Pade approximations

The magnetohydrodynamic (MHD) boundary layer flow of a Casson fluid over an exponentially permeable shrinking sheet has been investigated. The analytical solution arising differential system has been computed by the Adomian Decomposition Method (ADM). Variations of interesting parameters on the velocity are observed by plotting graphs.

In this paper the Proportional integral controllers based on Internal Model Control principles tuning technique for a pure integrating process with dead time is designed. The performance of these controllers with different closed loop time constants is analyzed and compared for its set point tracking capability. The controller with best performance for the selected time constant is decided.

The question of pole placement for glass capillary tube drawing process is considered. It is used a frequency domain approach to an arbitrary finite spectrum assignment for multivariable time delay systems in order to control glass capillary tube drawing process. The Padé approximation is used for the system of third order and time delay is eliminated from the transfer function of the process. The responses are shown as well the transfer function of the closed loop after applying finite spectrum method. By choosing state variables it is obtained non degenerative transfer function of process model. The time delays in open loop remained the same as in the closed loop. The all poles are located in the left half plane and system is stable. Appropriate program support for this type of problems is developed in Maple language.

In this paper, the modern nonlinear theory is applied to a third order phase locked loop (PLL) with a feedback time delay. Due to this delay, different behaviors that are not accounted for in a conventional PLL model are identified, namely, oscillatory instability, periodic doubling and chaos. Firstly, a Pade approximation is used to model the time delay where it is utilized in deriving the state space representation of the PLL under investigation. The PLL under consideration is simulated with and without time delay. It is shown that for certain loop gain (control parameter) and time delay values, the system changes its stability and becomes chaotic. Simulations show that the PLL with time delay becomes chaotic for control parameter value less than the one without time delay, i.e., the stable region becomes narrower. Moreover, the chaotic region becomes wider as time delay increases. 1. INTRODUCTION A phase-locked loop (PLL) is a versatile device used mainly in carrier synchronization, frequency synthesis, clock recovery, wireless communications and phase inverters [1-4]. When the PLL operates in the phase-locked state, the dynamic behavior of the loop is studied using linear theory. Unfortunately, the PLL may operate in the out-of-lock and in this case the dynamics behavior of the loop follows the nonlinear theory and analyzing this behavior becomes tedious [5-12]. Chaos and complex bifurcations are inherent to nonlinear systems due to dynamical instabilities. Chaos induced in phase locked loop was investigated by many researchers. Endo and Chua [13] proved the existence of horseshoe chaos in second order PLL using Melnikov's method. Later, Bradley and Straub [14] showed that chaotic PLL circuits sometimes can be useful. In fact, they utilized chaotic PLL to broaden the capture range of the PLL. Harb and Harb [15] applied modern nonlinear theory to analyze the chaotic behavior observed in a third order PLL with sinusoidal phase detector characteristics. Sarkar and Chakraborty [16] studied self-oscillations of a third order PLL in periodic and chaotic mode. Fortuna, et al. [17] used chaotic pulse position modulation to improve the efficiency of sonar sensors. In recent years, many researchers studied the dynamic instabilities induced in feedback systems due to the time delay of signals [18-20]. This delay effect causes an oscillatory behavior which has been reported

Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that would be pointwise convergent as the degrees of the numerator and denominator increase to ∞, unlike traditional Padé approximants, which converge only in measure or capacity.

In this paper the Proportional integral controllers based on Internal Model Control principles tuning technique for a pure integrating process with dead time is designed. The performance of these controllers with different closed loop time constants is analyzed and compared for its set point tracking capability. The controller with best performance for the selected time constant is decided.

In this paper the Proportional integral controllers based on Internal Model Control principles tuning technique for a pure integrating process with dead time is designed. The performance of these controllers with different closed loop time constants is analyzed and compared for its set point tracking capability. The controller with best performance for the selected time constant is decided.

In this paper the Proportional integral controllers based on Internal Model Control principles tuning technique for a pure integrating process with dead time is designed. The performance of these controllers with different closed loop time constants is analyzed and compared for its set point tracking capability. The controller with best performance for the selected time constant is decided.

Bloch system is mainly employed with attainment of the basic explanation of relaxation T1 and T2 and nuclear magnetic resonance (NMR). This equation is said to be the heart of both magnetic resonance imaging (MRI) and NMR spectroscopy. The major goal of this paper is to obtain an analytical solution of Fractional Nuclear Magnetic Resonance (NMR) flow equations. In this study, the use of Laplace transform and the Pade approximation has been ascertained to handle the truncated series that were obtained through the Perturbation Iteration Method for further advancement of the approximation. The fractional derivatives in this work are understood to be in the Caputo sense. The results achieved through this paper will prove that the algorithm is fit to be used for more general kinds
of fractional differential equations.

Another version of the classic Sumudu Transform called the Elzaki Transform, was put forward as closely
related to the Laplace Transform. In the following paper, the Elzaki Transform Algorithm, which has been built
on the Decomposition Method, is presented to be applied to find approximate solution of a class of non-linear, initial value problems. This method gives an approximate solution in a Convergent-Series form with easily computable components necessitating no linearization or a low perturbation criterion. The most important part of this paper is the error analysis conducted between exact solutions and pade approximate solutions; it proves that our approximate solutions narrow in rapidly to the exact solutions. Moreover, as we will discuss after the results
are resented, this algorithm can also be applicable to more general classes of linear and nonlinear differential
equations.

The magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet is examined analytically. The series solution is obtained using the Adomian decomposition method (ADM) coupled with Padé approximants to handle the condition at infinity. The numerical solutions agree very well with the results by the homotopy analysis method.