The Klein–Fock–Gordon equation (KFGE), defined as the equation of relativistic wave related to NLEEs, has numerous implications for energy particle physics and is useful as a model for several types of matter, with deviation in the basic stuffs of particles and in crystals. In this work, the Sardar subequation method (SSM) is used for finding the solution of this KFGE. The advantage of SSM is that it provides many different kinds of solitons, such as dark, bright, singular, periodic singular, combined dark–singular and combined dark–bright solitons. The results show that the SSM is very reliable, simple and can be functionalized to other nonlinear equations. It is verified that all the attained solutions are stable by modulation instability process. To enhance the physical description of solutions, some 3D, contour and 2D graphs are plotted by taking precise values of parameters using Maple 18.

The purpose of this article is to achieve new soliton solutions of the Gilson–Pickering equation (GPE) with the assistance of Sardar’s subequation method (SSM) and Jacobi elliptic function method (JEFM). The applications of the GPE is wider because we study some valuable and vital equations such as Fornberg–Whitham equation (FWE), Rosenau–Hyman equation (RHE) and Fuchssteiner–Fokas–Camassa–Holm equation (FFCHE) obtained by particular choices of parameters involved in the GPE. Many techniques are available to convert PDEs into ODEs for extracting wave solutions. Most of these techniques are a case of symmetry reduction, known as nonclassical symmetry. In our work, this approach is used to convert a PDE to an ODE and obtain the exact solutions of the NLPDE. The solutions obtained are unique, remarkable, and significant for readers. Mathematica 11 software is used to derive the solutions of the presented model. Moreover, the diagrams of the acquired solutions for distinct values of parameters were demonstrated in two and three dimensions along with contour plots.

Modern manufacturing has to cope with dynamic and changing circumstances. Market fluctuations, the effects caused by unpredictable material shortages, highly variable product demand, and worker availability all require system robustness, flexibility, and resilience. To adapt to these new requirements, manufacturers should consider investigating, investing in, and implementing system autonomy. Autonomy is being adopted in multiple industrial contexts, but divergences arise when formalizing the concept of autonomous systems. To develop an implementation of autonomous manufacturing systems, it is essential to specify what autonomy means, how autonomous manufacturing systems are different from other autonomous systems, and how autonomous manufacturing systems are identified and achieved through the main features and enabling technologies. With a comprehensive literature review, this paper provides a definition of autonomy in the manufacturing context, infers the features of autonomy from different engineering domains, and presents a five-level model of autonomy — associated with maturity levels for the features — to ensure the complete identification and evaluation of autonomous manufacturing systems. The paper also presents the evaluation of a real autonomous system that serves as a use-case and a validation of the model.