配置法求解非标准Volterra积分方程
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摘要
本文研究了高阶的配置法求解非标准Volterra积分方程(nonstandard Volterra inte-gral equation简称NVIE)(?)其中u(t)为未知函数, g,K为已知函数.
现实生活中很多物理和生物现象都可以通过方程(0.1)来描述.例如(?)这个方程是人口增长模型,见文献[5].由于这类方程有着广泛的应用,近年来,方程(0.1)得到了大量的关注和研究,见文献[4, 14, 19, 17, 18].从数值分析角度来看,方程(0.1)只有一些特例得到了解决,如Volterra-Hammerstein积分方程(见文献[12],[13])(?)和非线性Volterra积分方程(?)方程(0.4)解的存在性理论和细节的分析参见文献[16, 8, 10].配置法求解方程(0.4)见文献[2, 1, 7, 3].配置法的超收敛性见文献[6, 9].Ma & Brunner研究了一类非标准Volterra微分积分方程,见文献[15](?)
受Ma & Brunner的启发,并且目前还没有关于这类NVIE的数值研究,我们决定进行以下研究:首先,得到方程(0.1)解的存在性,唯一性和正则性.然后,用配置法求解这类方程,分析分片多项式配置解的收敛性.最后,我们研究配置法的超收敛性.
我们的研究主要面临三个难点:
第一,不能直接应用Picard迭代法得到方程(0.1)的解的存在性理论.
第二,右端项包含u(t)对于配置法求解积分方程来说是一个新的挑战.
第三,不能确定配置法的超收敛性依然存在.我们通过在子区间上应用Banach不动点理论得到解的存在性,通过添一项减一项的方法把u(t)移到左端克服第二个难点,最后通过迭代解得到在配置点处的超收敛性.
本文中,我们首先给出了Condition A的定义,然后在此条件下讨论了非标准Volterra积分方程的解的存在唯一性,解的正则性,以及配置法求解非标准Volterra积分方程的收敛性和超收敛性. Condition A的定义如下定义.我们称K满足Condition A,如果(?)其中D := {(t, s) : 0≤s≤t≤T}, 0 L(t, s) M0且(?)
在满足Condition A的条件下,我们得到了非标准Volterra积分方程的解的存在唯一性,如下定理定理. (存在唯一性)如果(i) g∈C(I), K∈C(D×R×R),(ii)函数K满足Condition A.则方程NVIE (1.1)的解u∈C(I)且存在唯一.
在得到非标准Volterra积分方程的解的存在唯一性后,我们证明了在给定条件下解具有正则性,如下定理定理. (解的正则性)假设(i) g∈Cm(I)且K∈Cm(D×R×R)这里m 1;(ii)函数K满足Condition A.则方程(1.1)的解u是Cm光滑的.然后,用配置法求解这类方程,分析配置解的收敛性,得到如下定理定理. (配置解的收敛性)假设(i)函数g∈Cm(I)并且对于整数m 1有K∈Cm(D×R×R) ;(ii)函数K满足Condition A;(?)条件下,由方程组(3.4)解出的方程(1.1)的配置解.则有(?)这里C只和{ci}有关,和h无关.
最后,我们研究了配置法的超收敛性,得到如下定理定理. (配置点Xh上的局部超收敛)假设(i)当整数1 m 2时,有g∈Cm+1(I)和K∈Cm+2(D×R×R);(ii)函数K满足Condition A;(?)是方程(1.1)的配置解,并且参数{ci}满足如下正交性条件(?)则在均匀剖分条件下配置解在配置点Xh上有如下超收敛性(?)
这里C只依赖于{ci}和u(m+1)∞不依赖于h.
This paper will be concerned with high-order collocation method for the nonstandardVolterra integral equation (NVIE)(?)where u(t) is the unknown function and g, K are given functions.Many physical and biological phenomena can be modeled via the integrals (0.1) on timescales. For instance,(?)this equation is a population growth model in [5]. As a response to the need of variousapplications, recently such equations have received a considerable amount of attention onthe qualitative properties of solutions of equation (0.1)(see [4],[14],[19], [17],[18]). Forthe numerical aspects, just some special cases of (0.1) was considered, such as Volterra-Hammerstein integral equation(see [12],[13])(?)and the nonlinear second-kind Volterra integral equation as follows(?)
