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rbfs.tex
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\begin{document}
\pagestyle{empty}
\pagenumbering{roman}
%================================================ Title Page ======================================================
\begin{center}
{THESIS TITLE MTSU MATHEMATICAL \\
[.10in]
SCIENCES DEPARTMENT THESIS FORMAT \\ [.07in]} \rm
\rule{1.25in}{.01in}\\[.0 in]
\vspace{.6in}
A Thesis \\ [.06 in]
Presented to the Faculty of the Department of Mathematical Sciences \\[.06in]
Middle Tennessee State University \\ [.06in]
\rule{1.25in}{.01in}\\
\vspace{.6in}
In Partial Fulfillment \\[.06 in]
of the Requirements for the Degree \\ [.06 in]
Master of Science in Mathematical Sciences \\ [.06 in]
\rule{1.25in}{.01in}\\
\vspace{.6in}
by \\ [.06in]
{ Name of Author} \\[.06in]
{August 2012}
\end{center}
%================================================ Approval Page ===================================================
\newpage
\pagestyle{plain}
\begin{center}
{\bf APPROVAL} \\ [.05in]
{\bf This is to certify that the Graduate Committee of }\\
Name of Author \\ [-.1in] met on the \\ [-.1in] 1st \ day of \
August, 2012.
\\ [.25in]
\end{center}
\baselineskip=20 pt
The committee read and examined his/her thesis,
supervised his/her defense of it in an oral examination, and decided
to recommend that his/her study should be submitted to the Graduate
Council, in partial fulfillment of the requirements for the degree
of Master of Science in Mathematics.
\vspace{.3in}
\noindent
\makebox[3.4in][l]{}\makebox[2.0in][l]{\rule{2.5in}{.01in}}\\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. A.Q.M.Khaliq}\\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{Chair, Graduate Committee } \\[.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. Zachariah Sinkala} \\[.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. Yuri Melnikov} \\[.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. James Hart} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ Graduate Coordinator,} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ Department of Mathematical Sciences} \\[.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. Don Nelson} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ Chair,} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ Department of Mathematical Sciences} \\[.1in]
\makebox[3.4in][l]{Signed on behalf of } \makebox[2.0in][l] {\rule{2.5in}{.01in}}\\[-.1in]
\makebox[3.4in][l]{the Graduate Council}\makebox[2.0in][l]{\it Dr. Michael Allen} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ Dean,} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ School of Graduate Studies}
%================================================ Abstract Page =================================================
\newpage
\begin{center}
{\bf ABSTRACT}\\
\end{center}
\baselineskip=24pt
Meshfree radial basis functions (RBF) is an interpolation technique
for constructing an unknown function from scattered data. In this
thesis we apply an efficient RBF method in evaluating the price of
standard European and American options. The analytical solution of
the European option exists and can be obtained by the Black-Scholes
formula. There is no exact solution of the American option problem
due to the existence of an early exercise constraint which leads to
a free boundary condition. We evaluate the American Option by adding
a small continuous nonlinear penalty term to the Black-Scholes model
to remove the free boundary condition. The application of RBFs leads
to a system differential equations which are solved by a time
integration scheme known as the $\theta$-method.The option price is
approximated with RBF with unknown parameters at each time step. We
compare the accuracy, efficiency and computation cost of three RBFs
Gaussian, Multiquadric and the Inverse-multiquadric.Finally a
comparison is made between the three RBFs and the solution obtained
by finite difference.\mathbb{R}
%================================================ Copyright Page =================================================
\newpage
\baselineskip=24 pt
\begin{center}
\ \ \
\vspace{3.in}
Copyright \copyright\ 2012, Nana Akwasi Abayie Boateng
\end{center}
%================================================ Dedication Page =================================================
\newpage
\begin{center}
{ \bf DEDICATION } \\ [.15in]
\end{center}
This thesis is dedicated to my parents, who has taught me,
encouraged me and supported me in my life. Thanks for all your
patience, love and unconditional support.
%================================================ Acknowledgments Page ==============================================
\newpage
\begin{center}
{ \bf ACKNOWLEDGMENTS} \\ [.15in]
\end{center}
Take this opportunity to thank your advisor, your thesis committee,
research collaborators and anyone who helped you in the process of
thesis accomplishment.
