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\begin{document}

\title {Stability of Variable Coefficients Runge-Kutta methods}

\author{By Masaharu Nakashima\\
Department of Mathematics, Faculty of Science, Kagoshima \\
University Korimoto cho 21-35, Kagoshima city 890, Japan\\
e-mail; naka @sci.kagoshima-u.ac.jp\\}\date{
Key word: Runge-Kutta methods\\
Subject Classification 65L06,65L07 }
\maketitle

\begin{abstract}
During the last years, there has been a considerable amount of research
on the numerical integration of stiff systems of ODE's.
A basic difficulty in the numerical solution of stiff systems is the
satisfying of
the requirement of stability.  J.C Butcher[1] provided Implicit Runge-Kutta
methods(abb;R-K methods) to overcome the stability problem.
However for solving the stiff equations by using explicit methods, the
author[2]
propose variable coefficients Runge-Kutta methods for  which  we have proved
stability for the diagonal system. We will propose the variable coefficients
algorithms of order 1 and study the stability analysis  for  2-dim system
differential equation: \\
$$\dot Y = AY,$$
$$ \quad with\quad A=\left(\matrix{a & b\cr
                                 c & d\cr}
     \right),\quad a,b,c,d \in R. \eqno (1)$$
whose eigenvalues have negative real parts. The methods which we propose are
$$ ^{1}y_{n+1}  = {^1}y_n + h\;{^{1}k_{1}}+h^2\sum_{i=1}^2{^{1}d_{i}},$$
$$ ^{2}y_{n+1}  = {^2}y_n + h\;{^{2}k_{1}}+h^2\sum_{i=1}^2{^{2}d_{i}},$$
$$^{1}k_1 = {^1}f(x_n,^1y_n,^2y_n),$$
$$^{2}k_1 = {^2}f(x_n,^1y_n,^2y_n).$$
Taking the  $^id_j,(i,j=1,2)$  suitably, we derive A-stable  for some
special case
in (1). We also present some numerical tests justifying the results.
 \begin{thebibliography}{99}
\bibitem{Butcher} J. C. Butcher,
\newblock  {\em Coefficients for the study of Runge-Kutta integration
processes,}
\newblock J. Austral. Math. Soc.,3(1963),185-201.
\bibitem{Nakashima}M. Nakashima,
\newblock {\em Variable Coefficients A-stable Explicit Runge-Kutta Methods,}
\newblock J,J,I,A,M,12(1995),285-308 .
\end{thebibliography}
\end{abstract}
\end{document}
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\begin{document}
\title {A study of Explicit Runge-Kutta methods for solving
the parabolic diffierntial Equation}
\author{By Masaharu Nakashima\\
Department of Mathematics, Faculty of Science, Kagoshima \\
University Korimoto cho 21-35, Kagoshima city 890, Japan\\
}
 \maketitle
\qquad It is well known that problem of stability arise in the solution
 of parabolic \\
 partial differential equations. If the equation is linear,for example,
 we study the numerucathe constant coefficients hyperbolic systems.
$${\partial u \over{ \partial t}} = \kappa  {\partial^2 u \over{\partial
x ^2}},\eqno (1)$$
subject to the initial and boundary conditions,
$$u(x,0)\;=\;f(x),\quad  0\le x \le 1, u(0,t)\;=\;g_1(t),
\;u(1,t)\;=\;g_2(t) \quad t\ge 0.$$
If the discretization of the partial operator is replaced by the difference\\
  $${\partial u^2 \over{{\partial t^2}}} \simeq
  {1\over{k^2}}\{u(x+k,t)-2u(x,t) + u(x-k,t)\},$$
  then the equation(1) is repalced by a set of simultaneous ordinary
  differential equation\\
  $$\dot x \;=\;Ax, \eqno (2)$$
Using Any explicit numerical methods of solving equation,the length of
time intervals \\
is thus introduced. To overcome those problem, we propose the variable
coefficients Runge-Kutta methods (explicit methods);
$$ y_{n+1}  = y_n + h\sum_{i=1}^rb_ik_i,\quad k_1 = f(x_n,y_n),$$
$$ k_i = f(x_n + c_ih,y_n + h\sum_{j=1}^{i-1}a_{ij}k_j),\;c_i =
\sum_{j=1}^{i-1}a_{ij}, \qquad\qquad (i = 2,..,r).$$
 where the coefficients $b_i$ is variable.
 Some numerical test justifying the results are also presented.
 \begin{thebibliography}{99}
\bibitem{ } M. Nakashima, Variable Coefficients A-stable
Explicit Runge-Kutta Methods,
  J,J,I,A,M,12(1995),285-308
  \end{thebibliography}
\end{document}