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<p>In this work, a sixth-order extrapolated optimal cubic spline collocation method is proposed for the numerical solution of various forms of the nonlinear Fisher’s equation. This equation arises in several applied fields, including nerve pulse propagation, neutron diffusion in nuclear reactions, and population genetics. The proposed approach employs cubic B-spline basis functions due to their high smoothness and the sparse banded structure of the resulting coefficient matrix, which leads to computational efficiency. To further enhance the spatial accuracy of the standard collocation scheme, an optimal extrapolation strategy is incorporated, resulting in a higher-order convergent method without increasing the stencil width. Time integration is carried out using the Crank–Nicolson scheme, which provides second-order accuracy and unconditional stability in the temporal direction. A rigorous convergence analysis of the fully discrete scheme is presented, and optimal error bounds are derived using Green’s function theory. Extensive numerical experiments are performed and the computed results demonstrate that the proposed method achieves significantly improved accuracy compared with several existing spline-based and meshfree numerical techniques. The close agreement between numerical and analytical solutions is illustrated through graphical comparisons and quantitative error analysis using the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_{\infty }\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L_{2}\)</EquationSource> </InlineEquation> norms, confirming the effectiveness and robustness of the proposed scheme.</p>

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Stability and Convergence Analysis of an Extrapolated Optimal Collocation Algorithm used for Solving Fisher’s Equation

  • 期刊论文

  • 发表日期：

    2026年5月25日

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  • Shallu,
  • Archna Kumari,
  • Vijay Kumar Kukreja

摘要

In this work, a sixth-order extrapolated optimal cubic spline collocation method is proposed for the numerical solution of various forms of the nonlinear Fisher’s equation. This equation arises in several applied fields, including nerve pulse propagation, neutron diffusion in nuclear reactions, and population genetics. The proposed approach employs cubic B-spline basis functions due to their high smoothness and the sparse banded structure of the resulting coefficient matrix, which leads to computational efficiency. To further enhance the spatial accuracy of the standard collocation scheme, an optimal extrapolation strategy is incorporated, resulting in a higher-order convergent method without increasing the stencil width. Time integration is carried out using the Crank–Nicolson scheme, which provides second-order accuracy and unconditional stability in the temporal direction. A rigorous convergence analysis of the fully discrete scheme is presented, and optimal error bounds are derived using Green’s function theory. Extensive numerical experiments are performed and the computed results demonstrate that the proposed method achieves significantly improved accuracy compared with several existing spline-based and meshfree numerical techniques. The close agreement between numerical and analytical solutions is illustrated through graphical comparisons and quantitative error analysis using the \(L_{\infty }\) and \(L_{2}\) norms, confirming the effectiveness and robustness of the proposed scheme.