A New Constructive Method for Solving the Schrödinger Equation

In this paper, a new method for the exact solution of the stationary, one-dimensional Schrödinger equation is proposed. Application of the method leads to a three-parametric family of exact solutions, previously known only in the limiting cases. The method is based on solutions of the Ricatti equation in the form of a quadratic function with three parameters. The logarithmic derivative of the wave function transforms the Schrödinger equation to the Ricatti equation with arbitrary potential. The Ricatti equation is solved by exploiting the particular symmetry, where a family of discrete transformations preserves the original form of the equation. The method is applied to a one-dimensional Schrödinger equation with a bound states spectrum. By extending the results of the Ricatti equation to the Schrödinger equation the three-parametric solutions for wave functions and energy spectrum are obtained. This three-parametric family of exact solutions is defined on compact support, as well as on the whole real axis in the limiting case, and corresponds to a uniquely defined form of potential. Celebrated exactly solvable cases of special potentials like harmonic oscillator potential, Coulomb potential, infinite square well potential with corresponding energy spectrum and wave functions follow from the general form by appropriate selection of parameters values. The first two of these potentials with corresponding solutions, which are defined on the whole axis and half axis respectively, are achieved by taking the limit of general three-parametric solutions, where one of the parameters approaches a certain, well-defined value.

New Perception of the Optical Solutions to the Fokas-Lenells Equation According to Two Different Techniques

In this article, the perturbed Fokas-Lenells equation (FLE)” which plays a vital role in modern asocial media and electronic communication” is employed. Two important different methods are invited to demonstrating new accurate solutions of this equation. The first method is the modified simple equation method (MSEM) that reduces large volume of calculations and realizes closed form solution. While the second is the modified extended tanh-function method (METFM) “which controlled by the auxiliary Ricatti equation” and used effectively to obtain accurate solutions Furthermore, few of the realized results are compatible with that obtained by previous authors elsewhere the others remains new. In addition to the varitional iteration method (VIM )is applied perfectly to achieved the numerical solution corresponding to the exact solution realized by each one of these methods individually.

Feedback stabilization for one sided Lipschitz nonlinear systems in reciprocal state space: Synthesis and experimental validation

This paper proposes a design of a stabilization control for one-sided Lipschitz (OSL) nonlinear systems in new reciprocal state space (RSS) framework. The main objective is to extend the state derivative feedback stabilization methods for a class of nonlinear systems where the nonlinearity of derivatives state satisfies the OSL properties in RSS. The presented controller is composed of a state derivative feedback approach in order to ensure asymptotic stability in the sense of Lyapunov. The first approach deals with the synthesis of a basic controller by adopting a simple transformation of Linear Matrix Inequality (LMI) to standard algebraic Ricatti equation (ARE). The second is an extension to adaptive version with adjustment parameters. High performances are shown through real-time implementation with a hardware in the loop (HIL) mode using digital signal processing (DSP) device (DSpace DS 1104).

Model predictive Controller for Mobile Robot

This paper proposes a Model Predictive Controller (MPC) for control of a P2AT mobile robot. MPC refers to a group of controllers that employ a distinctly identical model of process to predict its future behavior over an extended prediction horizon. The design of a MPC is formulated as an optimal control problem. Then this problem is considered as linear quadratic equation (LQR) and is solved by making use of Ricatti equation. To show the effectiveness of the proposed method this controller is implemented on a real robot. The comparison between a PID controller, adaptive controller, and the MPC illustrates advantage of the designed controller and its ability for exact control of the robot on a specified guide path.

On nonlinear horizontal dynamics and vibrations control for high-speed elevators

In this work, the horizontal nonlinear response of a three-degree-of-freedom vertical transportation model excited by guide rail deformations is investigated. The equation of motion contains nonlinearities in the form of Duffing stiffness for the translational spring in tilting motion of the cabin. In order to improve the comfort for passengers a control strategy based on the State-dependent Ricatti Equation (SDRE) is proposed. Numerical simulations are performed to study the nonlinear behavior of the adopted mathematical model. In addition, we test the robustness of the SDRE control technique considering parametric errors and noise. The obtained results confirm that the proposed strategy can be effective in controlling the response of the system.

Acoustic boundary conditions at an impedance lining in inviscid shear flow

The accuracy of existing impedance boundary conditions is investigated, and new impedance boundary conditions are derived, for lined ducts with inviscid shear flow. The accuracy of the Ingard–Myers boundary condition is found to be poor. Matched asymptotic expansions are used to derive a boundary condition accurate to second order in the boundary layer thickness, which shows substantially increased accuracy for thin boundary layers when compared with both the Ingard–Myers boundary condition and its recent first-order correction. Closed-form approximate boundary conditions are also derived using a single Runge–Kutta step to solve an impedance Ricatti equation, leading to a boundary condition that performs reasonably even for thicker boundary layers. Surface modes and temporal stability are also investigated.

On CR volume growth estimate in a complete pseudohermitian 3-manifold

In this paper, we first derive the CR Reilly's formula and a CR Ricatti equation for sub-Laplacian of the Carnot–Carathéodory distance in a complete pseudohermitian 3-manifold. As a consequence, we obtain the CR volume growth estimate in a complete pseudohermitian 3-manifold under a lower bound of pseudohermitian curvature tensors. This is a generalization of Nagel, Stein and Wainger's volume growth estimate for the Heisenberg ball in the standard Heisenberg group.

Dynamic Finite Element Analysis of a Highly Parallel Robotic Surface

A novel three-dimensional robotic surface is devised using triangular modules connected by revolute joints that mimic the constraints of a spherical joint at each triangle intersection. The finite element method (FEM) is applied to the dynamic loading of this device using three dimensional (6 degrees of freedom) beam elements to not only calculate the cartesian displacement and force, but also the angular displacement and torque at each joint. In this way, the traditional methods of finding joint forces and torques are completely bypassed. An effiecient algorithm is developed to linearly combine local mass and stiffness matrices into a full structural stiffness matrix for the easy application of loads. An analysis of optimal dynamic joint forces is carried out in Simulink® with the use of an algebraic Ricatti equation.