<p>In this study, we investigate the one-dimensional Chafee–Infante model, where heat diffusion serves as the primary mechanism of energy transfer. The Shehu transform is employed to derive solutions for three distinct forms of the Chafee–Infante equation, incorporating the time derivative in the Caputo fractional sense, and the resulting transformed systems are solved with the nonlinear terms using the Adomian decomposition method. The Chafee–Infante equation is widely used to model nonlinear phenomena in population dynamics, chemical reactions, heat transfer and neural activity. This study provides a new contribution by applying the Shehu decomposition method (SDM) to the time-fractional CI equation. The key advantage of the Shehu decomposition method (SDM) over transform-based hybrid methods such as Laplace–ADM, Sumudu–ADM and Elzaki–ADM lies in its direct handling of nonlinear problems without requiring inverse transforms, which significantly simplifies both analysis and computation. The present method also demonstrates accurate and convergent solutions, supported by error norms and rate of convergence analysis. A detailed comparison between the approximate and exact solutions demonstrates that the proposed hybrid approach is both highly accurate and computationally efficient for a broad class of nonlinear Chafee–Infante problems. For each case, the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${L}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${L}_{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> norms error norms are evaluated to assess accuracy. Numerical simulations, convergence investigations and comprehensive error and stability analyses consistently confirm excellent agreement between the analytical approximations and the exact solutions. The behavior of the solutions is further illustrated through three-dimensional surface plots and two-dimensional line graphs. Additionally, the numerical results obtained in this work are compared with existing findings in the literature, showing strong consistency and improved performance in terms of computational efficiency cost (CPU time).</p>

Efficient hybrid approach for time-fractional (1 + 1)-dimensional Chafee–Infante equation using the Shehu–Adomian decomposition method in the Caputo environment

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发表日期：

2026年5月20日

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Sunita Kumari Nayak,
Saumya Ranjan Jena

摘要

In this study, we investigate the one-dimensional Chafee–Infante model, where heat diffusion serves as the primary mechanism of energy transfer. The Shehu transform is employed to derive solutions for three distinct forms of the Chafee–Infante equation, incorporating the time derivative in the Caputo fractional sense, and the resulting transformed systems are solved with the nonlinear terms using the Adomian decomposition method. The Chafee–Infante equation is widely used to model nonlinear phenomena in population dynamics, chemical reactions, heat transfer and neural activity. This study provides a new contribution by applying the Shehu decomposition method (SDM) to the time-fractional CI equation. The key advantage of the Shehu decomposition method (SDM) over transform-based hybrid methods such as Laplace–ADM, Sumudu–ADM and Elzaki–ADM lies in its direct handling of nonlinear problems without requiring inverse transforms, which significantly simplifies both analysis and computation. The present method also demonstrates accurate and convergent solutions, supported by error norms and rate of convergence analysis. A detailed comparison between the approximate and exact solutions demonstrates that the proposed hybrid approach is both highly accurate and computationally efficient for a broad class of nonlinear Chafee–Infante problems. For each case, the ${L}_{2}$ $L_{2}$ and ${L}_{\infty }$ $L_{\infty}$ norms error norms are evaluated to assess accuracy. Numerical simulations, convergence investigations and comprehensive error and stability analyses consistently confirm excellent agreement between the analytical approximations and the exact solutions. The behavior of the solutions is further illustrated through three-dimensional surface plots and two-dimensional line graphs. Additionally, the numerical results obtained in this work are compared with existing findings in the literature, showing strong consistency and improved performance in terms of computational efficiency cost (CPU time).