%================================================ Table of Content =================================================
\newpage
\tableofcontents
%%================================================ Chapter 1 ==============================================================
%{INTRODUCTION}
%%================================================ Chapter 2 ==============================================================
%{\uppercase{Option Pricing}}
%%--
%%================================================ Chapter 3 ==============================================================
%\uppercase{Finite Difference Methods}\\
%%-------------------------------------------------------------------------------------------------------------------------
%%================================================ Chapter 4 ==============================================================
%\uppercase{RBF-Meshfree Methods}\\
%%-------------------------------------------------------------------------------------------------------------------------
%
%%================================================ Chapter 5 ==============================================================
%\uppercase{Discretization And Algorithms}\\
%%-------------------------------------------------------------------------------------------------------------------------
%%================================================ Chapter 7 ==============================================================
%\uppercase{ Numerical Methods and Stability Analysis }\\
%%================================================ List of Tables ===================================================
%%================================================ Chapter 6 ==============================================================
%\uppercase{Numerical Experiments And Results}\\
%%-------------------------------------------------------------------------------------------------------------------------
\newpage
\addcontentsline{toc}{section}{\rm LIST OF TABLES}
\listoftables
%================================================ List of Figures ====================================================
\newpage
\addcontentsline{toc}{section}{\rm LIST OF FIGURES}
\listoffigures
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\la{\langle}
\def\ra{\rangle}
\def\dist{{\rm dist}}
\def\X{{\bf X}}
\def\C{{\bf C}}
\def\D{{\bf D}}
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\def\argmin{{\rm argmin}}
\def\diag{{\rm diag}}
\def\sgn{{\rm sgn}}
\def\supp{{\rm\rm supp}}
\vspace*{1cm}
%============================================= Appendix Separation Page ===============================================
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APPENDICES
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%================================================ Chapter 1 ==============================================================
\sect{INTRODUCTION}
\pagestyle{myheadings} \markboth{ } { }
\pagenumbering{arabic}
%-------------------------------------------------------------------------------------------------------------------------
An option is a financial contract which gives the holder of the
option the right to purchase or sell a prescribed asset at a
prescribed time in the future known as the expiry date at a
prescribed amount which the exercise or strike price.
In 1973,Fisher Black and Myron Scholes showed that the option value
of the European call option can be modeled by a lognormal diffusion
partial differential equation.There are two categories of options
namely, standard options(European and American options) and non
standard options.
Hon and Mao\cite{Hon} introduced a numerical scheme in which by
applying global radial basis function as a spatial approximation
for the numerical solution of the value of the value of the option
and it's derivative in the Black-Scholes equation.From their
numerical results,they showed that the use of RBFs does not require
the generation of a rectangular grid and also the computational
domain is composed of scattered data points.
Khaliq \textit{et al}\cite{KK} investigated meshfree RBF approximation to
options with non-smooth payouts.By taking advantage of parallel
architecture,they developed a strongly stable time stepping fourth
order method which was a linear combination of four Backward
Euler-like solver on four concurrent processors.
Khaliq \textit{et al}\cite{ADS06} considered a penalty method
approach to solving American options.They observed that by
introducing a carefully chosen continuous penalty term to the
Black-Scholes equation,the free and moving boundary condition can
be removed and allow the problem to be solved on a fixed
domain.They introduced a linearly implicit scheme with superior
accuracy and stability by solving the nonlinear term explicitly.
The remaining chapters of this thesis is organized as follows.
We introduce option pricing and discuss two standard option,
the European and American options in chapter 2.In chapter 3 we adopt an Exponential
Time Differencing version of the Backward Euler finite difference
scheme to evaluate the price of the option. The theory and development of RBF
Meshfree methods are presented in chapter 4.In chapter 5, we elaborate on the
discretization and algorithms of the methods. The stability analysis and the
results of the numerical experiments are presented in chapters 6 and 7.
Finally all results are discussed and interpreted in chapter 7.
\subsection{Sample Section} This is a sample section.
The study of algebraic structures using its associate graphs is a
very exciting field which generates many fascinating results,
conjectures and questions \cite{Abdollahi2006}. There are various
ways to associate graphs to algebraic objects such as groups and
rings. For instance, the prime graph defined in\cite{Williams1981},
the conjugacy class graph defined in\cite{Bertram}, the
non-commuting graph defined in\cite{Abdollahi2006}, and the nonzero
divisor graph defined in\cite{DavidAnderson}
%\subsubsection{Sample Sub Section}
%This is a sample equation.
%\begin{align}
% \frac{\partial}{{\partial}t}\int\int\limits_{system}
% (V_{A_{r}} + V_{A_{s}})dxdy = 0 \label{eq: equation1}\\
% \frac{\partial}{{\partial}t}\int\int\limits_{system} V_B dxdy = 0
% \label{eq:equation2}
%\end{align}
%\begin{theorem}{\rm(\cite{Wang2008})}\label{thm1}
%Let $G$ be a group such that $\nabla{(G)} \cong\nabla(A_{n})$, where
%$n\ge 5$. If $n=5,6$ or at least one of $n,n-1,n-2$ is prime then
%$G\cong A_{n}$.
