Collocation Method For A Class Of Volterra Integral Functional Equations With Two Proportional Delays

In this paper, we study the collocation method for a class of Volterra integral func-tional equations with two proportional delays(TDVIFEs). Delay Equations have numerous applications in science and engineering. With the development of modern society, Delay Equations become an important topic of modern mathematics. Numerical methods based on finite difference methods, discontinuous Galerkin methods and spectral methods etc., have also been developed for various Delay Equations. In this paper,we mainly discuss the col-location method to a class of Volterra integral functional equations with two proportional delays(TDVIFEs).Firstly we introduce the general form of Volterra integral functional equations with multiple delays. (TDVIFEs)where the delay functionθi(t),i=1,2 are assumed to have the following properties: (P1)θi-(0)=0, andθi is strictly increasing onⅠ; (P2)θi(t)≤qit on I for some qi∈(0,1); (P3)θ∈Cpi(Ⅰ) for some integer pi≥0.An important special case(proportional delay) is the linear delay functionθi(t)=qit= t-(1-qi)t=t-τi(t)(0q2Ⅱ≥q2Ⅰ.The key point is that the overlap of the mages q1t and q2t of the collocation points t=tn,i:=tn+cih(i=1,…,m)with the interval(tn,tn+1].All the cases are clear when we understand this.Next,we present the existence and uniqueness of the collocation solution and the con-vergence analysis.The key to the proof of the existence and uniqueness of the collocation solution is that the matrix in front of Un is nonsingular.We can refer to [8]for the details about the similar proof. In the convergence, Volterra integral functional equations with two proportional delays accord with general Volterra integral equations, we obtain where the constant C depend on{ci}and q1,q2 but not on h.We both use the traditional analysis and operator method to prove it. About the su-perconvergence, no matter what collocation points we choose (such as m-Gauss points), TDVIFEs no longer have the superconvergence as VIEs.Finally we use the collocation method to four TDVIFEs examples with different choice of the delay coefficients q1,q2 and the functions a1(t),a2(t),f(t),K0(t,s), K1(1, s),K2(t, s). Re-sults of the numerical simulations verify our convergence analysis.