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<p>This study successfully introduces a new class of fractional integro-differential equations using the <InlineEquation ID="IEq2"><EquationSource Format="MATHML"><math><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$(k, \psi )$</EquationSource></InlineEquation>-Hilfer fractional derivative (HFD). A hybrid computational technique for solving these equations is developed for the first time, by introducing the airfoil wavelets as an appropriate set of wavelet functions. To do this, the <InlineEquation ID="IEq3"><EquationSource Format="MATHML"><math><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$(k, \psi )$</EquationSource></InlineEquation>-Hilfer fractional integro-differential equations (HFIDEs) are reformulated as an equivalent fractional integral equation (FIE) by utilizing the properties of the <InlineEquation ID="IEq4"><EquationSource Format="MATHML"><math><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$(k, \psi )$</EquationSource></InlineEquation>-Riemann-Liouville fractional (RLF) integral and the <InlineEquation ID="IEq5"><EquationSource Format="MATHML"><math><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$(k, \psi )$</EquationSource></InlineEquation>-HFD. The unknown function is approximated by applying an activation function on a finite expansion of airfoil wavelets and unknown coefficients. The solution to the problem at hand is then found by solving a system whose components are algebraic equations by employing the Gauss-Legendre numerical integration (GLNI) and collocation scheme. Furthermore, the convergence of the proposed method is shown in Hilbert space. A series of computational examples are provided to illustrate the accuracy and effectiveness of the mentioned strategy.</p>

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Numerical simulation of \((k, \psi )\)-Hilfer fractional integro-differential equations using airfoil wavelets

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月31日

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  • Parisa Rahimkhani,
  • Seydi Battal Gazi Karakoc,
  • Morteza Rajabzadeh

摘要

This study successfully introduces a new class of fractional integro-differential equations using the (k,ψ)$(k, \psi )$-Hilfer fractional derivative (HFD). A hybrid computational technique for solving these equations is developed for the first time, by introducing the airfoil wavelets as an appropriate set of wavelet functions. To do this, the (k,ψ)$(k, \psi )$-Hilfer fractional integro-differential equations (HFIDEs) are reformulated as an equivalent fractional integral equation (FIE) by utilizing the properties of the (k,ψ)$(k, \psi )$-Riemann-Liouville fractional (RLF) integral and the (k,ψ)$(k, \psi )$-HFD. The unknown function is approximated by applying an activation function on a finite expansion of airfoil wavelets and unknown coefficients. The solution to the problem at hand is then found by solving a system whose components are algebraic equations by employing the Gauss-Legendre numerical integration (GLNI) and collocation scheme. Furthermore, the convergence of the proposed method is shown in Hilbert space. A series of computational examples are provided to illustrate the accuracy and effectiveness of the mentioned strategy.