%\end{theorem}
%================================================ Chapter 2 ==============================================================
\sect{\uppercase{Option Pricing}}
% \numberwithin{Option Pricing}
%-------------------------------------------------------------------------------------------------------------------------
%\subsection{What is an Option?}
An option is a financial contract which gives the holder of the
option the right to purchase or sell a prescribed asset at a
prescribed time in the future known as the expiry date at a
prescribed amount which the exercise or strike price\cite{Wilmot}.
The most common kinds of prescribed assets which are traded on
financial markets are stocks,bonds,currency and commodities.An
option is a derivative product because it is traded on an underlying
asset.The holder of a call option makes profit if the price of the
underlying asset rises on the market whereas the holder of a put
option does so when the price of the underlying asset falls on the
financial market.The two primary uses of option are for hedging and
speculation\cite{Wilmot}.
There are numerous kinds of options which are
traded on financial markets.Vanilla options are options which do
not possess any special features or characteristics.Examples are the
European and American options.Exotic options possess special
features.Examples include Asian options,Barrier options,Basket
options.In this We consider The European and American options.
%\subsubsection{The European Options}
\subsection{The European Options}
%\numberwithin{equation}{The European Options}
The European option is an option
which can only be exercised at its maturity time.The exact or
analytical formula for estimating a fair price for European
options exist. In 1973 Black and Scholes by making a set of
explicit assumptions including the risk-neutrality of the
underlying asset price showed that the value European call option
satisfies a backward -in-time lognormal partial differential
equation of diffusion type which has come to be known as the
Black-Scholes equation\cite{BS73}.
Let the $V(S,t)$ be the price of an option which is function of both
asset price and time.This option satisfies the following
Black-Scholes equation.
\begin{equation} \label{BSS}
\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
P}{\partial S^2} +rS\frac{\partial P}{\partial S}-rP=0, \quad S
> \overline{S}(t),\ 0 \le t < T.
\end{equation}
where $r$ is the risk-free interest interest rate,$\sigma$ is the
volatility of the asset price,$S$ is the asset price.The Final
condition is given by\cite{Wilmot}
\[V(S,t) = \left\{
\begin{array}{l l}\label{FNC}
max\{E-X,0\} & \quad \mbox{ for a put option}\\
max\{S-E,0\}& \quad \mbox{for a call option}\\ \end{array} \right. \]
where E is the strike price.\\
The Boundary condition of the European call option is given as
follows:
\begin{equation}
C(S,t)\thicksim S \quad
as S\rightarrow\infty ,\quad C(0,t)=0.
\end{equation}
where $C(S,t)$ is the value of the European call option satisfying
\ref{BSS}. The Boundary condition at of the European put option is
given as follows:
% \begin{equation}\label{FC}
%P(S,t)\rightarrow 0 as S\rightarrow\infty \quad as ,\quad
%P(0,t)=E\exp^{\int^{T}_{t}\tau(\tau)d\tau}}.
%\end{equation}
where $P(S,t)$ is the value of the European put option
satisfying equation \ref{BSS} for a time dependent interest rate.\\
Equation \ref{BSS} can be transformed exponentially by making the
substitution $S=e^y$ to
\begin{equation}
\frac{\partial U}{\partial
t}+\frac{1}{2}\sigma^2\frac{\partial^2U}{\partial
y^2}+(r-\frac{1}{2}\sigma^2\frac{\partial U}{\partial y})-rU=0
\end{equation}
\[U(y,T) = \left\{
\begin{array}{l l}\label{INC}
max\{E-e^y,0\} & \quad \mbox{ for a put option}\\
max\{e^y-E,0\}& \quad \mbox{for a call option}\\ \end{array} \right. \]
The analytical solution of the Black- Scholes partial differential
equation \ref{BSS} with corresponding final and initial conditions
\ref{FNC} and \ref{FC} with a constant volatility and interest rate
for the European call option is given as\cite{Wilmot}
\begin{equation}
C(S,t)=SN(d_1)-E\exp^{-r(T-t)}N(d_2)
\end{equation}
where $N(.)$ is the cumulative distribution function for the
standardized normal random variable given by
\begin{equation}
N(x)=\frac{1}{\sqrt{2\pi}}\int^{x}_{-\infty}\exp^{-\frac{1}{2}y^2dy}
\end{equation}
The corresponding analytical solution of the European put option is
given by
\begin{equation}
P(S,t)=E\exp^{-r(T-t)}N(-d_2)-SN(-d_1)
\end{equation}
where\\
\begin{equation*}
d_1=\frac{\log(\frac{S}{E}+(r+\frac{1}{2}\sigma^2)(T-t))}{\sigma\sqrt{T-t}}
\end{equation*}
\begin{equation*}
