\magnification = \magstephalf \hsize=14truecm \hoffset=1truecm \parskip=5pt \nopagenumbers \font\eightrm=cmr8 \font\eighti=cmti8 \font\eightbf=cmbx8 \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1995/15\hfil Approximate General Solution \hfil\folio} \def\leftheadline{\folio\hfil Kazuo Amano \hfil EJDE--1995/15} \voffset=2\baselineskip \vbox {\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations, Vol. {\eightbf 1995}(1995) No.\ 15, pp. 1--14.\hfill\break ISSN 1072-6691. URL: http://ejde.math.swt.edu (147.26.103.110)\hfil\break telnet (login: ejde), ftp, and gopher access: ejde.math.swt.edu or ejde.math.unt.edu} \footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt % 1991 {\eighti Subject Classification:} 35K65, 65M99. \hfil\break {\eighti Key words and phrases:} Degenerate parabolic, numerical-symbolic method. \hfil\break \copyright 1995 Southwest Texas State University and University of North Texas.\hfil\break Submitted February 20, 1995. Published October 20, 1995.} } \bigskip\bigskip \centerline{APPROXIMATE GENERAL SOLUTION OF DEGENERATE} \centerline{PARABOLIC EQUATIONS RELATED TO POPULATION GENETICS} \smallskip \centerline{Kazuo Amano} \bigskip\bigskip {\eightrm\baselineskip=10pt \narrower \centerline{\eightbf Abstract} The author constructs an approximate general solution to a degenerate parabolic equation related to population genetics and implements a computational procedure. The result gives a theoretical foundation to the computer algebraic approach for degenerate partial differential equations and introduces a new numerical symbolic hybrid method. The techniques are likely to have wide applicability, since the key idea of the algorithm is a rearrangement of the finite difference method. \bigskip} \def\Proof{\smallbreak\noindent{\bf Proof.\enspace}} \def\QED{\ \hbox{\vrule height 1.6ex width 1ex depth .3ex}\ } \def\pd{\partial} \def\ds{\displaystyle} \bigbreak \centerline{\bf \S 1. Introduction} \medskip\nobreak As is well-known, if a given partial differential equation is very simple, we can compute its general solution with arbitrary functions by using arithmetic and elementary calculus. However, most equations require hard and abstract mathematical technicalities. An explicit and concrete representation of the solution may turn out to be utterly beyond our reach. It seems that researchers have already given up constructing explicit general solutions; they are either trying to find solutions in abstract function spaces or working out numerical algorithms. In this paper we shall show new possibilities for {\it approximate general solutions\/}; though an explicit representation is already at deadlock, an approximate one is able to break through obstacles and gives a new viewpoint. In fact, we prove that, for a certain initial value problem, there exists a simple algebraic representation of an approximate general solution, i.\ e., a symbolic combination of additions, subtractions and multiplications of initial data solves the problem. Our procedure of construction of a general solution is quite different from classical ones; we use a new type of {\it numerical-symbolic hybrid method\/}. Our numerical-symbolic hybrid computation totally depends on LISP and its result is expressed in C language, since the size of desired formula is more than 5.4M bytes. Such a formula is too big for classical pen and paper calculation. It is to be noted that a remarkably fast algorithm is derived from our formula of approximate general solution. The purpose of this paper is to construct an approximate general solution of the initial value problem for the degenerate parabolic equation $$ {\pd u\over\pd t} ={1\over2}{\pd^2\over\pd x^2}\bigl(V(x)u\bigr) -{\pd\over\pd x}\bigl(M(x)u\bigr)\qquad (t>0,\ 00,\ 00\hbox{ for } {1\over 2}-{1\over\sqrt{2h^2+4}}0\ ,\quad h^2>{1\over 2}-{1\over\sqrt{2h^2+4}}\ . $$ Hence, we obtain $$ f(x)>0\hbox{ for } h^2\leq x\leq 1\ ; $$ this implies $$ x\pm\sqrt{a(x)}\,h\geq {h^2\over 4}\hbox{ for } h^2\leq x\leq 1\ . $$ Similarly, we obtain $$ x\pm\sqrt{a(x)}\,h\leq 1-{h^2\over 4}\hbox{ for } 0\leq x\leq 1-h^2\ .