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<p>In this paper, we utilized a semi-analytical technique—Adomian Decomposition Shehu Transform Method (ADShTM)—for solving two distinct forms of the nonlinear Kaup–Kupershmidt equations involving Caputo fractional derivatives. The method constructs approximate solutions in the form of rapidly converging series and is supported by a rigorous convergence theorem, which ensures the mathematical validation. To assess the effectiveness of the proposed method, we perform a comparative analysis with the established q-Homotopy Analysis Transform Method (q-HATM) and Homotopy Perturbation Laplace Transform Method (HPLTM). Numerical simulations, including 2D plots across various fractional orders, demonstrate that ADShTM yields superior accuracy and closer alignment with exact solutions. In addition, 3D visualizations for the classical case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\gamma = 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> further validate the precision and adaptability of the method. The results confirm that ADShTM not only outperforms existing methods in terms of accuracy and convergence, but also offers a reliable and generalized framework for handling a broad class of nonlinear fractional differential equations.</p>

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On the solution of nonlinear fractional Kaup–Kupershmidt equations using a modified adomian technique

  • 期刊论文

  • 发表日期：

    2025年12月9日

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  • Rahul M. Makwana,
  • Sagar R. Khirsariya

摘要

In this paper, we utilized a semi-analytical technique—Adomian Decomposition Shehu Transform Method (ADShTM)—for solving two distinct forms of the nonlinear Kaup–Kupershmidt equations involving Caputo fractional derivatives. The method constructs approximate solutions in the form of rapidly converging series and is supported by a rigorous convergence theorem, which ensures the mathematical validation. To assess the effectiveness of the proposed method, we perform a comparative analysis with the established q-Homotopy Analysis Transform Method (q-HATM) and Homotopy Perturbation Laplace Transform Method (HPLTM). Numerical simulations, including 2D plots across various fractional orders, demonstrate that ADShTM yields superior accuracy and closer alignment with exact solutions. In addition, 3D visualizations for the classical case \(\gamma = 1\) γ = 1 further validate the precision and adaptability of the method. The results confirm that ADShTM not only outperforms existing methods in terms of accuracy and convergence, but also offers a reliable and generalized framework for handling a broad class of nonlinear fractional differential equations.