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<p>This work develops the extension of classical Smoluchowski’s coagulation equation into time-fractional framework. Thanks to the Variational Principle, the idea of a non-integer order coagulation model is studied under three kernels, namely constant, sum, and product, incorporating Caputo and Riesz-Caputo type fractional derivatives to embed long-range temporal memory into the aggregation dynamics. The proposed model captures non-Markovian kinetics, anomalous diffusion, and history-dependent gelation phenomena observed in viscoelastic colloids, polymerization processes, and astrophysical clustering. We employ the Laplace-accelerated homotopy perturbation method (LAHPM) to derive rapidly convergent series solutions that respect mass conservation. The convergence analysis and error estimation are provided to justify the novelty of the proposed scheme. Further, numerical validations, by taking three physical examples, confirm the high accuracy and robustness of our approach across a wide range of fractional orders <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((0 &lt; \alpha \leqslant 1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>&lt;</mo> <mi>α</mi> <mo>⩽</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and initial conditions. Numerical simulations indicate how the time-fractional order influences the dynamics of particle coagulation for each considered kernel.</p>

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Mathematical Study of the time-fractional non-linear Smoluchowski coagulation equation

  • 期刊论文

  • 发表日期：

    2026年5月1日

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  • Shantanu,
  • Shweta,
  • Bappa Ghosh,
  • Rajesh Kumar

摘要

This work develops the extension of classical Smoluchowski’s coagulation equation into time-fractional framework. Thanks to the Variational Principle, the idea of a non-integer order coagulation model is studied under three kernels, namely constant, sum, and product, incorporating Caputo and Riesz-Caputo type fractional derivatives to embed long-range temporal memory into the aggregation dynamics. The proposed model captures non-Markovian kinetics, anomalous diffusion, and history-dependent gelation phenomena observed in viscoelastic colloids, polymerization processes, and astrophysical clustering. We employ the Laplace-accelerated homotopy perturbation method (LAHPM) to derive rapidly convergent series solutions that respect mass conservation. The convergence analysis and error estimation are provided to justify the novelty of the proposed scheme. Further, numerical validations, by taking three physical examples, confirm the high accuracy and robustness of our approach across a wide range of fractional orders \((0 < \alpha \leqslant 1)\) ( 0 < α ⩽ 1 ) and initial conditions. Numerical simulations indicate how the time-fractional order influences the dynamics of particle coagulation for each considered kernel.