Phenol formaldehyde (phenolic resin) has a wide range of moldings. However, it has immense consumption in manufacturing electrical equipment due to its insulating property. Phenolic resin retains properties at the freezing point, and also its age cannot be determined. Due to its insulator property, it has wide use in electrical equipment. In this article, degree-based topological indices of phenol formaldehyde are determined with the help of M-polynomial. We calculate the Zagreb index, Randić index, K-Banhatti indices, modified K-Banhatti indices, atom-bond connectivity index, geometric arithmetic index, symmetric index, inverse sum index, and harmonic index.

In this study, we introduced a novel scheme to attain approximate and closed-form solutions of conformable Newell-Whitehead-Segel (NWS) equations, which belong to the most consequential amplitude equations in physics. The conformable Shehu transform (CST) and the Adomian decomposition method (ADM) are combined in the proposed method. We call it the conformable Shehu decomposition method (CSDM). To assess the efficiency and consistency of the recommended method, we demonstrate 2D and 3D graphs as well as numerical simulations of the derived solutions. As a result, CSDM demonstrates that it is a useful and simple mathematical tool for getting approximate and exact analytical solutions to linear-nonlinear fractional partial differential equations (PDEs) of the given kind. The convergence and absolute error analysis of the series solutions is also offered.

Phenol-formaldehyde resin has a wide range of moldings. The phenolic resin retains properties at the freezing point; hence, it is difficult to determine its age. However, it has immense consumption in manufacturing electrical equipment due to its insulating property. There are many types of topological indices such as degree-based topological indices, distance-based topological indices, etc. Topological indices correlate some physiochemical properties of chemical compounds. In this article, the degree-based topological indices of phenol-formaldehyde resin have been determined. Furthermore, the Revan index, hyper Revan index, modified Revan index, sum connectivity Revan index, harmonic Revan index, and inverse Revan index have been calculated.

In this study, the highly accurate analytical Aboodh transform decomposition method (ATDM) in the sense of Caputo fractional derivative is used to determine the approximate and exact solutions of both linear and nonlinear time-fractional Schrodinger differential equations (SDEs) with zero and nonzero trapping potential that describe the nonrelativistic quantum mechanical activity. The Adomian decomposition method (ADM) and the Aboodh transform of Caputo’s fractional derivative are combined in this method. The recurrence and absolute error of the four problems are analyzed to evaluate the efficiency and consistency of the presented method. In addition, numerical results are also compared with other methods such as the fractional reduced differential transform method (FRDTM), the homotopy analysis method (HAM), and the homotopy perturbation method (HPM). The results obtained by the proposed method show excellent agreement with these methods, which indicates its effectiveness and reliability. This technique has the benefit of not requiring any minor or major physical parameter assumptions in the problem. As a result, it may be used to solve both weakly and strongly nonlinear problems, overcoming some of the inherent constraints of classic perturbation approaches. To solve nonlinear fractional-order differential equations, just a few computations are necessary. As a consequence, it outperforms homotopy analysis and homotopy perturbation approaches significantly. The procedure is quick, precise, and easy to implement. Convergence analysis of the series solution is also offered.

The molecular topology of a graph is described by topological indices, which are numerical measures. In theoretical chemistry, topological indices are numerical quantities that are used to represent the molecular topology of networks. These topological indices can be used to calculate several physical and chemical properties of chemical compounds, such as boiling point, entropy, heat generation, and vaporization enthalpy. Graph theory comes in handy when looking at the link between certain topological indices of some derived graphs. In the ongoing research, we determine ve-degree, ev-degree, and degree-based (D-based) topological indices of fenofibrate’s chemical structure. These topological indices are the Zagreb index, general Randić index, modified Zagreb index, and forgotten topological index. These indices are very helpful to study the characterization of the given structure.