Numerical Solution of the Intuitionistic Fuzzy Complex Heat Equation with Intuitionistic Complex Dirichlet Boundary Conditions via the Explicit Finite Difference Method

Abstract

Recent developments in complex fuzzy (CF) sets have extended the classical fuzzy framework from the unit interval [0,1] to the unit disk in the complex plane C, allowing for the modeling of uncertainties in both magnitude and phase. Building upon this foundation, this study introduces-for the first time-the use of complex intuitionistic fuzzy (CIF) numbers to solve partial differential equations, specifically focusing on the CIF heat equation. The CIF framework integrates both membership and non-membership functions with a complex-valued representation, enabling a more expressive treatment of uncertainty, including hesitation. An explicit finite difference method, namely the Forward Time Central Space (FTCS) scheme, is employed to discretize and solve the CIF heat equation. The model considers fuzziness in the initial and boundary conditions, where uncertainty impacts amplitude and phase terms. To represent this uncertainty, triangular fuzzy numbers are used for the real and imaginary parts within the complex unit disk. The proposed approach demonstrates numerical stability and achieves second-order spatial and first-order temporal accuracy, validating its reliability and effectiveness. A numerical example confirms the feasibility of the method, showing strong alignment with theoretical predictions. This work generalizes existing CF heat equation models and provides a foundation for solving more complex systems involving higher-order and bipolar uncertainties in future studies.