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<p>This paper presents a Laplace-Residual Power Series Method (L-RPSM) for obtaining accurate semi-analytical solutions of the nonlinear time-fractional Kuramoto-Sivashinsky (KS) equation formulated in the Caputo sense. The proposed approach operates entirely in the Laplace domain, where the Caputo derivative is transformed into an algebraic expression, allowing the solution to be constructed as a fractional power series in inverse powers of the Laplace variable. The unknown coefficients are determined systematically through an asymptotic residual cancellation procedure, eliminating the need for repeated fractional differentiation and significantly reducing computational complexity.</p><p>The convergence of the method is rigorously established through truncation error estimates and uniform convergence analysis. Three representative parameter configurations of the fractional KS equation are investigated to assess the effectiveness of the method. In the classical case <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><mi>α</mi><mo>=</mo><mn>1</mn></math></EquationSource><EquationSource Format="TEX">$\alpha = 1$</EquationSource></InlineEquation>, the obtained solutions recover the known exact results, validating the correctness of the formulation. For fractional orders <InlineEquation ID="IEq2"><EquationSource Format="MATHML"><math><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mn>1</mn></math></EquationSource><EquationSource Format="TEX">$0 &lt; \alpha &lt; 1$</EquationSource></InlineEquation>, the method demonstrates rapid convergence and very small absolute errors. The accuracy and stability of the proposed scheme are further illustrated through two-dimensional and three-dimensional graphical representations, as well as detailed tabulated error comparisons. Comparative analysis with several recent analytical techniques, including NTDM, CHPETM, Cq-HATM, and q-HATM, confirms the superior accuracy and efficiency of the proposed framework.</p><p>The numerical results further reveal the significant influence of the fractional order on the dynamical behavior of the system, where decreasing <i>α</i> enhances memory effects and modifies the dissipative-dispersive balance. Overall, the L-RPSM provides a robust, accurate, and computationally efficient tool for solving nonlinear time-fractional evolution equations.</p>

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Recursive transform-space construction for nonlinear time-fractional Kuramoto-Sivashinsky equations: theoretical analysis and numerical validation

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月23日

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  • Fares Bekhouche,
  • Ouidad Boulakour,
  • Gul Agha Jan Assar,
  • Fatemah Mofarreh,
  • Ahmad Shafee,
  • Imran Khan

摘要

This paper presents a Laplace-Residual Power Series Method (L-RPSM) for obtaining accurate semi-analytical solutions of the nonlinear time-fractional Kuramoto-Sivashinsky (KS) equation formulated in the Caputo sense. The proposed approach operates entirely in the Laplace domain, where the Caputo derivative is transformed into an algebraic expression, allowing the solution to be constructed as a fractional power series in inverse powers of the Laplace variable. The unknown coefficients are determined systematically through an asymptotic residual cancellation procedure, eliminating the need for repeated fractional differentiation and significantly reducing computational complexity.

The convergence of the method is rigorously established through truncation error estimates and uniform convergence analysis. Three representative parameter configurations of the fractional KS equation are investigated to assess the effectiveness of the method. In the classical case α=1$\alpha = 1$, the obtained solutions recover the known exact results, validating the correctness of the formulation. For fractional orders 0<α<1$0 < \alpha < 1$, the method demonstrates rapid convergence and very small absolute errors. The accuracy and stability of the proposed scheme are further illustrated through two-dimensional and three-dimensional graphical representations, as well as detailed tabulated error comparisons. Comparative analysis with several recent analytical techniques, including NTDM, CHPETM, Cq-HATM, and q-HATM, confirms the superior accuracy and efficiency of the proposed framework.

The numerical results further reveal the significant influence of the fractional order on the dynamical behavior of the system, where decreasing α enhances memory effects and modifies the dissipative-dispersive balance. Overall, the L-RPSM provides a robust, accurate, and computationally efficient tool for solving nonlinear time-fractional evolution equations.