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Abstract Partial Differential Equations (PDEs) are one of the most important tools in many fields of practical mathematics. Many mathematical models in physics and medicine are based on PDEs. As numerical methods for handling scientific and engineering issues gain attention, many of these problems do not have an analytical solution, giving the numerical solution a very useful option. The major goal of this thesis is investigating several forms of analytical and numerical solutions for one of the most complicated nonlinear PDEs; the Sharma Tasso Oliver (STO). Quintic B-spline approach, Adomian Decomposition method (ADM) and Quartic non-polynomial spline method are provided for STO equation. Von Neumann stability is discussed for present numerical methods to investigate the system efficiency. The accuracy of proposed methods is demonstrated by test problems with comparison. Finally, I construct new variable delay Stochastic Differential Equations (SDDEs) model in order to apply in stock price branch, which aim to improve the accuracy of the prediction for stock price based real data compared with the existing stochastic models. This chapter introduces some fundamental definitions and concepts required to establish the thesis. We also go over PDEs in detail, as well as some numerical methods (B-Spline, ADM and quartic non-polynomial method) and the main concepts for using these methods to solve PDEs. Finally, in this chapter, we go over some basics for stochastic differential equations In this chapter, we solve nonlinear PDEs using the B-spline approach based on a quintic spline polynomial, particularly the STO problem. We use the Von Neumann approach to investigate our system’s stability. We also plotted some of the generated approximate solutions and compared them to the exact solution to demonstrate the method’s accuracy. The proposed approach is demonstrated to be beneficial for dealing with various related broad types of nonlinear PDEs. Chapter 3: In this chapter, the semi analytic approach based on ADM is utilized to estimate the semi analytic solution of the nonlinear problem STO. The obtained solution suggests that the method’s application is a highly promising mathematical tool for nonlinear problems emerging in mathematical physics. We also plotted some of the generated approximate solutions and compared them to the exact solution to demonstrate the method’s accuracy. In this chapter, we solve the STO problem using the quartic non-polynomial spline combined with finite difference method. The Von Neumann technique is applied to investigate the stability of the solution method. We also presented some figures to the obtained approximate solution and compared the findings to the actual solution to illustrate the method’s correctness. The proposed approach is demonstrated to be beneficial for dealing with various related broad types of nonlinear PDEs. In this chapter, an new approach of Stochastic Delay Differential Equations, namely the Stochastic Pantograph Differential Equation for modeling stock prices (SPDEs). SPDEs is a special type of past-dependence equation with many special properties, such as unbounded memory and a variable delay time , which is namely the pantograph delay and can be written as . The main motivations for the proposed stock price SP-SPDE model is the estimation of the volatility function using a past dependency with respect to variable delay time. In addition, the numerical solution for this model using the step theta miliston technique is described in this chapter. Furthermore, we demonstrate nonnegativity approximation solutions for stochastic models that meet positivity solutions has received increased interest in recent times for use in financial mathematics. We apply the SP-SPDE on a real data for some firms from “Yahoo finance” and comparing the results with other models as (BC Model and SDDE Model) |