The dynamics of a composite consisting of the nonlinear multilayer beam structure, interacting through elastic intermediate layers, under mobile point loading is investigated. This study finds a direct application in transport engineering technologies, more precisely in railways, where the moving point load is the train, and the multilayer beam, the rails interacting with the ballast, the foundation and base layers. From the Lagrange formulations, the system of damping partial differential equations of the model is found, and by considering the non-dissipative case with weak nonlinearity and constant charge they are used to find the eigen modes and the natural vibration frequencies of the system. Then the dissipative case with nonlinearity is studied, with a particular attention carried on the temporal part, which is reduced to a system of coupled nonlinear differential equations, where the first line is forced. This system of equation is used to determine the equilibrium points, after which they are subsequently solved analytically through the multiple time scale method for harmonic resonance case, showing the formation of hysteresis more and more complex as the number of cells increases. The coupled nonlinear equations of the system is next solved numerically, with the transition of the system towards chaos analyzed through the bifurcation diagram and the maximum Lyapunov exponent, which show strong sensitivity to the coupling parameter λ₂ as well as the system frequency. The results show for N = 2 and for some parameters the periodic behavior and the crisis for ω = 0.5. When the frequency is low; that is ω = 0.05 the chaotic band is considerably reduced, chaos appearing around the nonlinearity parameter γ₂ = 0.5 and also for γ₂ > 0.85. The time trace shows chaotic pulses and bursting type behavior, for some choices of the coupling parameter. The synchronization curves are also plotted and it is shown that q₂ doesn’t synchronizes with q₁ for some frequencies, while for others parameters, they synchronize, but fairly. For N = 3, the dynamics is more complex and the time traces plots show regular impulse for ω = 0.5 and bursting for weak frequency, ω = 0.05.

In this research paper, a spectral method is used for numerically solving the time-fractional diffusion equation as the time fractional diffusion equations are a powerful tool for simulating physical systems. We employ the Lucas polynomials (LPs) with Petrov-Galerkin for the linear combination basis. The main idea of the proposed technique is to convert the governed boundary-value problem into a system of linear algebraic equations by applying the Petrov-Galerkin method. Many procedures can solve the resulting linear system. The method’s accuracy is shown through several examples.

This note delves into the convergence analysis of several iterative methods and elucidates their behaviors. Furthermore, we demonstrate that the findings presented in “A new three-step fixed point iteration scheme with strong convergence and applications” are not entirely novel. In particular, some of the results either overlap with or restate previously established methods without introducing significant innovations.