关于非线性时滞积分方程和微分方程的数值解法
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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。
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 From Equation (0.1). we can find the difficulties are
The 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 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 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 have
Ⅲ(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 follows
Theorem 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 properties
θ(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.
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。
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
The 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
Ⅱ(transition phase or partial overlap) For qⅠ≤n
Ⅲ(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 follows
Theorem 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 properties
θ(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.
引文
[1]Hermann B., Collocation methods for Volterra integral and related functional differential equations. Cambridge Monographs on Applied and Computational Mathematics,15.Cambridge University Press, Cambridge,2004.
[2]Marzban H.R. and Tabrizidooz H.R., Solution of nonlinear delay optimal con-trol problems using a composite pseudospectral collocation method. Commun. Pure Appl. Anal.9 (2010), no.5,1379-1389.
[3]Rashidinia J., Ghasemi M. and Jalilian R., A collocation method for the solu-tion of nonlinear one-dimensional parabolic equations. Math. Sci. Quart. J.4 (201.0), no.1,87-104.
[4]Bellen A., Brunner H., Maset S. and Torelli L., Superconvergence in colloca-tion methods on. quasi-graded meshes for functional differential equations with vanishing delays. BIT 46 (2006), no.2; 229-247.
[5]Hermann B., The discretization of Volterra functional integral equations with proportional delays. (English summary)Differences and differential equations, pp 3-27, Fields Inst. Commun.,4.2, Amer. Math. Soc., Providence, RI,2004.
[6]Mustafa Nizami and Khalilov Elnur H. The collocation method for the solution of boundary integral equations. Appl. Anal.88 (2009), no.12,1665-1675.
[7]Auzinger Winfried, Koch Othmar and Weinmuller Ewa, Analysis of a new error estimate for collocation methods applied to singular boundary value problems. SIAM J. Numer. Anal.42 (2005), no.6,2366-2386 (electronic).
[8]Hermann B., Xie H.H. and Zhang R., Analysis of collocation solutions for a class of functional equations with proportional delays. IMA J. Numer. Anal., online.
[9]He J.H., Wu X.H., Variational iteration, method:new development and appli-cations. Comput.Math.Appl.,54(7-8):881-894,2007.
[10]He J.H., Wazwaz A.M. and Xu L., The variational iteration:reliable, efficient, and promising. Comput.Math.Appl.,54(7-8):879-880,2007.
[11]Chen X.M. and Wang L.J., The variational iteration method for solving a neu-tral functional-differential equation with proportional delays. Comput. Math. Appl.59 (2010), no.8,2696-2702.
[12]Yu Z.H., Variational iteration method for solving the multi-pantograph delay equation. Phys.Lett.A,372(43):6475-6479,2008.
[13]Wang S.Q and He J.H., Variational iteration method for solving integro-differential equations. Phys. Lett. A 367 (2007), no.3,188-191.
[14]He J.H., Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons Fractals,19(4):847-851,2004.
[15]黄得建,变分迭代算法及其应用.长沙理工大学,2009.
[16]王吉安.赵新主,利用变分迭代技术解时滞微分方程.数学的实践与认识,40卷14期,2010.
[2]Marzban H.R. and Tabrizidooz H.R., Solution of nonlinear delay optimal con-trol problems using a composite pseudospectral collocation method. Commun. Pure Appl. Anal.9 (2010), no.5,1379-1389.
[3]Rashidinia J., Ghasemi M. and Jalilian R., A collocation method for the solu-tion of nonlinear one-dimensional parabolic equations. Math. Sci. Quart. J.4 (201.0), no.1,87-104.
[4]Bellen A., Brunner H., Maset S. and Torelli L., Superconvergence in colloca-tion methods on. quasi-graded meshes for functional differential equations with vanishing delays. BIT 46 (2006), no.2; 229-247.
[5]Hermann B., The discretization of Volterra functional integral equations with proportional delays. (English summary)Differences and differential equations, pp 3-27, Fields Inst. Commun.,4.2, Amer. Math. Soc., Providence, RI,2004.
[6]Mustafa Nizami and Khalilov Elnur H. The collocation method for the solution of boundary integral equations. Appl. Anal.88 (2009), no.12,1665-1675.
[7]Auzinger Winfried, Koch Othmar and Weinmuller Ewa, Analysis of a new error estimate for collocation methods applied to singular boundary value problems. SIAM J. Numer. Anal.42 (2005), no.6,2366-2386 (electronic).
[8]Hermann B., Xie H.H. and Zhang R., Analysis of collocation solutions for a class of functional equations with proportional delays. IMA J. Numer. Anal., online.
[9]He J.H., Wu X.H., Variational iteration, method:new development and appli-cations. Comput.Math.Appl.,54(7-8):881-894,2007.
[10]He J.H., Wazwaz A.M. and Xu L., The variational iteration:reliable, efficient, and promising. Comput.Math.Appl.,54(7-8):879-880,2007.
[11]Chen X.M. and Wang L.J., The variational iteration method for solving a neu-tral functional-differential equation with proportional delays. Comput. Math. Appl.59 (2010), no.8,2696-2702.
[12]Yu Z.H., Variational iteration method for solving the multi-pantograph delay equation. Phys.Lett.A,372(43):6475-6479,2008.
[13]Wang S.Q and He J.H., Variational iteration method for solving integro-differential equations. Phys. Lett. A 367 (2007), no.3,188-191.
[14]He J.H., Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons Fractals,19(4):847-851,2004.
[15]黄得建,变分迭代算法及其应用.长沙理工大学,2009.
[16]王吉安.赵新主,利用变分迭代技术解时滞微分方程.数学的实践与认识,40卷14期,2010.