Special solutions of Huxley differential equation

Abstract

The conditions when solutions of Huxley equation can be expressed in special form and the procedure of finding exact solutions are presented in this paper. Huxley equation is an evolution equation that describes the nerve propagation in biology. It is often useful to obtain a generalized solitary solution for fully understanding its physical meanings. It is shown that the solution produced by the Exp-function method may not hold for all initial conditions. It is proven that the analytical condition describing the existence of the produced solution in the space of initial conditions (or even in the space of the system's parameters) can not be derived by the Exp-function method because the question about the existence of that solution is omitted. The proposed operator method, on the contrary, brings the load of symbolic computations before the structure of the solution is identified. The method for the derivation of the solution is based on the concept of the rank of the Hankel matrix constructed from the sequence of coefficients representing formal solution in the series form. Moreover, the structure of the algebraic-analytic solution is generated automatically together with all conditions of the solution's existence. Computational experiments are used to illustrate the properties of derived analytical solutions.

Keyword : Nonlinear differential equation, Hankel matrix, algebraic-analytical solution