[13] I. Steinwart and A. Christmann, Support Vector Machines. Springer Sci. & Business Media, 2008.Search in Google Scholar
[14] H. Wendland, Scattered Data Approximation, Vol. 17. Cambridge University Press, 2004.10.1017/CBO9780511617539Search in Google Scholar
© 2021 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA–volume 24–6–2021)
- Research Paper
- Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces
- B-spline collocation discretizations of caputo and Riemann-Liouville derivatives: A matrix comparison
- A strong maximum principle for the fractional laplace equation with mixed boundary condition
- Difference between Riesz derivative and fractional Laplacian on the proper subset of ℝ
- Some properties of the fractal convolution of functions
- Continuous dependence of fuzzy mild solutions on parameters for IVP of fractional fuzzy evolution equations
- Discrete fractional boundary value problems and inequalities
- On the generalized fractional Laplacian
- Recent developments on the realization of fractance device
- Explicit representation of discrete fractional resolvent families in Banach spaces
- Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems
- Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator
- An inverse problem approach to determine possible memory length of fractional differential equations
Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA–volume 24–6–2021)
- Research Paper
- Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces
- B-spline collocation discretizations of caputo and Riemann-Liouville derivatives: A matrix comparison
- A strong maximum principle for the fractional laplace equation with mixed boundary condition
- Difference between Riesz derivative and fractional Laplacian on the proper subset of ℝ
- Some properties of the fractal convolution of functions
- Continuous dependence of fuzzy mild solutions on parameters for IVP of fractional fuzzy evolution equations
- Discrete fractional boundary value problems and inequalities
- On the generalized fractional Laplacian
- Recent developments on the realization of fractance device
- Explicit representation of discrete fractional resolvent families in Banach spaces
- Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems
- Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator
- An inverse problem approach to determine possible memory length of fractional differential equations