The existence theorem and a more detailed analysis can be found in [16] or [10]([8] forlinear case). The collocation method applied to equation (0.4) can be found in [2],[1],[7],[3].Super-convergence of the collocation method was analyzed in [6](linear case [9]).
The equation called nonstandard Volterra integro-di?erential equation below was stud-ied by Ma and Brunner (see [15]):(?)
Motivated by the work in [15], and the fact that there is no work on the numericalsolution of this NVIE. The aims of this paper are as follows. Firstly we establish the exis-tence, uniqueness and regularity of the solution for (0.1). Secondly we apply the collocationmethod to approach this nonstandard Volterra integral equation and analyze the optimal orderof convergence of piecewise polynomial collocation approximations to its solution. Finally,we study super-convergence for the method.
There are three main challenges for this NVIE:
We can not directly apply the Picard iteration method to get the existence theorem asin [8].
The right-hand side that includes u(t) is a new challenge for the collocation methodapplied in integral equations.
If the super-convergence still exists for this special nonlinear case.We overcome these di?culties by using the Banach fixed point theorem in subintervals,adding terms to move u(t) to the left-hand side, and using the iterated solution to prove thesuper-convergence at the collocation points which is di?erent from [6], respectively.
In this paper, we firstly gave the definition of Condition A, then we got the existenceand uniqueness of the solution of nonstandard Volterra integral equation. And we got theregularity properties of the solutions under certain conditions. At last, we applied collocationmethod to solve nonstandard Volterra integral equations, and obtained the convergence andsuperconvergence of collocation method.
Firstly, we gave the Condition A as followsDefinition. The function K is said to satisfy Condition A, if(?) where D := {(t, s) : 0≤s≤t≤T}, 0 L(t, s) M0 and(?)
Under Condition A, we obtained the theorem of existence and uniquenessTheorem. (Existence and Uniqueness) Assume(i) g∈C(I), K∈C(D×R×R),(ii) the function K satisfies Condition A.
Then there exists a unique function u∈C(I) of the NVIE (0.1) on I.Under certain conditions we got the theorem of regularity as followsTheorem. (Regularity)Assume(i) g∈Cm(I) and K∈Cm(D×R×R) for some integer m 1;(ii) the function K satisfies Condition A.Then the solution u of (0.1) is Cm smooth.Then we applied collocation method to solve nonstandard Volterra integral equation andobtained the theorem of convergenceTheorem. (Convergence) Assume that(i) the functions g∈Cm(I) and K∈Cm(D×R×R) for some integer m 1;(ii) the function K satisfies Condition A;(iii) uh∈S m(??11)(Ih) is the collocation solution to (0.1) defined by (3.4) with h∈(0, hˉ).Then(?)where C depends on {ci} but not on h.
At last, we got the theorem of local superconvergenceTheorem. (Local super-convergence on Xh) Assume that(i) the functions satisfy g∈Cm+1(I) and K∈Cm+2(D×R×R) for some integer 1 m 2;(ii) function K satisfies Condition A; (iii) uh∈S m(??11)(Ih)(h∈(0, hˉ)) is the collocation solution for (0.1), with collocationparameters {ci} satisfying the orthogonality condition(?)
Then with uniform mesh the collocation solution is superconvergent on Xh , withmax(?)where C depends on the {ci} and on u(m+1)∞but not on h.
现实生活中很多物理和生物现象都可以通过方程(0.1)来描述.例如(?)这个方程是人口增长模型,见文献[5].由于这类方程有着广泛的应用,近年来,方程(0.1)得到了大量的关注和研究,见文献[4, 14, 19, 17, 18].从数值分析角度来看,方程(0.1)只有一些特例得到了解决,如Volterra-Hammerstein积分方程(见文献[12],[13])(?)和非线性Volterra积分方程(?)方程(0.4)解的存在性理论和细节的分析参见文献[16, 8, 10].配置法求解方程(0.4)见文献[2, 1, 7, 3].配置法的超收敛性见文献[6, 9].Ma & Brunner研究了一类非标准Volterra微分积分方程,见文献[15](?)
