<p>The objective of this work is to develop numerical schemes based on finite differences and cubic B-splines for solving the time fractional Nagumo equation, which is used to model the transmission of nerve impulses, circuit theory, and population genetics. First, we propose semi-implicit and fully implicit schemes using finite differences for spatial discretization. The fractional order time derivative is approximated with the Caputo derivative of both orders <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1907_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$O\left( \tau ^{2-\gamma }\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mfenced close=")" open="("> <msup> <mi>τ</mi> <mrow> <mn>2</mn> <mo>-</mo> <mi>γ</mi> </mrow> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1907_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$O\left( \tau ^{3-\gamma }\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mfenced close=")" open="("> <msup> <mi>τ</mi> <mrow> <mn>3</mn> <mo>-</mo> <mi>γ</mi> </mrow> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1907_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\gamma <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>γ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove that the semi-implicit schemes are unconditionally stable and positivity preserving. Also, we find the invariant region and perform the convergence analysis. Next, another new scheme is constructed based on cubic B-spline approximation for spatial discretization and performed its stability and convergence analysis. Some numerical examples are presented to demonstrate a comparative study and check the effectiveness of these schemes.</p>

Some Robust Numerical Schemes and Analysis for Solving Time Fractional Nagumo Equation

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发表日期：

2025年5月3日

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Sobin Thomas,
S. Rama Chandra Karthik,
Suresh Kumar Nadupuri

摘要

The objective of this work is to develop numerical schemes based on finite differences and cubic B-splines for solving the time fractional Nagumo equation, which is used to model the transmission of nerve impulses, circuit theory, and population genetics. First, we propose semi-implicit and fully implicit schemes using finite differences for spatial discretization. The fractional order time derivative is approximated with the Caputo derivative of both orders $O\left( \tau ^{2-\gamma }\right) $ $O (τ^{2 - γ})$ and $O\left( \tau ^{3-\gamma }\right) $ $O (τ^{3 - γ})$ , where $0<\gamma <1$ $0 < γ < 1$ . We prove that the semi-implicit schemes are unconditionally stable and positivity preserving. Also, we find the invariant region and perform the convergence analysis. Next, another new scheme is constructed based on cubic B-spline approximation for spatial discretization and performed its stability and convergence analysis. Some numerical examples are presented to demonstrate a comparative study and check the effectiveness of these schemes.