<abstract><p>In this paper, we used the Natural decomposition approach with nonsingular kernel derivatives to explore the modified Boussinesq and approximate long wave equations. These equations are crucial in defining the features of shallow water waves using a specific dispersion relationship. In this research, the convergence analysis and error analysis have been provided. The fractional derivatives Atangana-Baleanu and Caputo-Fabrizio are utilised throughout the paper. To obtain the equations results, we used Natural transform on fractional-order modified Boussinesq and approximate long wave equations, followed by inverse Natural transform. To verify the approach, we focused on two systems and compared them to the exact solutions. We compare exact and analytical results with the use of graphs and tables, which are in strong agreement with each other, to demonstrate the effectiveness of the suggested approaches. Also compared are the results achieved by implementing the suggested approaches at various fractional orders, confirming that the result comes closer to the exact solution as the value moves from fractional to integer order. The numerical and graphical results show that the suggested scheme is computationally very accurate and simple to investigate and solve fractional coupled nonlinear complicated phenomena that exist in science and technology.</p></abstract>

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. The fractional derivative is defined in Caputo manner. The proposed method is simple, effective, and straightforward as compared to other numerical techniques. To check the validity and accuracy of the proposed method, some illustrative examples are solved by using the present scenario. The obtained results have confirmed the greater accuracy than the modified Laguerre wavelet method, the Chebyshev wavelet method, and the modified wavelet-based algorithm. Moreover, based on the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations.

In this research, we provide tools to overcome the information loss limitation resulting from the requirement to estimate the results in the discrete initial expression domain. Through the use of 2-tuples, which are made up of a linguistic term and a numerical value calculated between [0.5,0.5), the linguistic information will be expressed. This model supports continuous representation of the linguistic data within its scope, permitting it to express any information counting received through an aggregation procedure. This study provides a novel approach to develop a linguistic multi-attribute group decision-making (MAGDM) approach with complex fractional orthotriple fuzzy 2-tuple linguistic (CFOF2TL) assessment details. Initially, the concept of a complex fractional orthotriple fuzzy 2-tuple linguistic set (CFO2TLS) is proposed to convey uncertain and fuzzy information. In the meantime, simple aggregation operators, such as CFOF2TL weighted average and geometric operators, are defined. In addition, the CFOF2TL Maclaurin’s symmetric mean (CFOF2TLMSM) operators and their weighted shapes are presented, and their attractive characteristics are also discussed. A new MAGDM approach is built using the developed aggregation operators to address managing economic crises under COVID-19 with the CFOF2TL information. As a result, the effectiveness and robustness of the developed method are accompanied by an empirical example, and a comparative study is carried out by contrasting it with previous approaches.

In this study, we focused on a subclass of bounded turning functions that are linked with a four-leaf-type domain. The primary goal of this study is to explore the limits of the first four initial coefficients, the Fekete-Szegö type inequality, the Zalcman inequality, the Kruskal inequality, and the estimation of the second-order Hankel determinant for functions in this class. All of the obtained findings have been sharp.

The approximate solutions of the time fractional advection-dispersion equation are presented in this article. The nonlocal nature of solute movement and the nonuniformity of fluid flow velocity in the advection-dispersion process lead to the formation of a heterogeneous system, which can be modeled using a fractional advection-dispersion equation, which generalizes the classical advection-dispersion equation and replaces the time derivative with the fractional Caputo derivative. Researchers use a variety of numerical techniques to study such fractional models, but the nonlocality of the derivative having fractional order leads to high computation complexity and complex calculations, so the task is to find an efficient technique that requires less computation and provides greater accuracy when numerically solving such models. A innovative techniques, homotopy perturbation method and new iteration method, are used in connection with the Elzaki transform to solve the “fractional advection-dispersion equation” which provides the solution in the convergent series form. When the homotopy perturbation method is used with the Elzaki transform, fast convergent series solutions can be obtained with less computation. By solving some cases of time-fractional advection-dispersion equation with varied initial conditions with the help of new iterative transform method and homotopy perturbation transform method demonstrates the usefulness of the proposed methods.