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Uncertainties present in the system parameters and/or external forces play a significant role in the static analysis of structures. Recently, fuzzy numbers have been used to model these uncertainties. It may be noted that when uncertainties are considered only for external loading in terms of type-1 or type-2 fuzzy numbers, the problem is converted to a fuzzy system of linear equations with crisp coefficients. In this regard, the purpose of this research is to propose a new technique to solve the fuzzy system of linear equations with crisp coefficients.

It is very interesting to propose effective techniques to solve such problems. A methodology based on the parametric forms of the type-1 and type-2 fuzzy numbers is used to convert the fuzzy system into interval systems of equations. Further, the concept of double and triple parametric forms is utilised to solve the interval systems.

The proposed technique has been successfully implemented to solve the fuzzy linear system of equations in type-1 as well as type-2 fuzzy environments. Accordingly, sample mathematical problems as well as application problems, namely 8-bar truss structure and uniform rectangular sheet structure, have been solved. Further, present results have been contrasted with the solution obtained by the existing approaches and found to be in good agreement.

To deal with the uncertainty in the external loadings, type-1 and type-2 fuzzy numbers are used here. Also, the approach presented in this work to solve the fuzzy linear system of equations shows the originality of this research.

Recently, fractional differential equations have been used to model various physical and engineering problems. One may need a reliable and efficient numerical technique for the solution of these types of differential equations, as sometimes it is not easy to get the analytical solution. However, in general, in the existing investigations, involved parameters and variables are defined exactly, whereas in actual practice it may contain uncertainty because of error in observations, maintenance induced error, etc. Therefore, the purpose of this paper is to find the dynamic response of fractionally damped beam approximately under fuzzy and interval uncertainty.

Here, a semi analytical approach, variational iteration method (VIM), has been considered for the solution. A newly developed form of fuzzy numbers known as double parametric form has been applied to model the uncertainty involved in the system parameters and variables.

VIM has been successfully implemented along with double parametric form of fuzzy number to find the uncertain dynamic responses of the fractionally damped beam. The advantage of this approach is that the solution can be written in power series or compact form. Also, this method converges rapidly to have the accurate solution. The uncertain responses subject to impulse and step loads have also been computed and the behaviours of the responses are analysed. Applying the double parametric form, it reduces the computational cost without separating the fuzzy equation into coupled differential equations as done in traditional approaches.

Uncertain dynamic responses of fuzzy fractionally damped beam using the newly developed double parametric form of fuzzy numbers subject to unit step and impulse loads have been obtained. Gaussian fuzzy numbers are used to model the uncertainties. In the methodology using the alpha cut form, corresponding beam equation is first converted to an interval-based fuzzy equation. Next, it has been transformed to crisp form by applying double parametric form of fuzzy numbers. Finally, VIM has been applied to solve the same for the general fuzzy responses. Various numerical examples have been taken in to consideration.

Vibration of large membranes has great utility in engineering application such as in important parts of drums, pumps, microphones, telephones and other devices. So, to obtain a numerical solution of this type of problems is necessary and important. In general, in existing approaches, involved parameters and variables are defined exactly. Whereas in actual practice, it may contain uncertainty owing to error in observations, maintenance-induced error, etc. So, the main purpose of this paper is to solve this important problem numerically under fuzzy and interval uncertainty to have an uncertain solution and to study its behaviour.

In this study, the authors have considered a new approach is known as double parametric form of fuzzy number to model uncertain parameters. Along with this a semianalytical approach, i.e. variational iteration method, has been used to obtain uncertain bounds of the solution.

The variational iteration method has been successfully implemented along with the double parametric form of fuzzy number to find the uncertain solution of the vibration equation of a large membrane. The advantage of this approach is that the solution can be written in a power series or a compact form. Also, this method converges rapidly to obtain an accurate solution. Various cases depending on the functional value involved in the initial conditions have been studied and the behaviour has been analysed. Applying the double parametric form reduces the computational cost without separating the fuzzy equation into coupled differential equations as done in traditional approaches.

The vibration equation of large membranes has been solved under fuzzy and interval uncertainty. Uncertainties have been considered in the initial conditions. New approaches, i.e. variational iteration method along with the double parametric form, have been applied to solve the vibration equation of large membranes.