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<p>This paper presents and establishes the Chebyshev collocation method, which generates numerical solutions for nonlinear fractional partial differential equations such as the fractional diffusion, wave, and Korteweg–De Vries equations. To obtain the novel fractional derivative operational matrix of the shifted Chebyshev polynomials, new theorems and lemmas have been proved. Caputo’s fractional-order derivative definition is used to represent the fractional-order terms. This approach transforms the problem under discussion into a nonlinear algebraic system of equations that Newton’s method is applied to solve numerically. The error analysis is determined to justify the proposed technique. Several numerical examples are provided to illustrate the accuracy and applicability of the suggested method. The estimated, residual, and absolute errors are computed for each numerical example, and comparisons with other approaches are shown to strengthen the reliability and effectiveness of the suggested method.</p>

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Computational study of fractional partial differential equations using the second-kind Chebyshev collocation technique with error analysis

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月13日

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  • Khaled M. Abdelgaber,
  • Mohamed Fathy,
  • Aya Hassan,
  • Emad A. Sayed

摘要

This paper presents and establishes the Chebyshev collocation method, which generates numerical solutions for nonlinear fractional partial differential equations such as the fractional diffusion, wave, and Korteweg–De Vries equations. To obtain the novel fractional derivative operational matrix of the shifted Chebyshev polynomials, new theorems and lemmas have been proved. Caputo’s fractional-order derivative definition is used to represent the fractional-order terms. This approach transforms the problem under discussion into a nonlinear algebraic system of equations that Newton’s method is applied to solve numerically. The error analysis is determined to justify the proposed technique. Several numerical examples are provided to illustrate the accuracy and applicability of the suggested method. The estimated, residual, and absolute errors are computed for each numerical example, and comparisons with other approaches are shown to strengthen the reliability and effectiveness of the suggested method.