Complex fuzzy sets (CFSs) have recently emerged as a potent tool for expanding the scope of fuzzy sets to encompass wider ranges within the unit disk in the complex plane. This study explores complex fuzzy numbers and introduces their application for the first time in the literature to address a complex fuzzy partial differential equation that involves a complex fuzzy heat equation under Hukuhara differentiability. The researchers utilize an implicit finite difference scheme, namely the Crank–Nicolson method, to tackle complex fuzzy heat equations. The problem’s fuzziness arises from the coefficients in both amplitude and phase terms, as well as in the initial and boundary conditions, with the Convex normalized triangular fuzzy numbers extended to the unit disk in the complex plane. The researchers take advantage of the properties and benefits of CFS theory in the proposed numerical methods and subsequently provide a new proof of the stability under CFS theory. A numerical example is presented to demonstrate the proposed approach’s reliability and feasibility, with the results showing good agreement with the exact solution and relevant theoretical aspects.

中文翻译：

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复狄利克雷条件下的复模糊热方程的修正曲柄-尼科尔森法解

复杂模糊集（CFS）最近已成为扩展模糊集范围以涵盖复平面单位圆盘内更广泛范围的有效工具。本研究探索了复模糊数，并在文献中首次介绍了它们的应用，以解决涉及 Hukuhara 可微分下的复模糊热方程的复模糊偏微分方程。研究人员利用隐式有限差分方案，即克兰克-尼科尔森方法来解决复杂的模糊热方程。问题的模糊性源于幅度和相位项以及初始条件和边界条件中的系数，凸归一化三角模糊数扩展到复平面中的单位圆盘。研究人员在提出的数值方法中利用了 CFS 理论的特性和优点，随后为 CFS 理论下的稳定性提供了新的证明。数值算例证明了该方法的可靠性和可行性，结果与精确解和相关理论方面吻合良好。