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<p>This study addresses the computational challenges in solving Bratu-type equations, particularly near critical parameters where traditional methods fail. We develop a robust numerical framework to achieve machine-precision accuracy across the entire solution spectrum. Building upon the homotopy analysis method (HAM) framework, an adaptive spectral homotopy analysis method (ASHAM) is proposed, integrating three innovations: (1) Chebyshev-Gauss-Lobatto spectral discretization for exponential convergence, (2) Homotopy deformation with Bell polynomial expansion of nonlinear terms, extending traditional homotopy approaches through spectral enhancement, and (3) Adaptive <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2250_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\hbar\)</EquationSource> </InlineEquation>-optimization via residual minimization using Brent’s method for robust convergence control. For Bratu’s boundary value problem, ASHAM achieves <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2250_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(10^{-12}\)</EquationSource> </InlineEquation> maximum absolute error with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2250_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda = 2\)</EquationSource> </InlineEquation> in 0.15 seconds, outperforming state-of-the-art methods by 3 orders of magnitude. Near criticality (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2250_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda _c = 3.513830719\)</EquationSource> </InlineEquation>), it maintains <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2250_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">\(5.7 \times 10^{-10}\)</EquationSource> </InlineEquation> accuracy where existing techniques diverge. The method also demonstrates 98% computational speedup versus wavelet approaches. ASHAM provides unprecedented accuracy and efficiency for Bratu-type equations through its convergence-optimized spectral framework that advances homotopy-based computation. The methodology establishes a new paradigm for singular nonlinear boundary value problems with broad applications in combustion theory and nanotechnology.</p>

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Adaptive spectral homotopy analysis with convergence control for high-precision solution of Bratu-type equations

  • 期刊论文

  • 发表日期：

    2025年10月22日

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  • Ujwal Warbhe

摘要

This study addresses the computational challenges in solving Bratu-type equations, particularly near critical parameters where traditional methods fail. We develop a robust numerical framework to achieve machine-precision accuracy across the entire solution spectrum. Building upon the homotopy analysis method (HAM) framework, an adaptive spectral homotopy analysis method (ASHAM) is proposed, integrating three innovations: (1) Chebyshev-Gauss-Lobatto spectral discretization for exponential convergence, (2) Homotopy deformation with Bell polynomial expansion of nonlinear terms, extending traditional homotopy approaches through spectral enhancement, and (3) Adaptive \(\hbar\) -optimization via residual minimization using Brent’s method for robust convergence control. For Bratu’s boundary value problem, ASHAM achieves \(10^{-12}\) maximum absolute error with \(\lambda = 2\) in 0.15 seconds, outperforming state-of-the-art methods by 3 orders of magnitude. Near criticality ( \(\lambda _c = 3.513830719\) ), it maintains \(5.7 \times 10^{-10}\) accuracy where existing techniques diverge. The method also demonstrates 98% computational speedup versus wavelet approaches. ASHAM provides unprecedented accuracy and efficiency for Bratu-type equations through its convergence-optimized spectral framework that advances homotopy-based computation. The methodology establishes a new paradigm for singular nonlinear boundary value problems with broad applications in combustion theory and nanotechnology.