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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]

In this paper, a new method is proposed to solve type‐2 fuzzy linear system of equations by converting the system into three interval linear systems and one crisp linear system of equations. In this work, three possible cases of type‐2 fuzzy linear system is considered depending upon uncertainty (as type‐2 fuzzy number) in right‐hand side vector B˜$$ \tilde{B} $$, in co‐efficient matrix A˜$$ \tilde{A} $$, and both in co‐efficient matrix A˜$$ \tilde{A} $$ as well as right‐hand side vector B˜$$ \tilde{B} $$. Triangular perfect quasi type‐2 fuzzy numbers are taken to propose the new method. The main significance of the proposed method is that it uses the arithmetic of interval theory, which makes this method easier to handle. Five numerical examples are solved to show the validation of the proposed technique. Further, two application problems related to 6‐bar truss and rectangular sheet are encountered in type‐2 fuzzy environment. Comparison among the results obtained by the present method and previously achieved results is also made in this work, which shows that the results are close enough and, in some cases, better bounds for the solution are obtained. [ABSTRACT FROM AUTHOR]

Researchers have recently examined several fractional-order prey–predator population models in a crisp or type-1 fuzzy environment. But incompleteness in the membership may also be present. In this regard, type-2 fuzzy numbers are employed in this inquiry to address the system's imprecision since they are more realistic than type-1 fuzzy numbers in light of the uncertain membership grades. To numerically estimate the solution, the Legendre wavelet method (LWM) is used in conjunction with the r 2 -cut of the r 1 -plane form of the type-2 fuzzy parameters. To test the applicability of the present method, three cases are examined. The present solutions are quantitatively compared with the RK4 solutions in the crisp case. Approximations of the type-2 fuzzy prey and predator populations are also included. In addition, two- and three-dimensional plots are depicted to examine the behaviour of prey and predator populations in crisp and type-2 fuzzy cases. To assess the LWM's efficacy, comparison graphs of the acquired solutions and the results of the RK4 approach are also provided. Numerical validation against the RK4 method in crisp cases demonstrates strong agreement, while revealing contrasting population behaviours in first two scenarios, whereas in the third case, both populations reduce with time. Further, the behaviour of the fuzzy solution has also been shown graphically using the footprint of uncertainty (FOU). • Fractional order Prey–Predator has been solved in this work in type-2 fuzzy uncertain scenario. • In this work, Legendre wavelet based numerical approach is used to find the approximate solution of the model. • Concept of Block Pulse function and integration operational matrix are used in this work to find out the solution. • Comparisons are also made in a special case viz. crisp case for integer order problem with the solution by RK4 method and found to be well agreed. • Use of type-2 fuzzy uncertainty in fractional order Prey–Predator model with wavelet technique is proposed for the first time in the literature. [ABSTRACT FROM AUTHOR]