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<p>In this paper, we study a coupled time-fractional nonlocal reaction–diffusion problem. The nonlocal reaction–diffusion framework is widely used in population dynamics. We discuss the existence and uniqueness of the solution at continuous level. A unified linearized numerical scheme is developed which uses weighted <i>b</i>-splines for spatial direction and <i>L</i>1,&#xa0; <i>L</i>2-<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1_\sigma\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mn>1</mn> <mi>σ</mi> </msub> </math></EquationSource> </InlineEquation> schemes for temporal direction. The weighted <i>b</i>-spline based mesh-free method has some advantages over standard finite element method. Moreover, we derive stability estimates and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-robust convergence estimates in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2(\varOmega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(H_0^1(\varOmega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> norms. The established results are validated through some numerical examples.</p>

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Mesh-Free Galerkin Approximation for Coupled Time-Fractional Nonlocal Reaction–Diffusion Problem Using Weighted b-Splines

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  • 发表日期：

    2025年9月18日

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  • Jitesh P. Mandaliya

摘要

In this paper, we study a coupled time-fractional nonlocal reaction–diffusion problem. The nonlocal reaction–diffusion framework is widely used in population dynamics. We discuss the existence and uniqueness of the solution at continuous level. A unified linearized numerical scheme is developed which uses weighted b-splines for spatial direction and L1,  L2- \(1_\sigma\) 1 σ schemes for temporal direction. The weighted b-spline based mesh-free method has some advantages over standard finite element method. Moreover, we derive stability estimates and \(\alpha\) α -robust convergence estimates in \(L^2(\varOmega )\) L 2 ( Ω ) and \(H_0^1(\varOmega )\) H 0 1 ( Ω ) norms. The established results are validated through some numerical examples.