\quad\QED $$ \proclaim{Lemma 3.2}. For $\,00\,$ for $\,\ds{h^2\over4}\leq h\leq h^2\,$, we have $$ \min_{ h^2/4\leq y\leq h^2}g(y)=g\Bigl({h^2\over4}\Bigr) ={2h^2\over4-h^2}\geq {h^2\over2}\ ; $$ this gives $$ \varepsilon(y)\geq {h\over\sqrt{2}}\hbox{ for }{h^2\over4}\leq y}{\tt\mc \kern0.500em}0{\tt\mc \kern0.500em}d\kern-.45em d\kern-.05em o\kern-.45em o\kern-.05em \noindent {\tt\mc \kern1.500em}{\tt <}{\tt <} \noindent {\tt\mc \kern3.000em}w\kern-.45em w\kern-.05em h\kern-.45em h\kern-.05em i\kern-.45em i\kern-.05em l\kern-.45em l\kern-.05em e\kern-.435em e\kern-.065em {\tt\mc \kern0.500em}length(list{\tt\_\kern.141em}in) {\tt\mc \kern0.500em}{\tt >}{\tt\mc \kern0.500em}0{\tt\mc \kern0.500em}d\kern-.45em d\kern-.05em o\kern-.45em o\kern-.05em \noindent {\tt\mc \kern3.000em}{\tt <}{\tt <} \noindent {\tt\mc \kern4.500em}tmp{\tt\mc \kern0.500em}:={\tt\mc \kern0.500em}first(list{\tt\_\kern.141em}in); \noindent {\tt\mc \kern4.500em}p{\tt\mc \kern0.500em}:={\tt\mc \kern0.500em}first(tmp); \noindent {\tt\mc \kern4.500em}s{\tt\mc \kern0.500em}:={\tt\mc \kern0.500em}first(rest(tmp)); \noindent {\tt\mc \kern4.500em}y{\tt\mc \kern0.500em}:={\tt\mc \kern0.500em}first(rest(rest(tmp))); \noindent {\tt\mc \kern4.500em}q{\tt\mc \kern0.500em}:={\tt\mc \kern0.500em}(1+h{\tt *}{\tt *}2{\tt *}c(y){\tt /}3+h{\tt *}{\tt *}4{\tt *}c(y){\tt *}{\tt *}2{\tt /}9){\tt *}p; \noindent {\tt\mc \kern4.500em}i\kern-.45em i\kern-.05em f\kern-.45em f\kern-.05em {\tt\mc \kern0.500em}domain{\tt\_\kern.141em}p(s){\tt\mc \kern0.500em}neq{\tt\mc \kern0.500em}0{\tt\mc \kern0.500em}t\kern-.44em t\kern-.06em h\kern-.45em h\kern-.05em e\kern-.435em e\kern-.065em n\kern-.45em n\kern-.05em \noindent {\tt\mc \kern4.500em}{\tt <}{\tt <} \noindent {\tt\mc \kern6.000em}list{\tt\_\kern.141em}tmp{\tt\mc \kern0.500em}:= {\tt\mc \kern0.500em}cons(\rm\mc $\kern.024em\{$q{\tt /}6,{\tt\mc \kern0.500em}s,{\tt\mc \kern0.500em}y+sqrt(a(y)){\tt *}h$\}\kern.024em$\tt\mc ,{\tt\mc \kern0.500em}list{\tt\_\kern.141em}tmp); \noindent {\tt\mc \kern6.000em}list{\tt\_\kern.141em}tmp{\tt\mc \kern0.500em}:= {\tt\mc \kern0.500em}cons(\rm\mc $\kern.024em\{$q{\tt /}6,{\tt\mc \kern0.500em}s,{\tt\mc \kern0.500em}y{\tt -}sqrt(a(y)){\tt *}h$\} \kern.024em$\tt\mc ,{\tt\mc \kern0.500em}list{\tt\_\kern.141em}tmp); \noindent {\tt\mc \kern6.000em}list{\tt\_\kern.141em}tmp{\tt\mc \kern0.500em}:= {\tt\mc \kern0.500em}cons(\rm\mc $\kern.024em\{$q{\tt /}3,{\tt\mc \kern0.500em}s,{\tt\mc \kern0.500em}y+b(y){\tt *}h{\tt *}{\tt *}2$\} \kern.024em$\tt\mc ,{\tt\mc \kern0.500em}list{\tt\_\kern.141em}tmp); \noindent {\tt\mc \kern6.000em}list{\tt\_\kern.141em}tmp{\tt\mc \kern0.500em}:= {\tt\mc \kern0.500em}cons(\rm\mc $\kern.024em\{$q{\tt /}3,{\tt\mc \kern0.500em}s{\tt -}h{\tt *}{\tt *}2,{\tt\mc \kern0.500em}y$\} \kern.024em$\tt\mc ,{\tt\mc \kern0.500em}list{\tt\_\kern.141em}tmp) \noindent {\tt\mc \kern4.500em}{\tt >}{\tt >} \noindent {\tt\mc \kern4.500em}e\kern-.435em e\kern-.065em l\kern-.45em l\kern-.05em s\kern-.45em s\kern-.05em e\kern-.435em e\kern-.065em \noindent {\tt\mc \kern6.000em}list{\tt\_\kern.141em}tmp{\tt\mc \kern0.500em}:= {\tt\mc \kern0.500em}cons(\rm\mc $\kern.024em\{$p,{\tt\mc \kern0.500em}s, {\tt\mc \kern0.500em}y$\}\kern.024em$\tt\mc ,{\tt\mc \kern0.500em}list{\tt\_\kern.141em}tmp); \noindent {\tt\mc \kern4.500em}list{\tt\_\kern.141em}in{\tt\mc \kern0.500em}:= {\tt\mc \kern0.500em}rest(list{\tt\_\kern.141em}in) \noindent {\tt\mc \kern3.000em}{\tt >}{\tt >}; \noindent {\tt\mc \kern3.000em}list{\tt\_\kern.141em}in{\tt\mc \kern0.500em}:= {\tt\mc \kern0.500em}list{\tt\_\kern.141em}tmp; \noindent {\tt\mc \kern3.000em}list{\tt\_\kern.141em}tmp{\tt\mc \kern0.500em}:= {\tt\mc \kern0.500em}\rm\mc $\kern.024em\{$$\}\kern.024em$\tt\mc ; \noindent {\tt\mc \kern3.000em}n{\tt\mc \kern0.500em}:={\tt\mc \kern0.500em}n{\tt -}1 \noindent {\tt\mc \kern1.500em}{\tt >}{\tt >}; \noindent {\tt\mc \kern1.500em}return{\tt\mc \kern0.500em}list{\tt\_\kern.141em}in \noindent e\kern-.435em e\kern-.065em n\kern-.45em n\kern-.05em d\kern-.45em d\kern-.05em ; \noindent \rm\mc %%%%%%%%%%%%%%%%%%%%%%%%% % end of REDUCE program % %%%%%%%%%%%%%%%%%%%%%%%%% Here {\tt\ domain\_p(t)\ } is a function defined in $\,[0, \infty)\,$ such that $$ {\tt domain\_p(t)}=\cases{ \ 1 &if\enspace$t\geq h^2$\cr \cr \ 0 &otherwise\ .\cr} $$ Functions $\,a(x),\ b(x)\,$ and $\,c(x)\,$ are coefficients defined by (2.6). To be more explicit, the parameter $\,h\,$ should be modified in each step of main loop of the above Reduce Program so that $$ y\pm\sqrt{a(y)}\,h,\ y+b(y)\,h^2\in [0,1] $$ is valid and also, we should identify any pair of list $\,(q_i\ t_i\ x_i)\,$ and $\,(q_j\ t_j\ x_j)\,$ with $\,(q_i+q_j\ t_i\ x_i)\,$ when $\,(t_i\ x_i)=(t_j\ x_j)\,$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % Section 4 % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigbreak \centerline{\bf \S4. Examples}\medskip\nobreak We shall solve the following two problems by using (3.19) : $$ \cases{ \ds{\pd u\over\pd t}={1\over4N}{\pd^2\over\pd x^2}\big(x(1-x)u\big) \qquad (00\,$. Then, since $\,c(x)=-NM'(x)=0\,$ implies $\,r=1\,$, Theorem 3.2 gives $$ |u(t,x)-u_k(t.x)|\leq C\,\Biggl(h^2+\sum_{\ell=0}^{[2t/h^2]+1}{\,k\,\choose\,\ell\,} \Bigl({1\over3}\Bigr)^\ell\Bigl({2\over3}\Bigr)^{k-\ell} +h^4k\Biggr) \leqno(4.3)$$ for $\,k\geq 2t/h^2\,$. Here we note that, seeing the proof of Lemma 2.3, we can estimate the above constant $\,C\,$ by using maximum principle and also, according to the fundamental property of binomial distribution, the second term of the right hand side of (4.3) tends to 0 when $\,k\to\infty\,$. Thus, if we choose $\,2t/h^2\ll k\ll 1/h^4\,$ appropriately, $\,|u(t,x)-u_k(t.x)|\,$ would become sufficiently small. The same argument remains true for the problem (4.2). The graphs of our approximate solutions of (4.1) and (4.2) are given in Figures 1 and 2. It is to be noted that our solutions are approximately equal to Kimura's exact ones and they lead to the same conclusion that Kimura obtained \ (cf.\ [$\,$2, Chapter 8$\,$]). %%%%%%%%%%%% % Figure 1 % %%%%%%%%%%%% \topinsert\vskip 8cm \special{psfile=sol1.eps hscale=70 vscale=70} \centerline{Figure 1} \endinsert %%%%%%%%%%%% % Figure 2 % %%%%%%%%%%%% \topinsert\vskip 11.5cm \special{psfile=sol2.eps hscale=60 vscale=60} \centerline{Figure 2} \endinsert %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % Acknowledgment % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent{\bf Acknowledgement}. The author would like to express his hearty thanks to the referee for his helpful criticisms, suggestions and encouragement. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % References % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vfill\eject \centerline{\bf References}\medskip\nobreak \item{[1]} K. Amano, Numerical-symbolic hybrid method for biharmonic Dirichlet problem, ({\it to appear\/}). \item{[2]} J. F. Crow and M. Kimura, An Introduction to Population Genetics Theory, {\sl Burgess Publishing Company}, 1970. \item{[3]} W. J. Ewens, Mathematical Population Genetics, {\sl Springer-Verlag}, 1989. \item{[4]} A. C. Hearn, REDUCE User's Manual, version 3.5, {\sl RAND Publication}, CP78 (Rev. 7/94), 1994. \item{[5]} O. A. Oleinik and E. V. Radkevich, Second Order Equations with Nonnegative Characteristic Form, {\sl Amer. Math. Soc., Province, Rhode Island and Plenum Press, New York}, 1973. {\sc Department of Mathematics, Josai University, Sakado, Saitama, JAPAN} E-mail: kamano@tansei.cc.u-tokyo.ac.jp \bye