American Journal of Theoretical and Applied Statistics 2017; 6(5-1): 30-39 http://www.sciencepublishinggroup.com/j/ajtas doi: 10.11648/j.ajtas.s.2017060501.15 ISSN: 2326-8999 (Print); ISSN: 2326-9006 (Online) Synchronization and Impulsive Control of Some Parabolic Partial Differential Equations Mahmoud M. El-Borai, Wagdy G. Elsayed, Turkiya Alhadi Aljamal Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt Email address: m_m_elborai@yahoo.com (M. M. El-Borai), Wagdygoma@alexu.edu.eq (W. G. Elsayed), trk8828@gmail.com (T. A. Aljamal) To cite this article: Mahmoud M. El-Borai, Wagdy G. Elsayed, Turkiya Alhadi Aljamal. Synchronization and Impulsive Control of Some Parabolic Partial Differential Equations. American Journal of Theoretical and Applied Statistics. Special Issue: Statistical Distributions and Modelling in Applied Mathematics. Vol. 6, No. 5-1, 2017, pp. 30-39. doi: 10.11648/j.ajtas.s.2017060501.15 Received: March 7, 2017; Accepted: March 8, 2017; Published: April 5, 2017 Abstract: Novel equi-attractivity in large generalized non-linear partial differential equations were performed for the impulsive control of spatiotemporal chaotic. Attractive solutions of these general partial differential equations were determined. A proof for existence of a certain kind of impulses for synchronization such that the small error dynamics that is equi-attractive in the large is established. A comparative study between these general non-linear partial differential equations and the existent reported numerical theoretical models was developed. Several boundary conditions were given to confirm the theoretical results of the general non-linear partial differential equations. Moreover, the equations were applied to Kuramoto– Sivashinsky PDE′s equation; Grey–Scott models, and Lyapunov exponents for stabilization of the large chaotic systems with elimination of the dynamic error. Keywords: Synchronization, Impulsive Control, Prabolic Partial Differential Equations 1. Introduction The ordinary differential equations (ODE′s) theory were applied in science and engineering researches [1, 2, 3], for mathematical modeling of many physical phenomena. The impulsive control on basis of these equations was successfully applied for stabilization of the systems with chaotic behavior using small control impulses even if the chaotic signals and noise are unpredictable. For example, autonomous systems of ODE′s Lorenz and Chua oscillator systems [4, 5, 6, 7], and non-autonomous systems such as Duffings oscillator [8, 9], and where practical stability of the system is achieved in a small region of phase space instead of controlling the approach of chaotic system to an equilibrium position. The impulsive synchronization of two identical chaotic systems by ODE′s [10, 11, 12, 13], involved autonomous drive system, and response system. Samples of the state variables (synchronization impulses) of drive system at discrete time intervals were used to: 1) drive the response system, 2) impulsively control error between the two systems, 3) minimizing the dynamic error, and 4) an upper bound on time intervals between impulses is obtained. This synchronization was generalized to vary impulse intervals [14, 15, 16], where less conservative conditions on Lyapunov function are obtained meaning that, it is required to be non- increasing along a subsequence of switching. The impulsive synchronization was applied in secure communications [17, 18], analysis of impulsive control, and synchronization of chaotic systems extending the theory of impulsive differential equations to PDE′s [19, 20, 21, 22], giving several differential inequalities, asymptotic stability, and first order partial differential-functional equations using Lyapunov energy functions, and the numerical analysis of first order PDE′s [23, 24, 25], The general application of impulsive control and impulsive synchronization on spatiotemporal chaotic systems generated by continuous extended systems including synchronization of spatiotemporal chaotic systems generated by coupled non-linear oscillators using ODE′s [26, 27, 28], and impulsive synchronization of spatiotemporal chaotic systems using PDE′s [29, 30, 31], using a finite number of local tiny perturbations selected by an adaptive technique [32, 33, 34], or using an extended time-delay auto synchronization algorithm [35, 36], or synchronizing using a finite number of coupling signals in terms of local spatial averages [37, 38, 39], frequency and phase synchronization of two non-identical PDE′s [40, 41, 42]. Using high