<p>This study presents a computational analysis of a fractional-order model for heat conduction in complex media with fading memory. The model incorporates Caputo time-fractional derivatives of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, accounts for heat flux memory effects, and includes a neutral delay. By representing the relaxation functions of heat flux and heat capacity as finite linear combinations of decaying exponentials, we derive a coupled system involving both fractional temporal operators and classical time derivatives, which extends the original fractional heat-conduction equation with two auxiliary equations. The stability estimate for the solution of the resulting system is established in a finite-dimensional Hilbert space, with respect to initial conditions and source terms. For the computational implementation, we first propose a difference scheme based on the L1 formula and rigorously investigate its unconditional stability, demonstrating a temporal convergence rate of order <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\min \{2-\alpha , 1+\alpha \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <mn>2</mn> <mo>-</mo> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. To achieve higher accuracy that is independent of the fractional order, an additional scheme based on the L2 formula is developed and proven to exhibit second-order temporal convergence. In addition, the methods are extended to graded non-uniform meshes to enhance their accuracy in cases where the solution possesses limited initial smoothness. Numerical simulations are conducted to validate the theoretical results.</p>

Computational analysis of fractional heat conduction with fading memory

期刊论文
发表日期：

2026年2月19日

查看全文

Anatoly A. Alikhanov,
Mohammad Shahbazi Asl,
Chengming Huang,
Adam A. Alikhanov

摘要

This study presents a computational analysis of a fractional-order model for heat conduction in complex media with fading memory. The model incorporates Caputo time-fractional derivatives of order $\alpha \in (0,1)$ $α \in (0, 1)$ , accounts for heat flux memory effects, and includes a neutral delay. By representing the relaxation functions of heat flux and heat capacity as finite linear combinations of decaying exponentials, we derive a coupled system involving both fractional temporal operators and classical time derivatives, which extends the original fractional heat-conduction equation with two auxiliary equations. The stability estimate for the solution of the resulting system is established in a finite-dimensional Hilbert space, with respect to initial conditions and source terms. For the computational implementation, we first propose a difference scheme based on the L1 formula and rigorously investigate its unconditional stability, demonstrating a temporal convergence rate of order $\min \{2-\alpha , 1+\alpha \}$ $min {2 - α, 1 + α}$ . To achieve higher accuracy that is independent of the fractional order, an additional scheme based on the L2 formula is developed and proven to exhibit second-order temporal convergence. In addition, the methods are extended to graded non-uniform meshes to enhance their accuracy in cases where the solution possesses limited initial smoothness. Numerical simulations are conducted to validate the theoretical results.