d_2=\frac{\log(\frac{S}{E}+(r-\frac{1}{2}\sigma^2)(T-t))}{\sigma\sqrt{T-t}}
\end{equation*}
\begin{figure}[h]
\begin{centering}
\vskip -0.5in
\includegraphics*[height=3in]{calloption.png}\
\caption{The pay-off of the European Call Option }
\includegraphics*[height=3in]{putoption2.png}\
\caption{The pay-off of the European Put Option } \vskip -0.5in
\end{centering}
\end{figure}
%\subsubsection{The American Option}
\subsection{The American Options}
The American option can be exercised at any time prior to expiry.The
American option is complicated because at at each time $t$ not only
is one interested in the value of the option but also for each asset
price $S$,whether it should be exercised or not.This creates a free
boundary problem\cite{Wilmot}.At each time $t$ there is a particular
value of $S$ which lies in the boundary between two regions:one
where early exercise is optimal to the other where one should hold
on to the option.The optimal exercise price$s_{f}(t)$ which in
general depends on time is not known \textit{priori} unlike the case
of European options.The American option valuation can be uniquely
specified by a set of constraints among which are the option value
must be greater than or equal to the payoff function, the option
value must be continuous function of $S$,replacing the Black-Scholes
equation by an inequality and lastly making the derivative of the
option with respect to the asset price(option delta) continuous.
The value $V(S,t)$ of the American option satisfies the following
inequality
\begin{equation}
\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
V}{\partial S^2} +rS\frac{\partial V}{\partial S}-rV\leq 0
\end{equation}
The Final condition is at expiry time $T$ given by\cite{Wilmot}
\[V(S,t) = \left\{
\begin{array}{l l}\label{FNC}
max\{E-X,0\} & \quad \mbox{ for a put option}\\
max\{S-E,0\}& \quad \mbox{for a call option}\\ \end{array} \right. \]
where E is the strike price.\\
In the region$0\leq S\leq S_{f}(t)$ where early exercise is optimal,the value of the
American put option satisfies the following inequality
\begin{equation}
\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
P}{\partial S^2} +rS\frac{\partial P}{\partial S}-rP< 0
\end{equation}
and
\begin{equation}
P=E-S
\end{equation}
In the other region,$S_{f}(t)< S<\infty$,early exercise is not
optimal and the value of the American put option satisfies the
Black-Scholes equation
\begin{equation}
\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
P}{\partial S^2} +rS\frac{\partial P}{\partial S}-rP= 0
\end{equation}
and
\begin{equation}
P>E-S
\end{equation}
The boundary condition at$S_{f}(t)= S$ are that $P$ and its
slope(delta) are continuous.
\begin{equation}\label{pp}
P(S_{f}(t),t)=max(E-S_{f}(t),0)
\end{equation}
\begin{equation}\label{dv}
\frac{\partial P}{\partial S}(S_{f}(t),t)=-1
\end{equation}
The boundary condition \ref{pp} determines the option value at the
free boundary,whereas \ref{dv}known as the \textit{smooth pasting condition} determines the location of the
free boundary and simultaneously maximizes the benefit to the
holder whiles avoiding arbitrage.
The value $C(S,t)$ of the American Call option satisfies the
corresponding equality in the holding region
$0\leq S\leq S_{f}(t)$
\begin{equation}
\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
C}{\partial S^2} +rS\frac{\partial C}{\partial S}-rC=0
\end{equation}
in the other region where early exercise is optimal $S_{f}(t)<
S<\infty$,the value $C(S,t)$ of the American call option satisfies
\begin{equation}
\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
C}{\partial S^2} +rS\frac{\partial C}{\partial S}-rC<0
\end{equation}
and
\begin{equation}
P=E-S
\end{equation}
\subsection{Penalty Method}
%An elegant transformation of this moving boundary
%problem to one on a fixed domain was suggested in \cite{ZFV98} and
%later refined in \cite{NST02}.
We introduce a penalty term into the Black-Scholes equation, to
obtain a parabolic nonlinear partial differential equation of the
form.The introduction of the penalty term changes the problem from
that of a constrained optimization problem to that of a series of
unconstrained optimization problem.The solution of the
unconstrained optimization problems converges to the original
constrained optimization problem.\cite{BOA08}
\begin{equation}\label{BSpenalty}
\frac{\partial P_\epsilon}{\partial
t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 P_\epsilon}{\partial S^2}
+rS\frac{\partial P_\epsilon}{\partial
S}-rP_\epsilon+\frac{_\epsilon C }{P_\epsilon + \epsilon - q(S)}=0,
\quad 0 \leq S \leq S_\infty, \ 0 \leq t < T.
