Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality

Thabet Abdeljawad, Qasem M. Al-Mdallal

Research output: Contribution to journalArticlepeer-review

90 Citations (Scopus)

Abstract

In this article, we studied the Caputo and Riemann–Liouville type discrete fractional difference initial value problems with discrete Mittag-Leffler kernels. The existence and uniqueness of the solution is proved by using Banach contraction principle. The linear type equations are used to prove new discrete fractional versions of the Gronwall's inequality. The nabla discrete Laplace transform is used to obtain solution representations. The proven Gronwall's inequality under a new defined α-Lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions. Numerical examples are discussed to demonstrate the reliability of the theoretical results.

Original languageEnglish
Pages (from-to)218-230
Number of pages13
JournalJournal of Computational and Applied Mathematics
Volume339
DOIs
Publication statusPublished - Sept 2018

Keywords

  • Discrete ABR and ABC fractional derivatives
  • Discrete Laplace transform
  • Discrete Mittag-Leffler function
  • Discrete fractional sum
  • Gronwall's inequality

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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