\documentstyle[12pt]{article} \topmargin=0cm\headheight=0cm\headsep=0cm \oddsidemargin=0truecm \evensidemargin=0truecm \textheight=24cm \textwidth=17cm \columnsep=0.4cm\baselineskip=0.4cm \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle[12pt]{article} \topmargin=0cm\headheight=0cm\headsep=0cm \oddsidemargin=0truecm \evensidemargin=0truecm \textheight=24cm \textwidth=17cm \columnsep=0.4cm\baselineskip=0.4cm \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} 