Investigation of some Statistical Characteristics of the Cauchy Problem for Burgers-Huxley Equation Solution

Аuthors

Vassilieva O. A.

Moscow State University of Civil Engineering, MSUCE, 26, Yaroslavskoe shosse, Moscow, 129337, Russia

e-mail: vasiljeva.ovas@yandex.ru

Abstract

Burgers-Huxley equation is applied in various areas of science such as nonlinear acoustics, fluid dynamics, biology, ecology chemistry etc. The Cauchy problem for Burgers-Huxley equation describes the evolution of the initial process with consideration of the effects of nonlinearity and diffusion. There are a lot of noise sources of different physical nature. This explains the importance of stochastic solutions of the Cauchy problem for Burgers-Huxley equation investigation.

The Cauchy problem for Burgers-Huxley equation with stationary normal process as initial condition is considered. Solution of this problem is a stationary stochastic process for any fixed value of time variable. As a rule, the practical interest is not stochastic solution itself but its statistical characteristics such as distribution, correlation function. Burgers-Huxley equation is nonlinear partial differential equation. There is no analytical solution of the equation in general case. So, numerical methods of statistical characteristics of Burgers-Huxley equation solution investigation are very important. For fixed value of time variable the solution correlation function is investigated. For some values of equation parameters and under rather general assumptions stochastic solution of Burgers-Huxley equation for fixed value time variables is ergodic in wide-sense process. Thus, the solution correlation function can be obtained from a single sufficiently long realization. This realization is a solution of the Cauchy problem for Burgers-Huxley equation for given value of time variable. The finite-difference method of second order is used for numerical solution of the Cauchy problem. The quadrature formula of second order is used for approximate calculation of the correlation function. Some numerical results are presented and discussed.

Keywords:

Burgers-Huxley equation, the Cauchy problem, stochastic process

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