Double Parametric Form Of Fuzzy Numbers Research Articles

Fourth-Order Numerical Solutions for a Fuzzy Time-Fractional Convection–Diffusion Equation under Caputo Generalized Hukuhara Derivative

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convection–diffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values.

Stability analysis of Timoshenko nanobeam with material uncertainties using a double-parametric form-based analytical approach and Monte Carlo simulation technique

This study is aimed at analyzing the effects of material uncertainties on the stability of Timoshenko nanobeam. Here, the uncertainties are considered to be associated with the diameter, length, and Young’s modulus of the nanobeam in terms of special fuzzy number, namely Symmetric Gaussian Fuzzy Number (SGFN). The governing equations for stability analysis of uncertain model are obtained by incorporating the Timoshenko beam theory along with Hamilton’s principle and double-parametric form of fuzzy numbers. The small-scale effect of the nanobeam is addressed by Eringen’s elasticity theory, and the double-parametric form-based Navier’s method is employed to calculate the results of the uncertain models for the lower bound (LB) and upper bound (UB) of the buckling loads. The results obtained by the uncertain model are validated with the deterministic model in particular case as well as with the results obtained by Monte Carlo simulation (MCS) technique in terms of the lower bound and upper bound of the buckling loads, exhibiting robust agreement. Further, a parametric study is conducted to investigate the fuzziness or spreads of the buckling loads with respect to different uncertain parameters.

Uncertain Dynamic Responses of Fuzzy Arbitrary-Order Damped Beam

Uncertain dynamic responses of fuzzy fractionally damped beams have been studied using the newly developed double parametric form of fuzzy numbers subject to unit step and impulse loads. Uncertainties are assumed to be present in the initial conditions, and these are modeled through triangular convex normalized fuzzy sets. Using the alpha-cut form, the corresponding beam equation is first converted to an interval-based fuzzy equation. Next, it has been transformed to crisp form by applying a double-parametric form of fuzzy numbers. Finally, homotopy perturbation method (HPM) has been used to solve the same for obtaining the general fuzzy responses. Various numerical examples are taken into consideration. The results are plotted.