Approximate General Solution of Degenerate Parabolic Equations Related to Population Genetics Kazuo Amano

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 diierential 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 nite diierence method. x1. Introduction As is well-known, if a given partial diierential 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 nd solutions in abstract function spaces or working out numerical algorithms. In this paper we shall show new possibilities for 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 diierent from classical ones; we use a new type of 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 (1:1) @u @t = 1 2 @ 2 @x 2