الفهرس 
Chapter 1 : General IntroductionOnly 14 pages are availabe for public view
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Abstract
The WHEP (Wiener Hermite expansion linked by the perturbation technique) technique applications are the main objectives of this thesis to find some statistical approximations for the solution stochastic processes of some random models of Navier- Stokes equations. This thesis contains six chapters as follows
Chapter 1 In this chapter, some basics items are discussed before applying the WHEP technique and they include the following points:
• A general introduction presents a simple survey over the thesis topics.
• Some basic definitions related to the random experiment, random variables, distributions of random variable, stochastic processes, Wiener process and white noise process.
• A simple survey on the stochastic Wiener Hermite expansion (WHE) and Wiener Hermit polynomials (WHPs).• Some statistical properties related to a mixed multiplication of the Wiener Hermit polynomials (WHPs).
• Some closed forms related to the statistical moments for any stochastic process which represented by WHE.• Some basics of the mathematical analysis related to the homptopy perturbation method.
• A simple survey on the Navier-Stokes equation.
• A scheme presents the arrangement of WHEP application steps which will be applied on the stochastic systems. Chapter 2
In this chapter, the 1-D perturbed Navier-Stokes equation which is exited randomly by the space white noise process is considered to study the application of the WHEP technique. The statistical properties of the approximate stochastic solution processes of the 1-D Navier-Stokes equation are discussed in terms of the kernels of the first and second order WHE. The perturbation method is used to solve the non-linear deterministic system of equations which are transformed into a family of iterative linear partial differential equations that can be solved using eigenfunctions expansion. The solutions of this family can be obtained by performing some mathematical analysis using of Mathematica 7 software. Some case studies are chosen to display graphically the reductive approximations of the statistical solutions by the WHEP technique. The out- puts of this chapter are accepted for publication in the journal of Applied Mathematical Modeling, (2012) (Journal homepage: http://www.elsevier.com/locate/apm) (Impact Factor 1.579).
Chapter 3
In this chapter, a second problem is analyzed by the WHEP technique which simulates a model to describe the perturbed 2-D Navier –Stokes equation under a stochastic excitation by the Wiener process and this model is formulated by the Cartesian coordinates system and defined over deterministic initial and boundary conditions. Some mathematical analyses are used to reduce the stochastic model into another single equation model. The WHEP technique is applied to give a deterministic system of equations in the kernels of the first order WHE for the stochastic solution process and it is expanded into a family of iterative linear differential equations under the application of the perturbation theory. By some mathematical analyses which are performed by the application of the eigenfunctions expansion technique with the aid of mathematica software, the solutions of this family can be obtained. Some case studies are selected to illustrate the method of analysis. The outputs of this chapter are presented in the second international conference on Mathematics and Information Science which was organized by Mathematics Department in September 2011 under the supervision of Sohag university and also, published in the Applied Mathematics and Information Science Journal No. 6-3S , PP:1095-1100. (Nov. 2012).
(Journal homepage: http://naturalspublishing.com/show.asp?JorID=1&pgid=0 )
(Impact Factor 0.624).
Chapter 4
In this chapter, a third problem related to the stochastic non-perturbed Navier-Stokes equation is formulated to perform some statistical analyses for its stochastic solution processes by the WHEP technique. The stochastic solution process of the problem is represented by the first order of WHE which has two deterministic kernels which are determined from a deterministic system of equations. This deterministic system is approximated according to the results of the homotopy perturbation method (HPM) which were discussed in chapter 1. Some case studies related to the statistical properties of the approximate solution process are considered after obtaining the solutions of the iterative family of linear partial differential equations. The outputs of this chapter were published in the Journal of Computer Sciences and Computational Mathematics.
(Journal homepage http://www.jcscm.net/cms/, the journal is under evaluation to be indexed by ProQuest, Math Sci Net, and EBSCO databases). Chapter 5
In this chapter, a fourth problem which simulates the statistical properties of the stochastic solution process of Navier - Stokes model using the WHEP technique. This model describes a formulation for the stochastic 3-D Navier-Stokes equation in the rectangle coordinates over deterministic initial and boundary conditions and its stochastic excitations are taken in different directions by the Wiener and noise processes. Some vectors analyses are performed to simplify this model into another stochastic single equation model. The solution process of the new model is obtained by different orders of the WHE. Some deterministic systems of equations due to make some statistical averages of the model’s form is obtained. The perturbation theory and the eigenfunction expansion method are used respectively to treat these deterministic systems and its outputs are displayed graphically through some chosen case studies. Chapter 6
In this chapter, a general summary is provided for the contents of the thesis and what has been done from the application of WHEP technique to the stochastic systems which simulate some models of Navier-Stokes equation through a quick narrative for the resulting equations from the WHEP treatments. Also, a plan for the future study is considered through some items related to the thesis filed.