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This paper investigates a robust adaptive second-order sliding mode control method for tracking problem of a class of uncertain linear systems with matched and unmatched disturbances. The fundamental idea of the suggested control method is that the discontinuous sign function is used for the time-derivative of the control signal and hence the smooth control input achieved after an integration process is continuous and removes the chattering problem. Using a PID sliding surface, the finite-time convergence of output tracking errors is obtained. The adaptive gain-tuning control law removes the necessity of gaining information about the upper bounds of the external disturbances. The control system is in the sliding mode and then, tracking errors converge to the origin in a finite time under the presence of the external disturbances. Simulation results on an uncertain numerical system and a turntable servo-system are presented to indicate the effectiveness and feasibility of the proposed scheme.

This work presents an investigation and analysis for the space-time Carleman equation (STCE) in nonlinear dynamical systems. We compute the point symmetries, similarity variable, similarity transformation for STCE with Riemann-Liouville (RL) derivative and reduce STCE to ordinary differential equation (ODE) of fractional order. The exact solutions with conformable derivative are obtained via the generalized Bernoulli (GB) sub-ODE method. The well known residual power series technique (RPST) is used to compute the corresponding approximate solutions for the obtained exact solution. We then verify the convergence analysis and error estimate of RPST. Numerical simulations of the results are shown with the aid of graphical illustrations and tables. We prove that the RPST is very efficient for investigating the numerical approximations for a system of fractional differential equations.

In this paper, we propose a design approach of composite nonlinear feedback control technique for the synchronization of master/slave nonlinear systems with time-varying delays, Lipschitz nonlinear functions and parametric uncertainties. Based on the Lyapunov–Krasovskii stabilization theory and linear matrix inequalities, a new sufficient condition is generated for the synchronization of chaotic systems with nonlinearities and perturbations on the master and slave systems. By using the Barbalat’s lemma, the proposed control method guarantees that the states of the master and slave systems are synchronized with an asymptotic convergence rate. Simulation results are demonstrated on two forms of Chua’s chaotic system, which illustrate that the suggested design technique yields satisfactory transient performance.

The aim of this paper is to obtain optical soliton solutions by nonlinear dispersion in parabolic law medium by using Jacobi elliptic functions. The presented problem is studied with Kerr law nonlinearity, and dark and bright solitons are acquired. Also, we develop the stability analysis for problem.

In this paper, the adaptive robust tracking and model following controller is designed for the uncertain nonlinear systems with disturbances, time-delays and input nonlinearities. A novel control law based on the linear matrix inequalities is proposed for the tracking control of the nonlinear systems with Lipschitz nonlinearities. The selection procedure of the nonlinear integral sliding surface and the existence of the proposed sliding mode are two significant subjects, which have been addressed. This controller guarantees that the tracking error decreases asymptotically to the origin in the presence of time-varying perturbations. By elimination of the reaching mode of the sliding surface, the robustness performance against perturbations is guaranteed right from the initial instant. Furthermore, the knowledge of the upper bounds of the uncertainties, disturbances and time-delays is not required to be determined. Simulation results are provided to illustrate the efficiency of the presented technique compared to the previous methods.

In this work, the coupled nonlinear Schrodinger’s equation (CNLSE) is studied with four forms of nonlinearity. The nonlinearities that are considered in this paper are the Kerr law, power law, parabolic law and dual-power law. Jacobi elliptic function solutions and also bright and dark optical soliton solutions are obtained for each law of the CNLSE. We will acquire constraint conditions for the existence of obtained solitons.

The current work provides comprehensive investigation for the time fractional third-order variant Boussinesq system (TFTOBS) with Riemann-Liouville (RL) derivative. Firstly, we obtain point symmetries, similarity variables, similarity transformation and reduce the governing equation to a special system of ordinary differential equation (ODE) of fractional order. The reduced equation is in the Erdelyi-Kober (EK) sense. Secondly, we solve the reduced system of ODE using the power series (PS) expansion method. The convergence analysis for the power series solution is analyzed and investigated. Thirdly, the new conservation theorem and the generalization of the Noether operators are applied to construct nonlocal conservation laws (CLs) for the TFTOBS. Finally, we use residual power series (RPS) to extract numerical approximation for the governing equations. Interesting figures that explain the physical understanding for both the explicit and approximate solutions are also presented.

In this paper, we analyze the dynamic behavior of the ill-posed Boussinesq equation (IPBE) that arises in nonlinear lattices and also in shallow water waves. Some solitary wave solutions are obtained by using the solitary wave ansatz method and the Bernoulli sub-Ode. By applying the technique of nonlinear self-adjoint, a quasi self-adjoint substitution for the IPBE is constructed. The classical symmetries of the equation are constructed. Then, we used along with the obtained nonlinear self-adjoint substitution to construct a set of new conservation laws (Cls).