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A discretization of Caputo derivatives with application to time fractional SDEs and gradient flows

Publication ,  Journal Article
Li, L; Liu, JG
Published in: SIAM Journal on Numerical Analysis
January 1, 2019

We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of the corresponding implicit schemes, are proved by its linkage to Volterra integrals with completely monotone kernels. We then apply the backward scheme corresponding to this discretization to two time fractional dissipative problems, and these implicit schemes are helpful for the analysis of the corresponding problems. In particular, we show that the overdamped generalized Langevin equation with fractional noise has a unique limiting measure for strongly convex potentials and we establish the convergence of numerical solutions to the strong solutions of time fractional gradient flows. The proposed scheme and schemes derived using the same philosophy can be useful for many other applications as well.

Duke Scholars

Published In

SIAM Journal on Numerical Analysis

DOI

ISSN

0036-1429

Publication Date

January 1, 2019

Volume

57

Issue

5

Start / End Page

2095 / 2120

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4903 Numerical and computational mathematics
  • 4901 Applied mathematics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

Citation

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Li, L., & Liu, J. G. (2019). A discretization of Caputo derivatives with application to time fractional SDEs and gradient flows. SIAM Journal on Numerical Analysis, 57(5), 2095–2120. https://doi.org/10.1137/19M123854X
Li, L., and J. G. Liu. “A discretization of Caputo derivatives with application to time fractional SDEs and gradient flows.” SIAM Journal on Numerical Analysis 57, no. 5 (January 1, 2019): 2095–2120. https://doi.org/10.1137/19M123854X.
Li L, Liu JG. A discretization of Caputo derivatives with application to time fractional SDEs and gradient flows. SIAM Journal on Numerical Analysis. 2019 Jan 1;57(5):2095–120.
Li, L., and J. G. Liu. “A discretization of Caputo derivatives with application to time fractional SDEs and gradient flows.” SIAM Journal on Numerical Analysis, vol. 57, no. 5, Jan. 2019, pp. 2095–120. Scopus, doi:10.1137/19M123854X.
Li L, Liu JG. A discretization of Caputo derivatives with application to time fractional SDEs and gradient flows. SIAM Journal on Numerical Analysis. 2019 Jan 1;57(5):2095–2120.

Published In

SIAM Journal on Numerical Analysis

DOI

ISSN

0036-1429

Publication Date

January 1, 2019

Volume

57

Issue

5

Start / End Page

2095 / 2120

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4903 Numerical and computational mathematics
  • 4901 Applied mathematics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics