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<p>Time-fractional diffusion (TFD) equations have emerged as powerful tools for modeling various physical and engineering processes, particularly in signal smoothing applications. This paper presents an efficient numerical method for solving TFD equations based on the operational matrix approach using Bernstein polynomials. The operational matrix of fractional derivatives is systematically derived and represented as a product of structured matrices to enhance computational efficiency. With the help of this matrix and the collocation method, the TFD equation is transformed into a system of algebraic equations, significantly simplifying the numerical procedure. A perturbation-based technique is employed to analyze the stability of the proposed scheme. Numerical results confirm the accuracy, stability, and effectiveness of the method, highlighting its potential for solving a broad class of fractional diffusion problems.</p>

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A computational framework for time-fractional diffusion equations via Bernstein polynomials

  • 期刊论文

  • 发表日期：

    2026年3月25日

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  • S Poojitha,
  • Ashish Awasthi

摘要

Time-fractional diffusion (TFD) equations have emerged as powerful tools for modeling various physical and engineering processes, particularly in signal smoothing applications. This paper presents an efficient numerical method for solving TFD equations based on the operational matrix approach using Bernstein polynomials. The operational matrix of fractional derivatives is systematically derived and represented as a product of structured matrices to enhance computational efficiency. With the help of this matrix and the collocation method, the TFD equation is transformed into a system of algebraic equations, significantly simplifying the numerical procedure. A perturbation-based technique is employed to analyze the stability of the proposed scheme. Numerical results confirm the accuracy, stability, and effectiveness of the method, highlighting its potential for solving a broad class of fractional diffusion problems.