\end{equation}
Here $0 < \epsilon \ll1$ is a small regularization parameter, $C
\geq rE$ is a positive constant, and $q(S) = E - S$ is the barrier
function. The value $S_\infty$ is a (relatively very large) price
for which the option is worthless. The following are terminal and
boundary conditions which accompany the nonlinear nonlinear partial
differential equation.
\begin{eqnarray}
P_\epsilon(S,T) & = & \max (E-S,0)\\
P_\epsilon(0,t) & = & E,\\
P_\epsilon(S_\infty,t) & = & 0.
\end{eqnarray}
%\begin{theorem}{\rm(\cite{Wang2008})}\label{thm1}
%Let $G$ be a group such that $\nabla{(G)} \cong\nabla(A_{n})$, where
%$n\ge 5$. If $n=5,6$ or at least one of $n,n-1,n-2$ is prime then
%$G\cong A_{n}$.
%\end{theorem}
%================================================ Chapter 3 ==============================================================
\sect{\uppercase{Finite Difference Methods}}
\subsection{Finite Difference Approximations}
The method of Finite Difference Approximation which is based on Taylor series expansions
of functions near the point of interest will be used to discretize
the Black-Scholes partial differential equation\cite{Wilmot}
\begin{equation}
\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^2
S^2\frac{\partial^2 P}{\partial ^2}+rS\frac{\partial P}{\partial
S}-rP +\frac{_{\epsilon}C}{P_{\epsilon}+\epsilon-q(s)}
\end{equation}
The first order partial derivative $\frac{\partial P}{\partial S}$ is approximated by central differencing
with spatial step size $h$=$\frac{S_{f}-S_{0}}{N}$ and time step size$k$=$\frac{T_{f}-T_{0}}{M}$ as
follows.We apply the ETD-BE scheme performing a Backward-Euler approximation on
$\frac{\partial P}{\partial t}$ treat the penalty$Q$ term explicitly.
\begin{equation*}
\frac{\partial P^2}{\partial
S^2}=\frac{P_{i+1,j}-2P_{i,j}+P_{i-1,j}}{h^2}
\end{equation*}
\vspace{10mm}
\begin{equation*}
\frac{\partial P}{\partial S}=\frac{P_{i+1,j}-P_{i-1,j}}{2h}
\end{equation*}
Using the approximations above ,the Black-Scholes partial differential
equation is then applied to mesh points $(nk,mk)$ ,$n=1,2...N-1$,at
the time level $t=mk$,$m=1,2,...,M$.At each $n$ we have
\begin{equation*}
\frac{P_{i,j}-P_{i,j-1}}{k}=\frac{1}{2}\sigma^2S^2\left[\frac{P_{i+1,j}-2P_{i,j}+P_{i-1,j}}{h^2}\right]+rs\left[\frac{P_{i+1,j}-P_{i-1,j}}{2h}\right]-rP_{i,j}+\frac{_{\epsilon}C}{P_{\epsilon}+\epsilon-q(s)}
\end{equation*}
\begin{equation*}
P_{i,j}-P_{i,j-1}=\frac{1}{2}\frac{\sigma^2S^2k}{h^2}\left[
P_{i+1,j}-2P_{i,j}+P_{i-1,j}\right]+\frac{krS}{2h}\left[
P_{i+1,j}-P_{i-1,j}\right]-rkP_{i,j}+\frac{_{\epsilon}Ck}{P_{\epsilon}+\epsilon-q(s)}
\end{equation*}
Let $\beta=\frac{1}{2}\frac{\sigma^2S^2k}{h^2}$ , $\frac{krS}{2h}$
and $Q=\frac{_{\epsilon}Ck}{P_{\epsilon}+\epsilon-q(s)}$
\begin{equation*}
P_{i,j}-P_{i,j-1}=\beta P_{i+1,j}-2\beta P_{i,j}+\beta
P_{i-1,j}+\alpha P_{i+1,j}-\alpha P_{i-1,j}-rkp_{i,j} +Q
\end{equation*}
\begin{equation*}
P_{i,j}+2\beta P_{i,j}+rkP_{i,j}-\beta P_{i+1,j}-\alpha
P_{i-1,j}-\beta P_{i-1,j}=P_{i,j-1}+Q
\end{equation*}
\begin{equation*}
(1+2\beta
+rk)P_{i,j}-(\alpha+\beta)P_{i+1,j}+(\alpha-\beta)P_{i-1,j}=P_{i,j-1}+Q
\end{equation*}
This leads to the following tridiagonal system
