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<p>This paper proposes and analyzes a class of time-fractional phase-field equations for describing shape transformation processes. Based on a modified Allen–Cahn model with the Caputo time-fractional derivative introduced, the equation aims to accurately characterize non-classical dynamic behaviors during phase transitions, particularly memory effects and anomalous diffusion mechanisms. For this nonlinear fractional evolution equation, we design an efficient and stable numerical algorithm: the L1 scheme based on the Caputo definition is employed for temporal discretization, a seven-point difference stencil is used to approximate the Laplace operator in the spatial direction, and an explicit prediction step is specially introduced to enhance the diagonal dominance and computational stability of the discrete system. In terms of theoretical analysis, the conditional stability of the algorithm under specific time step constraints is proved based on the discrete energy method, a modified energy dissipation law is established, and under enhanced regularity conditions on the solution, optimal convergence order estimates with respect to time and space steps are derived using the generalized discrete Gronwall inequality, confirming that the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation> error bound of the numerical solution is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(O\left( \Delta t^{2-\beta }+h^2\right) \)</EquationSource> </InlineEquation>. In the numerical experiments section, the influence of the fractional order parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\beta \)</EquationSource> </InlineEquation> on the evolution dynamics is systematically investigated through four typical cases. The numerical results show that a smaller fractional order leads to a stronger memory effect and faster movement of the phase interface. This behavior differs from classical integer-order models. The study provides mathematical analysis and convergence results and suggests new methods to control microstructure evolution and shape changes.</p>

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A time-fractional phase-field equation for shape transformation

  • 期刊论文

  • 发表日期：

    2026年4月28日

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  • Jilong He,
  • Junseok Kim

摘要

This paper proposes and analyzes a class of time-fractional phase-field equations for describing shape transformation processes. Based on a modified Allen–Cahn model with the Caputo time-fractional derivative introduced, the equation aims to accurately characterize non-classical dynamic behaviors during phase transitions, particularly memory effects and anomalous diffusion mechanisms. For this nonlinear fractional evolution equation, we design an efficient and stable numerical algorithm: the L1 scheme based on the Caputo definition is employed for temporal discretization, a seven-point difference stencil is used to approximate the Laplace operator in the spatial direction, and an explicit prediction step is specially introduced to enhance the diagonal dominance and computational stability of the discrete system. In terms of theoretical analysis, the conditional stability of the algorithm under specific time step constraints is proved based on the discrete energy method, a modified energy dissipation law is established, and under enhanced regularity conditions on the solution, optimal convergence order estimates with respect to time and space steps are derived using the generalized discrete Gronwall inequality, confirming that the \(L^2\) error bound of the numerical solution is \(O\left( \Delta t^{2-\beta }+h^2\right) \) . In the numerical experiments section, the influence of the fractional order parameter \(\beta \) on the evolution dynamics is systematically investigated through four typical cases. The numerical results show that a smaller fractional order leads to a stronger memory effect and faster movement of the phase interface. This behavior differs from classical integer-order models. The study provides mathematical analysis and convergence results and suggests new methods to control microstructure evolution and shape changes.