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<p>We propose a simple yet efficient iterative scheme for obtaining approximate solutions to a class of nonlinear parabolic partial differential equations. The proposed approximation method is based on the residual error function and the multiple power series expansion. Error estimate is derived and convergence of the proposed method is established. Several benchmark problems are then considered to validate the theoretical results and demonstrate the computational ease of the developed residual power series method when applied to multi-dimensional nonlinear Burgers’ equations. It is observed that the proposed approximation method accurately captures shock wave propagation and steep gradient formation, and produces more accurate results than existing methods in higher dimensions. Finally, recommendations are provided regarding the handling of discontinuous initial data, as well as strategies for improving accuracy and extending the convergence region.</p>

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Dynamics of the nonlinear multi–dimensional Burgers’ equations via an iterative strategy with an error estimation

  • 期刊论文

  • 发表日期：

    2026年7月6日

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  • Manzor Hussain,
  • Musaad S. Aldhabani,
  • Lubna Farooq,
  • Abdul Ghafoor,
  • Ahmed M. Zidan,
  • Ahmad Shafee

摘要

We propose a simple yet efficient iterative scheme for obtaining approximate solutions to a class of nonlinear parabolic partial differential equations. The proposed approximation method is based on the residual error function and the multiple power series expansion. Error estimate is derived and convergence of the proposed method is established. Several benchmark problems are then considered to validate the theoretical results and demonstrate the computational ease of the developed residual power series method when applied to multi-dimensional nonlinear Burgers’ equations. It is observed that the proposed approximation method accurately captures shock wave propagation and steep gradient formation, and produces more accurate results than existing methods in higher dimensions. Finally, recommendations are provided regarding the handling of discontinuous initial data, as well as strategies for improving accuracy and extending the convergence region.