\[A = \left( \begin{array}{cccc}
1+2\beta +rk& -(\alpha+\beta) & ...& 0\\
\alpha-\beta & 1+2\beta +rk & & \\
& \ddots & \ddots & \\
& & & \\
%0 &...&\alpha-\beta & 1+2\beta +rk &-(\alpha+\beta)\\
0&......& \alpha-\beta&1+2\beta+rk -(\alpha+\beta) \\
0 &...&\alpha-\beta & 1+2\beta
+rk \end{array} \right) .\] The boundary condition vector
$\textbf{b}_j$ is given by
\begin{equation*}
\textbf{b}_j= \left[
(\alpha-\beta)P_{1},0,.......0,-(\alpha+\beta)P_{N,}\right]^{T}
\end{equation*}
\begin{equation*}
AV_{i,j} + b_{j}=V_{i,j-1}+Q
\end{equation*}
The eigenvalues of the matrix A can be shown to
%\begin{equation*}
% \begin{array}
%
% \lambda_{s}&=&1+2\beta+rk+2\sqrt{-(\alpha+\beta)(\alpha-\beta)}\cos\frac{S\pi}{N+1}
%
% \end{array}
% \end{equation*}
\begin{equation*}
\begin{array}{lcl} \lambda_{s} & = &1+2\beta+rk+2\sqrt{-(\alpha+\beta)(\alpha-\beta)}\cos\frac{S\pi}{N+1} \\ S & = & 1\cdots N-1 \end{array}
\end{equation*}
\begin{equation*}
\lambda_s=1+2\beta+rk+2\sqrt{\beta^2-\alpha^2}\cos\frac{S\pi}{N+1}
\end{equation*}
If $0<\alpha \leq \beta$ then the eigenvalues are real numbers
satisfying
\begin{equation*}
-(1+2\beta+rk+2\sqrt{\beta^2-\alpha^2})\leq \lambda_s \leq
1+2\beta+rk+2\sqrt{\beta^2-\alpha^2}
\end{equation*}
If $\beta \leq \alpha$,then the eigenvalues are complex numbers
satisfying
\begin{equation*}
\begin{array}{cccllllc}
\lambda_s&=&1+2\beta+rk+2\sqrt{-(-\beta^2-\alpha^2)}\\
&=&1+2\beta+rk+2\sqrt{\alpha^2-\beta^2}i\\
&=&1+2\beta+rk+2i\gamma_j\\
\end{array}
\end{equation*}
where
\begin{equation*}
-\sqrt{\alpha^2-\beta^2}\leq \gamma_j\leq\sqrt{\alpha^2-\beta^2}
\end{equation*}
The eigenvalues of \textbf{A} are essential in determining the
stability of the numerical scheme above.The system is stable if
$max|\lambda_s | \leq 1$ for $S=1...N-1$
\subsection{Time Stepping Scheme}
We Consider the following nonlinear parabolic
initial-boundary value problem:
\begin{eqnarray}\label{tms1}
u_t + A u &=& F(t,u) \qquad \textrm {in} \; \Omega,\qquad t \in \left( 0,T \,\, \right], \label{Eq1}\\
u &=& v \qquad \qquad \; \, \textrm {on} \;\partial \Omega,\quad t \in \left( 0,T \,\, \right], \nonumber \\
u(\cdot,0) &=& u_0 \qquad \qquad \textrm {in} \;\Omega, \nonumber
\end{eqnarray}
where $\omega$ is a bounded domain in $\mathbb{R}^{d}$ which also
has Lipschitz continuous boundary,we let $A$ represents a uniformly
elliptic operator, and $F$ is a sufficiently smooth, nonlinear
reaction term. We represent a differential operator as follows:
\begin{eqnarray}
A := -\sum_{j,k=1}^d \frac{\partial}{\partial x_{j}} \left (
a_{j,k}(x)\frac{\partial}{\partial x_{k}} \right )+\sum_{j=1}^d
b_{j}(x)\frac{\partial}{\partial x_{j}}+b_{0}(x),
\end{eqnarray}
where the coefficients $a_{j,k}$ and $b_{j}$ are $C^{\infty}$ (or
sufficiently smooth) functions on $\overline{\Omega}$,
$a_{j,k}=a_{k,j}$, $b_{0} \ge 0$, and for some $c_0 > 0$
%\begin{eqnarray*}
% \sum_{j,k=1}^{d}a_{j,k}(\cdot)\xi_j\xi_k \geq
%c_0 \mid \xi \mid^2$,\hspace{5mm}on $\={\Omega},\hspace{5mm}for all
%\xi \epsilon\Re^d
%\end{eqnarray*}
\begin{eqnarray}
\sum_{j,k=1}^d a_{j,k}(\cdot) \xi_{j} \xi_{k} \geq c_{0}
|\xi|^2,\quad \rm on \; \overline{\Omega}, \quad \rm{for\; all} \;
\xi \in \mathbb{R}^{d}.
