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Boundary problems of sequential fractional differential equations having a monomial coefficient.

Debao Yan1

  • 1School of Mathematics and Statistics, Heze University, Heze City, Shandong Province, 274000, PR China.

Heliyon
|September 16, 2024
PubMed
Summary

This study investigates nonlinear boundary value problems for sequential fractional-order differential equations. We establish existence conditions and analyze Ulam-Hyers stability for these complex mathematical models.

Keywords:
34A0834B1034B1534D20Boundary value problemExistence of solutionsMonomial coefficientSequential fractional differential equationStability of Ulam-Hyers

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Area of Science:

  • Mathematics
  • Applied Mathematics
  • Differential Equations

Background:

  • Fractional-order differential equations are increasingly used to model complex phenomena.
  • Nonlinear boundary value problems present significant analytical challenges.

Purpose of the Study:

  • To analyze sequential fractional-order differential equations with monomial coefficients.
  • To establish existence criteria for solutions to these nonlinear boundary problems.
  • To investigate the Ulam-Hyers stability of the solutions.

Main Methods:

  • Conversion of the differential equation problem into an equivalent integral equation.
  • Application of the contraction mapping principle.
  • Utilization of Krasnoselskii's fixed-point theorem.

Main Results:

  • Two distinct existence conditions for the solutions were derived.
  • The Ulam-Hyers stability of the solutions was successfully investigated.
  • A practical example demonstrated the applicability of the theoretical findings.

Conclusions:

  • The study provides a rigorous mathematical framework for analyzing a specific class of fractional-order differential equations.
  • The established existence and stability results contribute to the theoretical understanding of these models.
  • The presented illustration confirms the practical relevance of the developed methods.