<p>Time-fractional singularly perturbed Burgers–Huxley (SPBH) equations arise in a variety of physical, biological, and engineering applications where nonlinear convection, diffusion, reaction kinetics, and memory effects interact. The presence of a fractional time derivative and a small perturbation parameter makes the problem highly stiff and leads to boundary-layer behavior, thereby reducing the effectiveness of classical numerical methods. In this work, we develop an efficient and parameter-robust numerical scheme for solving the time-fractional SPBH equation, where the time derivative is defined in the Caputo sense with 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>. The fractional derivative is approximated using the high-accuracy <i>L</i>1-2 scheme, while the nonlinear term is handled through a quasi-linearization technique, resulting in a tractable semi-discrete system. To resolve spatial boundary layers uniformly with respect to the perturbation parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>, a non-standard finite difference (NSFD) method is employed in the spatial direction. The proposed fully discrete scheme is shown to be <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-uniformly convergent with an overall accuracy of order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$O(N^{-1} + \Delta t^{\,3-\alpha })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow> <mspace width="0.166667em" /> <mn>3</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The effectiveness and reliability of the method are demonstrated through the numerical experiments, which confirm the theoretical findings.</p>

An efficient numerical approach for time-fractional singularly perturbed Burger-Huxley equation

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发表日期：

2026年2月3日

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Pramod Chakravarthy Podila,
Rahul Mishra,
Kamalesh Kumar,
Higinio Ramos

摘要

Time-fractional singularly perturbed Burgers–Huxley (SPBH) equations arise in a variety of physical, biological, and engineering applications where nonlinear convection, diffusion, reaction kinetics, and memory effects interact. The presence of a fractional time derivative and a small perturbation parameter makes the problem highly stiff and leads to boundary-layer behavior, thereby reducing the effectiveness of classical numerical methods. In this work, we develop an efficient and parameter-robust numerical scheme for solving the time-fractional SPBH equation, where the time derivative is defined in the Caputo sense with order $\alpha \in (0,1)$ $α \in (0, 1)$ . The fractional derivative is approximated using the high-accuracy L1-2 scheme, while the nonlinear term is handled through a quasi-linearization technique, resulting in a tractable semi-discrete system. To resolve spatial boundary layers uniformly with respect to the perturbation parameter $\epsilon $ $ϵ$ , a non-standard finite difference (NSFD) method is employed in the spatial direction. The proposed fully discrete scheme is shown to be $\epsilon $ $ϵ$ -uniformly convergent with an overall accuracy of order $O(N^{-1} + \Delta t^{\,3-\alpha })$ $O (N^{- 1} + Δ t^{3 - α})$ . The effectiveness and reliability of the method are demonstrated through the numerical experiments, which confirm the theoretical findings.