Homotopy Analysis Method Analytical Scheme for Developing a Solution to Partial Differential Equations in Fuzzy Environment

Abstract

Partial differential equations are known to be increasingly important in today’s research, and their solutions are paramount for tackling numerous real-life applications. This article extended the analytical scheme of the homotopy analysis method (HAM) to develop an approximate analytical solution for Fuzzy Partial Differential Equations (FPDEs). The scheme used its powerful tools, the auxiliary function and convergence-control parameter, in the analysis and optimization, which ensures the convergence of the approximate series solution in addition to considering all necessary concepts from fuzzy set theory to provide high precision in the fuzzy environment. Furthermore, the efficiency was shown by applying the proposed scheme to linear and nonlinear cases of Fuzzy Reaction–Diffusion Equation (FRDE) and Fuzzy Wave Equation (FWE).

Untitled

“…Future studies should investigate disease prevention methods and drug effects in the context of fractional SEIR models [53,54,55,56]. To better understand influenza dynamics and increase forecast accuracy, additional models such as those mentioned in references [57,58,59] can be further investigated. Overall, this study advances influenza modeling and sheds light on influenza transmission dynamics in Saudi Arabia.…”

Solving Fuzzy System of Boundary Value Problems by Homotopy Perturbation Method with Green’s Function

“…Fuzzy systems of differential equations appear when modeling these problems are incomplete and questionable in the case of inaccurate data. Fuzzy differential equation systems are applicable mathematical models [6]. Dynamic models of dynamic systems with uncertainties or ambiguous models, represented by fuzzy differential equations, are widely utilized across a spectrum of fields, such as geometry and population modeling.…”

“…The concept of fuzziness has experienced a surge in popularity and application across various fields since its inception, including learning theory, automata, decision-making processes, algorithms, pattern classification, and linguistics. The idea behind fuzzy concepts involves converting differential equations into differential inclusions and accepting the solution as the α−cut of the fuzzy solution, leading to the transformation of the fuzzy differential equation (FDE) into a equivalent fuzzy integral equation [6]. In the last years, the researchers Visuvasam et al [23], Shanthi et al [19], Vijayalakshmi et al [21] and Vijayalakshmi et al [22] have applied HPM to solve the mathematical models as (porous rotating disk electrodes, ECE reactions at rotating disk electrodes, magnetohydrodynamic, etc.).…”