Mathematical models, especially complex nonlinear systems, are always difficult to analyze and synthesize, and researchers need effective and suitable control methods to address these issues. In the present work, we proposed a hybrid method that combines the well-known Takagi-Sugeno fuzzy model with wavelet decomposition to investigate nonlinear systems characterized by the presence of mixed nonlinearities. Here, one nonlinearity is super-linear and convex, and other is sub-linear, concave, and singular at zero, which leads to difficulties in the analysis, as is known in PDE theory. Linear and polynomial fuzzy models were combined with wavelets to ensure an improvement in both methods for investigating such problems. The results showed a high performance compared with existing methods via error estimates and Lyapunov theory of stability. The model was applied to a prototype nonlinear Schrödinger dynamical system.

Clean water is a necessity for many organisms, especially human life. Due to many factors, there is a significant shortage of potable water. This has led to efforts involving recovering water from wastewater or the sea through different technologies. Recently, the desalination of seawater via the reverse osmosis system has shown to be a promising method for drinking water treatment and recovery. Such a technique relies on mathematical models based on many parameters, resulting in special PDEs to model the reverse osmosis system. This paper develops a numerical method to solve a reverse osmosis model. The governing PDE is converted into a Sylvester equation that is proved to be uniquely solvable, stable, consistent, and convergent. The numerical scheme developed is validated with experimental data from the literature, and some numerical simulations.