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<p>This study aims to develop an operational matrix method of integration based on Haar wavelets to approximate the solutions of linear and nonlinear fractional differential equations. The fractional derivative is considered in the Atangana-Baleanu-Caputo sense. The operational matrix of fractional order integration is utilized to transform fractional differential equations into objective functions, which are then solved using simulated annealing optimization. In the context of error analysis, an upper bound for error is established to demonstrate the convergence of the proposed method. Illustrative examples are provided to demonstrate the simplicity, applicability, and effectiveness of the proposed method. The performance measures are the root mean square error, mean absolute deviation, error in the Nash–Sutcliffe efficiency, Theil’s inequality coefficient, and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({L}^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi>L</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({L}^{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi>L</mi> </mrow> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> norms, verifying the efficiency, and accuracy of the proposed method. The applicability of the proposed method was further confirmed by comparing it with exact solutions and other existing methods.</p>

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A Haar Wavelet Operational Matrix Method for Fractional Derivatives with Non-Singular Kernel

  • 期刊论文

  • 发表日期：

    2025年8月13日

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  • Najeeb Alam Khan,
  • Mumtaz Ali,
  • Asmat Ara,
  • Ibrahim Alraddadi,
  • Hijaz Ahmad

摘要

This study aims to develop an operational matrix method of integration based on Haar wavelets to approximate the solutions of linear and nonlinear fractional differential equations. The fractional derivative is considered in the Atangana-Baleanu-Caputo sense. The operational matrix of fractional order integration is utilized to transform fractional differential equations into objective functions, which are then solved using simulated annealing optimization. In the context of error analysis, an upper bound for error is established to demonstrate the convergence of the proposed method. Illustrative examples are provided to demonstrate the simplicity, applicability, and effectiveness of the proposed method. The performance measures are the root mean square error, mean absolute deviation, error in the Nash–Sutcliffe efficiency, Theil’s inequality coefficient, and \({L}^{2}\) L 2 and \({L}^{\infty }\) L ∞ norms, verifying the efficiency, and accuracy of the proposed method. The applicability of the proposed method was further confirmed by comparing it with exact solutions and other existing methods.