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<p>In this paper, a comparative study between two analytical methods: the Residual Power Series Transform Method (RPSTM) and the Optimal Homotopy Asymptotic Transform Method (OHATM) is conducted. The RPSTM combines the Residual Power Series Method with the Laplace Transform, while the OHATM integrates the Optimal Homotopy Asymptotic Method with it. Both methods are applied to time-fractional Cauchy reaction-diffusion equations with uncertainty, where the fractional derivative is considered in the Caputo sense. Approximate solutions, specifically the upper and lower bound solutions of the equations, are obtained by both RPSTM and OHATM. The study includes theoretical analysis, graphical illustrations, and numerical evaluations. The effectiveness and accuracy of both methods are demonstrated by the comparisons between the exact solutions and the approximate solutions in a fuzzy environment. It is revealed through the findings that the Residual Power Series Transform Method is simpler, more expedient, and more effective in obtaining approximate solutions of time-fractional Cauchy reaction-diffusion equations compared with the Optimal Homotopy Asymptotic Transform Method.</p>

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Solution of time-fractional Cauchy reaction diffusion equations with uncertainty

  • 期刊论文

  • 发表日期：

    2026年1月21日

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  • Shreya Mukherjee,
  • Amit Kumar,
  • Ankur Kanaujiya

摘要

In this paper, a comparative study between two analytical methods: the Residual Power Series Transform Method (RPSTM) and the Optimal Homotopy Asymptotic Transform Method (OHATM) is conducted. The RPSTM combines the Residual Power Series Method with the Laplace Transform, while the OHATM integrates the Optimal Homotopy Asymptotic Method with it. Both methods are applied to time-fractional Cauchy reaction-diffusion equations with uncertainty, where the fractional derivative is considered in the Caputo sense. Approximate solutions, specifically the upper and lower bound solutions of the equations, are obtained by both RPSTM and OHATM. The study includes theoretical analysis, graphical illustrations, and numerical evaluations. The effectiveness and accuracy of both methods are demonstrated by the comparisons between the exact solutions and the approximate solutions in a fuzzy environment. It is revealed through the findings that the Residual Power Series Transform Method is simpler, more expedient, and more effective in obtaining approximate solutions of time-fractional Cauchy reaction-diffusion equations compared with the Optimal Homotopy Asymptotic Transform Method.