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<p>In recent years, the inherent numerical difficulties in the study of two-dimensional fractional differential equations (T-FDEs) make them challenging for investigation. In the present contribution, we introduce a new general class of T-FDEs to be solved. Their solution is addressed by the use of unique polynomials of the generalized shifted Legendre functions. For this purpose, we evaluate the association of new fractional and ordinary operational matrices using the Lagrange multipliers algorithm. The analysis of the convergence is done and the existence of a unique solution for the T-FDEs is proved. Three problems are considered to test the proposed methodology. It is pointed out that our methodology can be considered as a promising strategy to solve two-dimensional variable-order fractional differential equations and 2D variable-order fractional optimal control problems.</p>

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Implementation of an Optimization Algorithm for a New General Class of Two-Dimensional Fractional PDEs

  • 期刊论文

  • 发表日期：

    2025年3月10日

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  • Zakieh Avazzadeh,
  • Hossein Hassani,
  • Mohammad Javad Ebadi,
  • Ali Bayati Eshkaftaki,
  • Ahmed Hendy

摘要

In recent years, the inherent numerical difficulties in the study of two-dimensional fractional differential equations (T-FDEs) make them challenging for investigation. In the present contribution, we introduce a new general class of T-FDEs to be solved. Their solution is addressed by the use of unique polynomials of the generalized shifted Legendre functions. For this purpose, we evaluate the association of new fractional and ordinary operational matrices using the Lagrange multipliers algorithm. The analysis of the convergence is done and the existence of a unique solution for the T-FDEs is proved. Three problems are considered to test the proposed methodology. It is pointed out that our methodology can be considered as a promising strategy to solve two-dimensional variable-order fractional differential equations and 2D variable-order fractional optimal control problems.