A Solution of the Complex Fuzzy Heat Equation in Terms of Complex Dirichlet Conditions Using a Modified Crank–Nicolson Method

Abstract

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.

Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable

“…Fuzzy fractional differential and integral equations have received significant attention in the physical sciences during the past few decades. Many researchers have obtained the solution of fuzzy differential equations using various analytical and numerical techniques such as homotopy perturbation method [ 13 ], Fuzzy Laplace transform method [ 14 ], Crank-Nicolson method [ 15 ], Natural adomian decomposition method [ 16 ], SM’s method [ 17 ]. Akram and Bilal [ 18 ] utilized homotopy perturbation scheme for the analytical solution of fuzzy heat problem.…”