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<p>This paper is to investigate the existence and various types of stability for an implicit Pantograph fractional order system that uses the Hilfer-Hadamard derivative and emphasize the implications of fractional calculus on system behavior. To demonstrate the existence and uniqueness of solutions, we utilize the Banach fixed-point theorem. Additionally, we examine the stability characteristics, focusing on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2547_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\( \mathfrak{P}\)</EquationSource> </InlineEquation>-Mittag-Leffler functions through the application of the Grönwall-Bellman inequality. Utilizing numerical results for analysis, we construct stability criteria and present results in detailed tables and figures. Numerical simulations demonstrate these effects graphically, illustrating parameter sensitivities and stability regions. Overall, our outcomes contribute to a deeper understanding of stability in fractional order systems for further research and applications in engineering and applied mathematics. Finally, we compare the findings to highlight the differences and similarities in the stability characteristics observed across different approaches.</p>

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Exploring stability in the Hilfer-Hadamard fractional order systems: insights from numerical analysis and simulations

  • 期刊论文

  • 发表日期：

    2025年6月24日

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  • Safoura Rezaei Aderyani,
  • Reza Saadati,
  • Chenkuan Li

摘要

This paper is to investigate the existence and various types of stability for an implicit Pantograph fractional order system that uses the Hilfer-Hadamard derivative and emphasize the implications of fractional calculus on system behavior. To demonstrate the existence and uniqueness of solutions, we utilize the Banach fixed-point theorem. Additionally, we examine the stability characteristics, focusing on \( \mathfrak{P}\) -Mittag-Leffler functions through the application of the Grönwall-Bellman inequality. Utilizing numerical results for analysis, we construct stability criteria and present results in detailed tables and figures. Numerical simulations demonstrate these effects graphically, illustrating parameter sensitivities and stability regions. Overall, our outcomes contribute to a deeper understanding of stability in fractional order systems for further research and applications in engineering and applied mathematics. Finally, we compare the findings to highlight the differences and similarities in the stability characteristics observed across different approaches.