The Numerical Modeling For A Class Of BBM Equations

This dissertation mainly talks about the initial boundary value problem of time fractional BBM equation,where ?,?,? and ? are non-negative parametes,? is the damping parameter,u is the horizontal velocity of fluid,OcDt1/2 is 1/2 Caputo time fractional term,?uxx is the viscous term,and ?uxxt is the dispersion term.This type of BBM equation with a time fractional is called the viscous water wave equation,which is a more appropriate to describe long-wave unidirection-al propagation in a weakly nonlinear dispersive medium than the classical water wave equations such as the kdV equation.The proposal of the solitary wave move-ment on a calm water surface by J.Scott-Russell,especially the discovery of the non-propagating solitary wave on the water surface has forcefully promoted the development of the nonlinear science.The reason lies in the fact that various non-linear dynamic physical phenomena can be described as a nonlinear mathematical model.The characteristic of the BBM equation is that the diffusive terms and the dispersive terms influence each other,which can be interpreted in physics as that the viscous boundary layer possesses diffusion and dispersion in the fluid at the same time,which has a wide range of applications in fields such as nonlinear optics,plasma physics and so on.A three-level linearied difference schemes?the two-level linearied difference schemes(C-N-L1)?C-N-L1-2)and a fourth order compact finite difference scheme are proposed for fractional BBM equation in this paper.The truncation error is obtained according to the Taylor expansion.And the prior estimate of the so-lution is obtained.Convergence and stablity of the finite difference schemes are prove by Discerte Sobolev's inquality and Discrete Gronwall's inequality.Numeri-cal experiments are presented to verify that theoretical analysis is accurate and to demonstrate that the numerical scheme is effective.