<p>This paper presents two high-order compact difference schemes to discuss the numerical solution of the one-dimensional and two-dimensional multi-term time-fractional convection-diffusion equation of the Sobolev type based on the Caputo fractional derivative. For this purpose, we employ the L2 formula for the temporal discretization of the Caputo fractional derivatives and introduce a new compact difference operator for the space discretization. The proposed schemes transform the original problem into a system of algebraic equation. We present a novel analysis of the convergence and theoretical stability of both methods. The difference schemes have fourth-order and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3077_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </InlineMediaObject> <EquationSource Format="TEX">$3-\max \{\alpha _r, \beta _r\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>-</mo> <mo movablelimits="true">max</mo> <mo stretchy="false">{</mo> <msub> <mi>α</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>β</mi> <mi>r</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> order of accuracy in space and time respectively, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3077_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha _r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>α</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3077_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta _r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>β</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> are the orders of the fractional derivatives in the multi-term convection-diffusion model. Additionally, we construct an L2-type numerical scheme on non-uniform graded meshes to address problems where the solution exhibits weak initial regularity. Some test functions are provided to demonstrate the accuracy and efficiency of the proposed schemes. The numerical results of inclusive examples confirm the proposed schemes’ theoretical results and illustrate their applicability and efficiency.</p>

A high-order compact difference scheme for the multi-term time-fractional Sobolev-type convection-diffusion equation

期刊论文
发表日期：

2025年1月11日

查看全文

Anatoly A. Alikhanov,
Poonam Yadav,
Vineet Kumar Singh,
Mohammad Shahbazi Asl

摘要

This paper presents two high-order compact difference schemes to discuss the numerical solution of the one-dimensional and two-dimensional multi-term time-fractional convection-diffusion equation of the Sobolev type based on the Caputo fractional derivative. For this purpose, we employ the L2 formula for the temporal discretization of the Caputo fractional derivatives and introduce a new compact difference operator for the space discretization. The proposed schemes transform the original problem into a system of algebraic equation. We present a novel analysis of the convergence and theoretical stability of both methods. The difference schemes have fourth-order and $3-\max \{\alpha _r, \beta _r\}$ $3 - max {α_{r}, β_{r}}$ order of accuracy in space and time respectively, where $\alpha _r$ $α_{r}$ and $\beta _r$ $β_{r}$ are the orders of the fractional derivatives in the multi-term convection-diffusion model. Additionally, we construct an L2-type numerical scheme on non-uniform graded meshes to address problems where the solution exhibits weak initial regularity. Some test functions are provided to demonstrate the accuracy and efficiency of the proposed schemes. The numerical results of inclusive examples confirm the proposed schemes’ theoretical results and illustrate their applicability and efficiency.