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<p>Tempered fractional derivatives generalize classical fractional operators by incorporating an exponential tempering factor, which enables more realistic modeling of nonlocal processes with finite memory. In particular, left-sided tempered fractional derivatives capture dynamics influenced by past states and are therefore well suited for initial value problems. However, achieving high accuracy is challenging because solutions of such problems typically exhibit mild singular behavior near the initial point, which significantly degrades the convergence of standard spectral schemes. To address this difficulty, we develop a new initial-point adaptive spectral method that enriches and refines the approximation space in a neighborhood of the initial singularity. The proposed approach attains high accuracy for both rational and irrational tempered fractional orders. We also provide a rigorous <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{2}_{\varkappa ^{\gamma -1,0,\nu }}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <msup> <mi>ϰ</mi> <mrow> <mi>γ</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>ν</mi> </mrow> </msup> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation>-error convergence analysis to establish the efficiency, and robustness of the method. Numerical experiments confirm the theoretical findings and demonstrate that the proposed framework offers a reliable tool for a broad class of left-sided tempered fractional differential equations.</p>

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A new adaptive spectral collocation method for tempered fractional differential equations with initial singularities

  • 期刊论文

  • 发表日期：

    2026年3月13日

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  • Mahmoud A. Zaky,
  • Eid H. Doha

摘要

Tempered fractional derivatives generalize classical fractional operators by incorporating an exponential tempering factor, which enables more realistic modeling of nonlocal processes with finite memory. In particular, left-sided tempered fractional derivatives capture dynamics influenced by past states and are therefore well suited for initial value problems. However, achieving high accuracy is challenging because solutions of such problems typically exhibit mild singular behavior near the initial point, which significantly degrades the convergence of standard spectral schemes. To address this difficulty, we develop a new initial-point adaptive spectral method that enriches and refines the approximation space in a neighborhood of the initial singularity. The proposed approach attains high accuracy for both rational and irrational tempered fractional orders. We also provide a rigorous \(L^{2}_{\varkappa ^{\gamma -1,0,\nu }}\) L ϰ γ - 1 , 0 , ν 2 -error convergence analysis to establish the efficiency, and robustness of the method. Numerical experiments confirm the theoretical findings and demonstrate that the proposed framework offers a reliable tool for a broad class of left-sided tempered fractional differential equations.