\end{eqnarray}
We chose $A$ and $F$ based on an abstract formulation for
convenience to facilitate the development of the numerical scheme
and its analysis. The initial value problem \ref{tms1} is reset to
be posed in a Hilbert space ${X}$, as follows. Consider now $A$ to
be a linear, self-adjoint, positive definite, closed operator with a
compact inverse, defined on a dense domain $D(A) \subset {X}$. The
operator $A$ could represent any of $\{A_h\}_{0< h \le h_0}$,
obtained through spatial discretization, and ${X}$ We assume the
resolvent set $\rho(A)$ satisfies, for some $\alpha \in
(0,\frac{\pi}{2} )$, $\rho(A) \supset \overline{\Sigma}_{\alpha},
\,\, $ where $\Sigma_{\alpha}:= { z \in := \alpha < |\arg(z)| \leq
\pi,z \neq 0}$. Also, assume there exists $M \geq 1$ such that
\begin{equation}
\|(zI-A)^{-1}\| \geq M |z|^{-1}, \: z \in \Sigma_\alpha.
%\label{assumption1}
\end{equation}
It follows that $-A$ is the infinitesimal generator of an analytic
semigroup $\{e^{-tA}\}_{t\ge 0}$ which is the solution operator for
\ref{tms1}, and $|e^{-tA}| \leq C$ for ${t \ge 0}$. Also, we assume
that $F(t,u(t))$ is Lipschitz on $[0,T] \times X$, i.e. it satisfies
the following assumption:
\textbf{Assumption1}
\label{assumption2}
\emph{ $F:[0,T] \times X
\rightarrow X$ and $U$ be an open subset of $[0,T] \times X$. For
every $(t,x) \in U$ there exists a neighborhood $V \subset U$ and a
real number $L_{T}$ such that}
\begin{eqnarray}
\|F(t_1,x_1) - F(t_2,x_2)\| \leq L_{T} ( |t_1-t_2| + \|x_1-x_2\|X)
\label{lipschitz}
\end{eqnarray}
for all $(t_i, x_i) \in V $. Using the standard representation:
\begin{equation*}
E(t) := e^{-tA} = \frac{1}{2\pi i}
\int_{\Gamma}e^{-tz}(zI-A)^{-1}dz, \label{Eq3}
\end{equation*}
where
\begin{equation*}
\Gamma := {z \in :\arg(z) = \theta}
\end{equation*}
, oriented so that
$\mbox{\rm Im}(z)$ decreases, for any
$\theta \in (\alpha, \frac{\pi}{2})$ and the Duhamel principle, the
exact solution can be written as
\begin{equation}
u(t)=E(t)v+\int_{0}^t E(t-s) F(s,u(s)) ds. \label{NL-DP}
\end{equation}
Let $0 < k \le k_0$, for some $k_0$, and $t_n = nk$, $0\leq n \leq N
$. Replacing $t$ by $t+k$, using basic properties of $E$ and by the
change of variable $s-t=k\tau$, we arrive at the following
recurrence formula for the exact solution:
\begin{equation}
u(t_{n+1})= e^{-kA}u(t_n) + k\,\int_{0}^1 e^{-kA(1-\tau)} F(t_n+\tau
k, u(t_n+\tau k))\,d\tau.\label{Ch7-Eq2}
\end{equation}
This underlines the basis for deriving ETD schemes.
{\em The ETD-BE Scheme}.
Denoting the semidiscrete approximation to $u(t_n)$ by $u_n$ (note
that only the time-variable is discretized) and $F(t_n, u_n)$ by
$F_n$, the simplest approximation to the integral is to impose that
$F$ is constant for $t \in [t_n, t_{n+1}]$, i.e. $F \approx F_n$.
