الفهرس 
The Aim of this ThesisOnly 14 pages are availabe for public view
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Abstract
The aim of this thesis is to develop numerical solutions in mean square sense for random ordinary and partial differential equations using the finite difference method (FDM) and the finite element method (FEM).Differential equations can describe the systems undergoing change.They are in science, engineering, etc.Often,differential equations describe a complex system in which the analytical solution to these equations isn’t tractable.Numerical methods and computer simulations are useful in these complex systems Differential equations involving random elements are called random differential equations. Also,the differential equation in which one or more of the terms is a stochastic process and its solution is also a stochastic process is called stochastic differential equation (SDE). Now,random differential equations deal with the uncertainties (e.g., general randomness and white noise). This thesis deals with some random models that describe a great many phenomena and widely are used in physics, chemistry, biology, sociology, economics, and finance, e.g.Mean square theory of random differential equation and stochastic calculus is very important and attractive as its applications to problems follow the corresponding deterministic procedure.Mean square convergence and stability of the solutions or schemes are of our interest. We show the mean square consistency and stability via multiple ways.The first used numerical method is the finite difference method (FDM) that is the oldest, and based upon the application of a local Taylor expansion [6]. The second one is the finite element method (FEM) which involving complicated geometries. What makes the FEM more powerful is the technique of subdividing the domain into subdomains (i.e., finite elements).The thesis is divided into three chapters as listed below:Chapter 1 introduces a brief review of some needed subjects in this thesis starting by the theory of probability passing by the mean square and the mean fourth calculus till the FDM and FEM.Chapter 2 shows that how the FDM works.By this method, we obtain an approximation solution for some random models in the sense of the mean square.Also, the consistency and stability of the constructed random finite difference schemes are studied under the mean square sense via multiple ways. Some numerical examples are shown for the theoretical results.Chapter 3 shows that how the variational methods and FEM work by applying them to some random models. Convergence of the random approximation solution is studied in mean square and some model problems are shown.