الفهرس 
Introduction to Spline polynomialsOnly 14 pages are availabe for public view
 |  | from | 135 |  |  |
| | | from | 135 | | |
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)