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<p>Fractional-order partial differential equations (PDEs) have become an essential tool in modeling complex phenomena across various scientific disciplines due to their ability to describe processes involving memory and hereditary effects. This paper explores the application of the Caputo fractional derivative in solving FOPDEs and systems, presenting a novel numerical method that combines the Laplace transform with the Least-Squares Residual Power-Series Method. This approach enables the efficient approximation of solutions to fractional differential equations, even under complex boundary conditions. Additionally, the paper addresses the Kolmogorov equation, commonly used in stochastic processes, by integrating fractional derivatives to model nonlocal and anomalous diffusion phenomena. The results demonstrate the effectiveness of the proposed method in solving both FOPDEs and Kolmogorov equations, providing valuable insights for future research in fractional calculus and its diverse applications in engineering, physics, and finance.</p>

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Caputo fractional derivative in the solution of Kolmogorov equations: a new numerical approach

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月28日

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  • Mohammad Alshammari,
  • Saleh Alshammari,
  • Manoj Singh,
  • Yousef Jawarneh,
  • Nedal M. Mohammed,
  • Khalid M. K. Alshammari

摘要

Fractional-order partial differential equations (PDEs) have become an essential tool in modeling complex phenomena across various scientific disciplines due to their ability to describe processes involving memory and hereditary effects. This paper explores the application of the Caputo fractional derivative in solving FOPDEs and systems, presenting a novel numerical method that combines the Laplace transform with the Least-Squares Residual Power-Series Method. This approach enables the efficient approximation of solutions to fractional differential equations, even under complex boundary conditions. Additionally, the paper addresses the Kolmogorov equation, commonly used in stochastic processes, by integrating fractional derivatives to model nonlocal and anomalous diffusion phenomena. The results demonstrate the effectiveness of the proposed method in solving both FOPDEs and Kolmogorov equations, providing valuable insights for future research in fractional calculus and its diverse applications in engineering, physics, and finance.