الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
Stochastic differential equations (SDEs), is an ordinary differential equation(ODE) with stochastic process that can model unpredictable real-life behavior of any continuous system and they are important tools in modeling complex phenomena. They arise in many physics and engineering applications such as wave propagation, diffusion through heterogeneous random media, randomly forced Burgers and Navier-Stokes equations. Additional examples can be found in materials science, chemistry, biology, and other areas. However, reaching a solution of these equations in a closed form is not always possible or even easy. Due to the complex nature of SDEs, numerical simulations play an important role in studying this class of DEs. For this reason, few numerical and analytical methods have been developed for simulating SDEs. This thesis introduces analytical and numerical solutions of the nonlinear Langevin’s equation under square nonlinearity with stochastic non homogeneity. The solution is obtained using WHEP technique and the results are compared with those of Picard iterations and HPM. The WHEP technique is used to obtain up to fourth order approximation for The mean and variance of the solution are obtained and compared among the different methods and some parametric studies are done using Matlab. Also, the Mathematica 9 program is used in WHEP technique and the results are compared with those of Matlab in order to make sure of the validity of the results. In Picard iterations we tried to use more than one arbitrary initial approximation were taken as three cases for studying and comparing between them to know which one of them is better. The HPM method is used to obtain the solution of Langevin’s equation up to fourth order approximation. The objective of this thesis is to identify an approximation solution of nonlinear Langevin’s equation using three techniques comparing between the three methods and trying to track the explosion which will be expected to occur from the checking of growth condition. Key Words: Nonlinear, stochastic D.E, Langevin’s equation, WHEP technique, Picard approximation HPM technique.