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.

This paper presents a novel technique to handle uncertainties in coupled systems arising in modelling biological systems. The methods are described and applied to two of the most common biological systems (one linear and another nonlinear) to illustrate the effect of imprecise coefficients in models. We have compared the results obtained with the crisp solution of the models to show the validity and applicability of the method.

Purpose: The vibration of large membranes is a powerful part of engineering applications, such as in the components of drums, pumps, telephones, microphones, and other equipment’s. Various researchers have investigated the above topic, and the variables and parameters have been provided in a crisp/exact manner. Although this may be accurate in theory, it is possible that, in practice, it contains uncertainty due to inaccuracies in observations, maintenance-induced inaccuracies, and other sources of error. So, the primary objective of this paper is to solve this important problem numerically under interval and fuzzy uncertainty to have an uncertain solution and to study its behaviour. Methods: In this study, we consider these uncertainties as fuzzy/intervals and employ a technique, namely the double parametric form of fuzzy numbers (DPFFNs), to solve the uncertain fractional vibration model of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \hspace{1mm} (1 < \eta \le 2)$$\end{document}. In this titled problem has been solved for forward and inverse in various cases using the Adomian decomposition method (ADM). Results: The Adomian decomposition method has been successfully implemented along with the double parametric form to find forward and inverse problems of the time-fractional vibration equation of large membranes in an uncertain environment. The solution is expressed in compact or power series form, which is an advantage of this technique. In addition, this approach converges quickly to a precise solution. In this work, moving to the forward case, we have found fuzzy displacements, and in the inverse case, we found fuzzy velocities of the model problem successfully. Conclusion: The forward and inverse problems of the time-fractional vibration equation of large membranes in an uncertain environment have been solved by the Adomian decomposition method. The present approach’s computational efficiency is good and reliable for providing an approximate numerical solution for various cases. The obtained results are illustrated graphically and compared to particular cases.

Purpose: The investigation of the time-fractional vibration equation (TFVE) of large membranes is significant because of its well-recognised practical applications. Several studies have been conducted on the above topic, and the variables and parameters have been provided in a crisp/exact manner. In practical application, these parameters may not be precisely understood but rather exist in an uncertain or approximate form. In this study, we analyse the uncertainties as an interval/fuzzy environment. Methods: In this study, we analyse the uncertainties as interval/fuzzy and suggest a novel approach called the double parametric form of fuzzy numbers to resolve the uncertain fractional vibration model of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1 < \Theta \le 2)$$\end{document}related to large membranes. The modified Adomian decomposition method (MADM) has finally been used to solve this problem. Results and Conclusions: The modified Adomian decomposition method has been successfully implemented along with the double parametric form to find vibration equation of large membranes in uncertain environment. Regarding computing efficiency, the present technique works well. It is shown that the approach is reliable for providing an approximate numerical solution for various cases. Plots are used to present the results, which are compared in particular cases.

This article presents a comprehensive analysis to investigate the static buckling stability and static deflection of axially single-walled (SW) functionally graded (FG) carbon nanotubes reinforced composite (CNTRC) plates with temperature-dependent material properties and graded by different functions, for the first time. The distribution of the carbon nanotubes is described by two functions, one for the x-direction CNTs distribution (CNTRC-xplate) and another for z-direction CNTs distribution (CNTRC-zplate). The graduation functions of CNTs are unidirectional (UD CNTRC), FG-X CNTRC, FG-O CNTRC, and FG-V CNTRC. The extended rule of mixture and the molecular dynamics simulations are exploited to evaluate the equivalent mechanical properties of FG-CNTRC plate. Equilibrium equations are formulated using principal of Hamilton and solved analytically using Galerkin method to cover various boundary conditions. New higher order shear deformation theory is proposed. The numerical results gained by the proposed solution are verified by comparing with those of published ones. Numerical results present influences of gradation function, inhomogeneity parameters, aspect ratio, thickness ratio, boundary conditions and temperature on the static buckling and deflection of FG-CNTRC plate using modified higher order shear theories.

The present paper proposed a new method to solve nthorder fuzzy differential equations using collocation type of method. In the solution procedure, Gram Schmidt orthogonalisation process is used with Legendre and Chebyshev polynomials. Known example problems are solved and compared with the exact results to illustrate the efficiency and reliability of the proposed method.

This article aims to investigate the flow of water-based nanofluid between two inclined planes that are permitted to stretch/shrink. The Optimal Homotopy Analysis Method (OHAM) has been employed here to handle governing coupled nonlinear Differential Equations (DEs). An efficient approach has been used to calculate the convergence controller parameters used in OHAM, which minimizes the complexity of the calculations. Convergence of the present method is observed numerically from the obtained numerical results for the different number of terms in the series solution. For validation, present results have been compared with the numerical results obtained from benchmark MATLAB solver BPV4C. Also, the Nusselt number and coefficient of skin friction obtained from the present method have been compared with the previously published results. Further, the accuracy of the obtained results has been seen by calculating residual errors for successive approximations. Computations show that the obtained results converge up to three decimal places, and residual errors are close enough to zero in four terms of approximations. Finally, the effects of different physical parameters on velocity and temperature profiles are illustrated graphically by varying a single parameter with fixed values of the remaining ones. All possible combinations are considered, such as shrinking wall/convergent channel, shrinking wall/divergent channel, stretching wall/convergent channel, and stretching wall/divergent channel. Effects of the stretching and shrinking parameters are of main importance in this article. The results show that velocity value increases and temperature value decreases by increasing the stretching/shrinking parameter in both cases viz. divergent and convergent cases. • Study of Jeffery-Hamel flow problem with stretching/shrinking walls. • An efficiency technique has been used to calculate convergence controller parameters. • Convergence of the used method has been studied numerically, and residual errors have been discussed. • The coefficient of skin fraction and Nusselt number have been used to observe the efficacy of the method. • Effects of all involved parameters on velocity and temperature are discussed. [ABSTRACT FROM AUTHOR]

This paper proposes a new method to solve n th order fuzzy differential equations using collocation type of method. In the solution procedure Legendre polynomials are used in the collocation method. Three different cases have been considered for the analysis. Known example problems are solved using the proposed procedure. Obtained results are compared with the exact solution in order to illustrate the efficiency and reliability of the proposed method. Solutions are depicted in term of figures and tables.

Dynamical behaviors of the time-fractional nonlinear model of the coupled spring-mass system with damping have been explored here. Fractional derivatives with singular and nonsingular kernels are used to assess the suggested model. The fractional Adams–Bashforth numerical method based on Lagrange polynomial interpolation is applied to solve the system with nonlocal operators. Existence, Ulam–Hyers stability, and uniqueness of the solution are established by using fixed-point theory and nonlinear analysis. Further, the error analysis of the present method has also been included. Finally, the behavior of the solution is explained by graphical representations through numerical simulations.

Much of the time, geological data are not accurately known. Instead, geoscientists work with a range of possible data. These data can have a range since they can vary spatially and temporally. In this context, we demonstrate how a well-established structural geological model of velocity profile of ductile simple shear, expressed as an equation, can be furthered by fuzzy set concept. The implementation of fuzzy set concepts in terms of fuzzification in (geo)scientific models takes into consideration the uncertainties of magnitudes of the parameters that define the model. The present study shows that the fuzzified velocity profile of a ductile simple shear zone with parallel boundaries is effectively independent of the dip of the shear zone. This article demonstrates how models of (structural) geological processes can be modelled by fuzzy set approach capable of incorporating a range of magnitudes of parameters.