This yields (from (22))
\begin{eqnarray}
u(t_{n+1}) &\approx & e^{-kA}u(t_n) + e^{-kA} k \,\int_{0}^1
e^{kA\tau} \,d\tau \,F_n \, = \, e^{-kA}u(t_n) - A^{-1}
\left(e^{-kA} - I\right) \,F_n. \label{ETD1_sd}
\end{eqnarray}
This semidiscrete scheme becomes useful after discretization of the
matrix exponentially is efficient.Observe that
\begin{eqnarray}
-A^{-1} \big( e^{-kA} -I \big) &=& -A^{-1} \big( \, (I+kA)^{-1} -I \big)\nonumber\\
&=& -A^{-1} \big( I-(I+kA) \,\big)(I+kA)^{-1}\nonumber\\
&=& k \, (I+kA)^{-1}\nonumber\\
&=& kR_{0,1}\big(kA\big),\label{derivation_ETD1}
\end{eqnarray}
this results in a fully discrete first order scheme, where $v$ now
denotes the fully discrete solution. The standard first order
linearly implicit scheme for solving nonlinear schemes is
equivalent to this method . The \textbf{ETD-BE} scheme is as
follows:
\begin{equation}
u_{n+1} = R_{0,1}(kA)u_n + kR_{0,1}\big(kA\big) \,F(t_n, u_n).
\label{ETD1}
\end{equation}
\begin{equation}\label{fdd}
(I+kA)u_{n+1}=u_n +kF(t_n,u_n)
\end{equation}
This method is known as the exponential time differencing scheme of
the Backward Euler scheme, this is considerably efficient than
backward Euler method in solving nonlinear problems. We shall adopt
this scheme as the initial damping scheme for problems with
irregular data.The \textbf{ETD-BE} scheme has the advantage over
the backward Euler in solving nonlinear systems because it is
explicit in the nonlinear part which does not require nonlinear
solvers at each iteration which can be very time consuming for
instance,a modified Newton's method.
\subsubsection{Stability Analysis}
Let $U^n$ denote the analytical solution of the finite
difference scheme\ref{fdd} and let $V^n$ be the numerical solution.\\
Setting $E^n=U^n-V^n$,we arrive at the following relationship:\\
$(I+kA)E^{n+1}=E^n +k(F(U^n)-F(V^n))$\\
We assume that $f$ is a smooth function satisfying
the relationship:\\
\begin{equation}
|f'(u)|\leq L, for \: u \, \epsilon \,T
\end{equation}
where $T$ is an interval of $\Re$ containing the solution of scheme\ref{fdd}.\\
The mean value theorem involves\\
\begin{equation*}
F(U^n)-F(V^n)=F'(W^n)E^n
\end{equation*}
with
\begin{equation*}
F'(W^n)=diag(f'(w_1^n)...f'(w^n_{N-1}))
\end{equation*}
Hence
\begin{equation*}
(I+kA)E^{n+1}=kA+kF'(W^n)E^n
\end{equation*}
Setting $R^n=(kA)^{-1}kA + kF'(W^n)$,the positivity of the matrix
$(I+kA)^{-1}$ implies the positivity of the matrix $R_n$.\\
with the previous notations,we have $|E^{n+1}|\leq R_n|E^n|$ whose
elements are $|E^n_1|$, for $i=1...N-1$ and Let $|R_n| $ be the
matrix with elements $|Rn_{ij}|$,for $i,j=1...N-1$.\\
From the positivity of the matrix $R_n$ the following relation
follows:
\begin{equation*}
|E^{n+1}|\leq R_n|E^n|
\end{equation*}
From condition(26) we deduce $F'(W^n)\leq L\cdot I $ and therefore:
\begin{equation*}
R_n\leq(I+kA)^{-1}(1+Lk)I
\end{equation*}
setting
\begin{equation*}
R=(I+kA)^{-1}(1+Lk)I
\end{equation*}
we have$|E^{n}|\leq R_n|E^0|$ with $|E^0|=|U^0-V^0|$.\\
Let $\rho(R)$ be the spectral radius of the matrix $R$.if
$\rho(R)<1$ then
\begin{equation*}
\lim_{n \rightarrow +\infty} \\mathbb{R}^{n} = 0
\end{equation*}
Therefore,if
\begin{equation*}
\rho(R)<1, \lim_{n \rightarrow +\infty} |E^n|= 0
\end{equation*}
and scheme (25) is numerically stable.
\subsubsection{Algorithm}
We use \textbf{ETD-BE} method for time stepping which leads to the following equation\\
\vspace{5mm}
\begin{equation}
[\Phi-kR]c^n=[\Phi +kR]c^{n+1}+kQ^{n+1}
\end{equation}
The terminal condition serves as an initial condition for the ODE
system. After collocation at the points $x_i$, $i=1,\ldots,N$, the
coefficients $c_j(T)$ are given as the solution of the linear system
\begin{equation*}
\Phi c(T)=\textbf{P}
\end{equation*}
where $\Phi$ is as above, and
$ \textbf{P }= [P_\epsilon(x_1,T),\ldots,P_\epsilon(x_N,T)]^T$.
Since radial basis functions do not satisfy the boundary conditions
automatically, they are satisfied by adding specific equations to