Search

Typically, the parameters of a practical problem are considered to be deterministic. Nevertheless, there may be possibility that these parameters are incomplete owing to observational or experimental errors. Fuzzy sets are capable of handling such problems. In addition, the majority of studies on fuzzy fractional heat equations have employed two spatial variables and fuzzy sets with deterministic membership. In some instances, however, there may be more than two spatial variables, and the membership grade associated with an uncertain parameter may also contain imprecise information. Consequently, dealing with such a situation may be fascinating. In this regard, the main objective of this research is to develop alternative methods for solving the n + 1 -dimensional fractional order heat equation with an external source term in an uncertain environment. The problem is addressed by using the Elzaki transformed homotopy perturbation method and the fractional reduced differential transformation method in the fuzzy environment. Caputo fuzzy fractional derivatives are considered here. In this study, triangular fuzzy numbers and Gaussian fuzzy numbers are utilised to handle type-1 fuzzy uncertainty, whereas triangularly perfect quasi type-2 fuzzy numbers and newly introduced Gaussian-triangular type-2 fuzzy numbers are used to handle type-2 fuzzy uncertainty. Numerical examples with graphical representations are provided to demonstrate the applicability of the approaches. Convergence plots are also provided. [ABSTRACT FROM AUTHOR]

Purpose: Investigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking model. Nonetheless, due to observational or experimental errors, or any other circumstance, it may contain some incomplete information. Fuzzy sets can be used to deal with uncertainty. Yet, there may be some inconsistency in the membership as well. As a result, the primary goal of this proposed work is to numerically solve the model in a type-2 fuzzy environment. Design/methodology/approach: Triangular perfect quasi type-2 fuzzy numbers (TPQT2FNs) are used to deal with the uncertainty in the model. In this work, concepts of r2-cut at r1-plane are used to model the problem's uncertain parameter. The Legendre wavelet method (LWM) is then utilised to solve the giving up smoking model in a type-2 fuzzy environment. Findings: LWM has been effectively employed in conjunction with the r2-cut at r1-plane notion of type-2 fuzzy sets to solve the model. The LWM has the advantage of converting the non-linear fractional order model into a set of non-linear algebraic equations. LWM scheme solutions are found to be well agreed with RK4 scheme solutions. The existence and uniqueness of the model's solution have also been demonstrated. Originality/value: To deal with the uncertainty, type-2 fuzzy numbers are used. The use of LWM in a type-2 fuzzy uncertain environment to achieve the model's required solutions is quite fascinating, and this is the key focus of this work. [ABSTRACT FROM AUTHOR]