Dissipativity Analysis Of The Numerical Methods For A Class Of Coupled System Of Functional Differential And Functional Equations

Functional differential and functional equations are a class of systems which are coupled by functional differential equations and functional equations.It can be used to describe many problems in physics and engineering.However,it is difficult to obtain the analytical solution of such a system.Therefore,it is very important to solve them numerically.Let X be a real(or complex)Hilbert space,(·,·)and ||·|| are the inner product in X and the corresponding inner product norm respectively.Consider the initial value problem of a class of nonlinear functional differential and functional equations(FDFEs)in X as follows initial function:y(t)=?(t),z(t)=?(t),t?0,where ?>0 is a real constant,?,? is a given continuous function,satisfying the consis-tency condition:?(0)=g(0,?(0),?(-?),?(-?)).Mapping f:[0,+?)× X × X × X?X and g:[0,+?)×X×X×X?X continuous,and for all t>0,y,u,v,w?X,f and g satisfy:2Re(f(t,u,v,w),u>??1+??u?2+??v?2+??w?2,?g(t,u,v,w)?2??2+Lu?u?2+Lv?v?2+Lw?w?2,here-?,?,?1,?2,Lu,Lv,Lw,? are non-negative real constands.The main work and results obtained in this paper are as follows:We study the dissipativity of the FDFEs problem,the sufficient conditions for the system are obtained,it is proved that under certain conditions,G(c,p,0)-algebraically stable one-leg methods and(k,l)-algebraically stable Runge-Kutta methods can inherit the dissipativity of the system,some numerical experiments have been carried out and the results also prove the correctness of the theoretical analysis.