【作者】 孙春雪；

【导师】 张然；

【作者基本信息】 吉林大学 ， 计算数学， 2011， 硕士

【摘要】 本文对两类非线性时滞方程的数值解法进行了研究。第一章针对一类非线性时滞Volterra积分方程,给出了方程解析解的存在唯一性证明,并利用配置法求解该方程,给出了相应数值格式的矩阵表示,证明了配置解的存在性及收敛性,并利用数值实验对理论结果加以佐证。第二章针对一类非线性时滞微分方程,利用变分迭代法,给出了迭代解序列,并证明了迭代解序列的收敛性,最后通过数值实验对理论结果进行验证。更多还原

【Abstract】 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 0<q<1,a,g and K are sufficient smooth.From Equation (0.1). we can find the difficulties areThe equation contains two delays, which makes the integral interval more complicated;Nonlinear equation, thus for nonlinear numerical format and theoretical proof comparing with linear equation are more complicated;The study of Volterra integral contains nonlinearity, thus numerical scheme may be implicit form.We have given equation (0.1) of the existence and uniqueness of analytic solutions and regularity.Theorem 0.1 Assume(ⅰ) a, g∈C(I), I=[0, T], K∈C(D), D={(t, s)|0≤t≤T, qt≤s≤T},(ⅱ)‖a‖∞<1. Then for any given g∈C(I), the analytic solution of equation (0.1) u∈C(I).Theorem 0.2 Assume(ⅰ) a, g∈Cv(I), K∈C(D), v≥lisaninteger, (ⅱ)‖a‖∞<1. Then the analytic solution of equation (0.1) u∈G Cv(Ⅰ).Given Ih and collocation points set Xh = {tn,i = tn + cjh. 0<c1 <…≤1,n = 0,…, N - 1}, collocation space Sm-1-1) = {v∈C(I) : v|en∈πm, (0≤n≤N -1)}. Applying equation (0.1), thus we have Un,i=uhn(tn,i) Considering the difficulty of solving (0.2) in proportion with time delay is given at first and integral items,the classification of {qtn,i}i=1m , n =0,…, N-1.Ⅰ(initial phase or complete overlap) For n = 0, we have {qtn,i}i=1m∈(tn,tn+1).Ⅱ(transition phase or partial overlap) For qⅠ≤n<qⅡ, there existsαn∈{1,…,m-1}, s.t.Ⅲ(pure delay phase or nonoverlap) For qⅡ≤n≤N-1, qtn,i≤tn, i = 1,…,m. Because {qtn,i}i=1m belong to one or two intervals, so we haveⅢ(a) {qtn,i}i=1m belong to two intervals, which mean {qtn,i}i=1m(?) (tqn,tqn+1], qn = qn,i,i =1,…,m;Ⅲ(b) {qtn,i}i=1m belong to one interval, which means there existsβn∈{1,…,m-1},s.t.Categorizing as above, we get the matrix expression of collocation equa-tion (0.2), what’s more, we obtain the collocation solution of (0.1). The existence, uniqueness and convergence theorem of the collocation solution are as followsTheorem 0.3 Assume in the equation (0.1) (1)a,g∈C(I), K∈C(D).(2)‖a‖∞<1,(3) Ih is uniformly meshes. Then there exists h>0, 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.更多还原