Introduction The time dependent Schrö dinger equation with arbitrary potential function is a fundamental equation of quantum mechanics. It is well known that Schrö dinger equation play an essential role in the relativistic and non-relativistic quantum mechanics. This equation arises in many other practical domains of physical and technological interest, e. g. optics, seismology and plasma physics. Many problems of solid state physics, atomic and molecular structure and spectra, molecular dynamics and quantum chemistry require the solution of the nonlinear time dependent Schrö dinger equation. ...

Point reactor kinetics equations with one group of delayed neutrons in the presence of the time-dependent external neutron source are solved analytically during the start-up of a nuclear reactor. Our model incorporates the random nature of the source and linear reactivity variation. We establish a general relationship between the expectation values of source intensity and the expectation values of neutron density of the sub-critical reactor by ignoring the term of the second derivative for neutron density in neutron point kinetics equations. The results of the analytical solution are in good agreement with the results obtained with numerical solution.

In this illuminating research, the classical Euler-Poisson equations (describing the dynamics of rigid body rotations over the fixed point) have been revisited for the case of a rigid body that is under the influence of a time-dependent temperature field. It is shown that two classical first integrals of motion remain the same whereas the integral of energy should be updated accordingly to slowly variable principal moments of inertia {Ii} (i = 1, 2, 3) stemming from the temperature-dependent sizes of a rigid body. The revisited Euler-Poisson equations are reduced to the system of 3 nonlinear ordinary differential equations of 1-st order in regard to 3 functions Ki = {Ii Ωi} (Ωi are the components of angular velocity along the principal axes) for which the elegant approximate semi-analytical solutions have been obtained with numerical findings supported by graphical plots. Such theoretical findings can be useful in taking into account the stabilization of gyroscopes supporting the system of orientations of spacecraft on orbits since regimes of drastically changing temperature for technical equipment do exist in outer space.

Error estimates for element schemes for time-dependent for convection-di, usionreaction equations with memory are derived and stated. For the spatially discrete scheme, optimal order error estimates in L2,H1,and W1, p norms for 2 ,p < 1,are obtained. In this paper, we also study the lumped mass modi, cation. Based on the Crank-Nicolson method, a time discretization scheme is discussed and related error estimates are derived.

The paper presents a new method for in site discharge estimation in pressured pipes. The method consists in using the water hammer equations solved with the method of characteristics with an unsteady friction factor model. The differential pressure head variation measured during a complete valve closure is used to derive the initial flow rate, similarly to the pressure-time (Gibson) method. The method is validated with a numerical experiment, and tested with experimental laboratory measurements. The results show that the proposed method can reduce the discharge estimation error by 0.6% compared to the standard pressure-time (Gibson) method for the flow rate investigation.

In this paper, we discuss the existence, uniqueness and sta-bility of mild solutions of time-dependent impulsive neutral stochastic partial integrodi, erential equations with fractional Brownian motion and Poisson jumps. The existence of mild solutions for the equations are discussed by means of semigroup theory and theory of resolvent op-erator. Next, certain su, cient conditions and results are obtained by using the method of successive approximation and Bihari's inequality. Finally, an example is provided to illustrate our results.

In this paper, we have applied an iterative method to the singular and nonlinear fractional partial differential of Emden– Fowler equations types. Haar wavelets operational matrix of fractional integration will be used to solve the problem with the Picard technique. The singular equations turn to Sylvester equations that will be solved so that numerically solvable is very cost-effective. Moreover, the proposed technique is reliable enough to overcome the difficulty of the singular point at x = 0. Numerical examples are providing to illustrate the efficiency and accuracy of the technique.

In this work, we investigate a timedependent dissipative scalar field. Inspired by the Bateman lagrangian, we showed a procedure for quantizatizing the massive scalar field with the time-dependent dissipative term. So, the properties of a quantum dissipative scalar field are analyzed by the Caldirola-Kanai model. Also, we construct the Hilbert space for the system and calculate the wave eigen-functions and probability distribution of the system.