An explicit solution of linear conformable systems with variable coefficients


Aydın M.

Sigma Journal of Engineering and Natural Sciences, vol.42, no.6, pp.1806-1812, 2024 (ESCI) identifier

  • Publication Type: Article / Article
  • Volume: 42 Issue: 6
  • Publication Date: 2024
  • Doi Number: 10.14744/sigma.2024.00029
  • Journal Name: Sigma Journal of Engineering and Natural Sciences
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, Academic Search Premier, Directory of Open Access Journals
  • Page Numbers: pp.1806-1812
  • Keywords: Confromable Derivative, Fractional Differential System, Generalized Peano-Baker Series, State-Transition Matrix, Variation of Constants
  • Van Yüzüncü Yıl University Affiliated: Yes

Abstract

This paper is mainly devoted to exact solutions to the initial value problem for linear conformable systems with variable coefficients. The famous method known as the generalized Pea-no-Baker series, which inholds the conformable integral, is exploited to acquire the state-transition matrix. A representation of an exact solution in a closed interval for linear confromable systems with variable coefficients is determined with the help of this matrix. It is verified by showing that the determined exact solution satisfies the systems step by step. Moreover, another exact solution in the same closed interval is identified thanks to the method of variation of parameters. The existence and uniqueness of the second exact solution to the systems are provided by the Banach contraction mapping principle. This provides that the representations of the two solutions coincide although they are obtained by completely different approaches and they have completely different structures. A couple of examples are presented to exmplify the use of the findings.