Numerical Solution of Fuzzy Heat Equation with Complex Dirichlet Conditions

Abstract

Recently, complex fuzzy sets have become powerful tools for generalizing the range of fuzzy sets to wider ranges that lie on a unit disk in the complex plane. In this study, complex fuzzy numbers are discussed and applied for the first time to solve a complex fuzzy partial differential equation involving a complex fuzzy heat equation under Hukuhara differentiability. Subsequently, an explicit finite difference scheme, referred to as the forward time-center space (FTCS), was implemented to solve the complex fuzzy heat equations. The imprecision of the issue is evident in the initial and boundary conditions, as well as in the amplitude and phase terms' coefficients, where the convex normalized triangular fuzzy numbers are extended to the unit disk in the complex plane. The proposed numerical methods utilized the properties and benefits of the complex fuzzy set theory. Furthermore, a new proof of consistency, stability, and convergence was established under this theory. A numerical example was provided to illustrate the reliability and feasibility of the proposed approach. The results obtained using the proposed approach are in adequate agreement with the exact solution and related theoretical aspects.

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

“…Fractional differential equations (FDEs) have garnered significant interest due to their wide range of real-life applications in chemistry, physics, biology, engineering, and other fields [1][2][3][4][5][6][7][8][9][10][11][12][13]. In conventional studies on natural processes modeled by FDEs, the vague parameters are considered to be precise and well defined.…”