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<p>In this study, we utilize the Caputo-type fractal–fractional derivative to formulate a fractal–fractional model for proportional delay pantograph differential equations with variable coefficients. The proposed structure combines fractal–fractional calculus and proportional delay pantograph differential equations to more accurately simulate complex dynamical systems that are not effectively represented by ordinary integer-order techniques. A collocation scheme based on an operational matrix of fractal–fractional integration and fractional-order Gegenbauer wavelets is developed to transform the considered model into a solvable algebraic system. Some lemmas and theorems established the reliability of the method. The accuracy and effectiveness of the method are demonstrated through numerical examples. The outcomes are presented using tables and graphical representations, including numerical solutions and error analysis. When <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1863_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(\xi = \zeta = 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>=</mo> <mi>ζ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the approximate wavelet solutions obtained for the considered model show excellent consistency with the corresponding exact solutions. Numerical experiments confirm that the method yields accurate results, as demonstrated by lower error values and better resolution of solution behavior. This makes the approach highly suitable for modeling systems with complex delays and multiscale dynamics.</p>

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Fractional-order Gegenbauer wavelets method to solve the proportional delay pantograph differential equations with variable coefficients

  • 期刊论文

  • 发表日期：

    2025年9月24日

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  • Deepak Singh,
  • Sag Ram Verma,
  • Deepika Patel,
  • Ashish Rayal

摘要

In this study, we utilize the Caputo-type fractal–fractional derivative to formulate a fractal–fractional model for proportional delay pantograph differential equations with variable coefficients. The proposed structure combines fractal–fractional calculus and proportional delay pantograph differential equations to more accurately simulate complex dynamical systems that are not effectively represented by ordinary integer-order techniques. A collocation scheme based on an operational matrix of fractal–fractional integration and fractional-order Gegenbauer wavelets is developed to transform the considered model into a solvable algebraic system. Some lemmas and theorems established the reliability of the method. The accuracy and effectiveness of the method are demonstrated through numerical examples. The outcomes are presented using tables and graphical representations, including numerical solutions and error analysis. When \(\xi = \zeta = 1\) ξ = ζ = 1 , the approximate wavelet solutions obtained for the considered model show excellent consistency with the corresponding exact solutions. Numerical experiments confirm that the method yields accurate results, as demonstrated by lower error values and better resolution of solution behavior. This makes the approach highly suitable for modeling systems with complex delays and multiscale dynamics.