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<p>This paper develops a high-order numerical scheme for the variable-order time-fractional generalized Burgers equation (VOTFGBE). The temporal discretization is based on the <i>L</i>2–<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1_{\sigma }\)</EquationSource> </InlineEquation> formula for the Caputo fractional derivative, providing second-order accuracy, while spatial derivatives are approximated using a Radial Basis Function–Hermite Finite Difference (RBF-HFD) method with fourth-order convergence. A rigorous Fourier analysis is carried out to establish the unconditional stability of the fully discrete scheme. The proposed method is validated through a series of numerical experiments and benchmark comparisons with existing techniques. The results show that the proposed scheme possesses satisfactory numerical accuracy, and its computational performance is reflected by measured CPU runtime.</p>

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A numerical study on the variable-order time-fractional generalized Burgers equation via the RBF-HFD method

  • 期刊论文

  • 发表日期：

    2026年6月29日

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  • Ziyu Guo,
  • Kaysar Rahman,
  • Shahid Hussain,
  • Junping Guan

摘要

This paper develops a high-order numerical scheme for the variable-order time-fractional generalized Burgers equation (VOTFGBE). The temporal discretization is based on the L2– \(1_{\sigma }\) formula for the Caputo fractional derivative, providing second-order accuracy, while spatial derivatives are approximated using a Radial Basis Function–Hermite Finite Difference (RBF-HFD) method with fourth-order convergence. A rigorous Fourier analysis is carried out to establish the unconditional stability of the fully discrete scheme. The proposed method is validated through a series of numerical experiments and benchmark comparisons with existing techniques. The results show that the proposed scheme possesses satisfactory numerical accuracy, and its computational performance is reflected by measured CPU runtime.