Numerical Method For Nonlinear Delay VIE And Delay ODE

In this paper, two kinds of nonlinear delay equations are studied. In re-cent years, with delay integral equation, delay differential equation in physics, biology, medicine, chemical, engineering and economics, and so the realm of science, more and more application to delay integral equation, ordinary dif-ferential equations of academic research attracted great attention.In the first chapter of this paper a class of nonlinear delay product with a Volterra integral equations with nonlinear delay are studied as follows with 00, such that for all h∈(0. h) and g∈(0,1), the systems possesses a unique solution uh. Hence, the collocation solution uh∈Sm-1(-1) exists and is unique for all meshes.Theorem 0.4 Assume in the equation (0.1)(1)a, g∈Cv1(I), K∈Cv2(D), v1,v2≥m,(2)‖a‖∞<1,(3)uh∈Sm-1-1(Ih) is the collocation solution. Then‖y-uh‖∞≤Chm, where C depends on {ci}i=1m and q, but not on h.In the second chapter, a class of nonlinear differential equation of vanish-ing delay was studied, in the form of equations where a(t), f(t)∈C[0,T], b satisfies Lipschicz continuous,θ(t) has the fol-lowing propertiesθ(0)=0 andθ(t) are monotonia increasing in I = [0, T];θ(t)≤qt, t∈I, q∈(0,1);θ(t)∈Cd(I), d≥1 is an integer. Consider a general nonlinear system Lu(t) + Nu(t)=g(t), where L is linear operator, N is a nonlinear operator,and g(t) is a known analytical function. Then usual format for variational iterative method as follows among themλ(t, s) is Lagrange multiplier. The main ideas of the iteration format is to produce the correction iterative solution value∫0l(t, s)(Lun(s)+ Nun(s)-g(s))ds. Taking the variational, the iterative solution convergent. Then we bring (0.3) in (0.4), making use of variational theory, finally, we can get Thus the Lagrange multiplier can be Then equation (0.3) for the variational iterative formula is We take u0(t) as the beginning value, according to (0.6), we can get the recurrence sequence{un(t)}n=0∞,t∈I. We can prove that the sequence is convergent, and the limit of the sequence will be the solution of original equation.Theorem 0.5 Assume a,f∈C[0,T], b(t,u) is Lipschitz continuous in I×B,θ(t) is vanishing delays, by the variational iterative we can get the solution sequence un(t) is convergent in I=[0,T].Finally, numerical tests are presented. And the results of the numerical experiments can illustrate our theoretical results.