A complete and partial integrability technique of the Lorenz system

In this paper we deal with the well-known nonlinear Lorenz system that describes the deterministic chaos phenomenon. We consider an interesting problem with time-varying phenomena in quantum optics. Then we establish from the motion equations the passage to the Lorenz system. Furthermore, we show that the reduction to the third order non linear equation can be performed. Therefore, the obtained differential equation can be analytically solved in some special cases and transformed to Abel, Dufing, Painlev\'{e} and generalized Emden-Fowler equations. So, a motivating technique that permitted a complete and partial integrability of the Lorenz system is presented.Comment: 10 pages, 2 figure

Investigation of TCSC Controller Effect on IDMT Directional Over-current Relay

AbstractWith the presence of FACTS controllers in power system transmission for enhancing the power system controllability and stability, the problem of coordinating protective relays becomes more challenging. The over-current relay is widely used in many protection applications throughout distribution network and theIDMT directional over-current Relay is one of the most important protection systems on transmission lines. Its function would generally be changed in presence of FACTS devices.In this paper a study for obtaining the direct effects of the varying reactance of the TCSC with respect to the firing angle alpha on short-circuit parameters of three phase fault and DOCR operating time is investigated. The simulation of the linear programming technique is performed in Matlab software environment. The case study is compared between compensated and uncompensated system

Generalized Thomas-Fermi equation: existence, uniqueness, and analytic approximation solutions

The existence and uniqueness theorem for the generalized boundary value problem of the Thomas-Fermi equation: \begin{eqnarray*} \left\{ \begin{array}{l} y''+f(x, y) = 0, \ 0<x <\infty, \\ y(0) = 1, \ y(\infty) = 0, \end{array} \right. \end{eqnarray*} where \begin{equation*} \label{6}f(x, y) = -y \left(\frac{y}{x}\right)^{\frac{p}{p+1}}, \ p>0, \ 0<x <\infty, \end{equation*} is proved. Also, highly accurate approximate solutions are obtained explicitly for this new boundary value problem which arises in particular studies of many-electron systems (atoms, ions, molecules, metals, crystals). To the best of our knowledge, the results obtained here are new and provide the lower and upper bounds approximate solutions for the generalized Thomas-Fermi problem

On a generalization of a coupled integrable dispersionless system

In this paper, we are concerned with a generalization of a coupled integrable dispersionless system, which plays a crucial role in nonlinear problems. We show with the appropriate transformations that the solutions can be expressed in terms of those of the sine-Gordon and sinh-Gordon equations involving an arbitrary function. The exact solutions are investigated