受Ma & Brunner的启发,并且目前还没有关于这类NVIE的数值研究,我们决定进行以下研究:首先,得到方程(0.1)解的存在性,唯一性和正则性.然后,用配置法求解这类方程,分析分片多项式配置解的收敛性.最后,我们研究配置法的超收敛性.
我们的研究主要面临三个难点:
第一,不能直接应用Picard迭代法得到方程(0.1)的解的存在性理论.
第二,右端项包含u(t)对于配置法求解积分方程来说是一个新的挑战.
第三,不能确定配置法的超收敛性依然存在.我们通过在子区间上应用Banach不动点理论得到解的存在性,通过添一项减一项的方法把u(t)移到左端克服第二个难点,最后通过迭代解得到在配置点处的超收敛性.
本文中,我们首先给出了Condition A的定义,然后在此条件下讨论了非标准Volterra积分方程的解的存在唯一性,解的正则性,以及配置法求解非标准Volterra积分方程的收敛性和超收敛性. Condition A的定义如下定义.我们称K满足Condition A,如果(?)其中D := {(t, s) : 0≤s≤t≤T}, 0 L(t, s) M0且(?)
在满足Condition A的条件下,我们得到了非标准Volterra积分方程的解的存在唯一性,如下定理定理. (存在唯一性)如果(i) g∈C(I), K∈C(D×R×R),(ii)函数K满足Condition A.则方程NVIE (1.1)的解u∈C(I)且存在唯一.
在得到非标准Volterra积分方程的解的存在唯一性后,我们证明了在给定条件下解具有正则性,如下定理定理. (解的正则性)假设(i) g∈Cm(I)且K∈Cm(D×R×R)这里m 1;(ii)函数K满足Condition A.则方程(1.1)的解u是Cm光滑的.然后,用配置法求解这类方程,分析配置解的收敛性,得到如下定理定理. (配置解的收敛性)假设(i)函数g∈Cm(I)并且对于整数m 1有K∈Cm(D×R×R) ;(ii)函数K满足Condition A;(?)条件下,由方程组(3.4)解出的方程(1.1)的配置解.则有(?)这里C只和{ci}有关,和h无关.
最后,我们研究了配置法的超收敛性,得到如下定理定理. (配置点Xh上的局部超收敛)假设(i)当整数1 m 2时,有g∈Cm+1(I)和K∈Cm+2(D×R×R);(ii)函数K满足Condition A;(?)是方程(1.1)的配置解,并且参数{ci}满足如下正交性条件(?)则在均匀剖分条件下配置解在配置点Xh上有如下超收敛性(?)
这里C只依赖于{ci}和u(m+1)∞不依赖于h.
This paper will be concerned with high-order collocation method for the nonstandardVolterra integral equation (NVIE)(?)where u(t) is the unknown function and g, K are given functions.Many physical and biological phenomena can be modeled via the integrals (0.1) on timescales. For instance,(?)this equation is a population growth model in [5]. As a response to the need of variousapplications, recently such equations have received a considerable amount of attention onthe qualitative properties of solutions of equation (0.1)(see [4],[14],[19], [17],[18]). Forthe numerical aspects, just some special cases of (0.1) was considered, such as Volterra-Hammerstein integral equation(see [12],[13])(?)and the nonlinear second-kind Volterra integral equation as follows(?)
The existence theorem and a more detailed analysis can be found in [16] or [10]([8] forlinear case). The collocation method applied to equation (0.4) can be found in [2],[1],[7],[3].Super-convergence of the collocation method was analyzed in [6](linear case [9]).
The equation called nonstandard Volterra integro-di?erential equation below was stud-ied by Ma and Brunner (see [15]):(?)
Motivated by the work in [15], and the fact that there is no work on the numericalsolution of this NVIE. The aims of this paper are as follows. Firstly we establish the exis-tence, uniqueness and regularity of the solution for (0.1). Secondly we apply the collocationmethod to approach this nonstandard Volterra integral equation and analyze the optimal orderof convergence of piecewise polynomial collocation approximations to its solution. Finally,we study super-convergence for the method.
There are three main challenges for this NVIE:
We can not directly apply the Picard iteration method to get the existence theorem asin [8].
The right-hand side that includes u(t) is a new challenge for the collocation methodapplied in integral equations.
If the super-convergence still exists for this special nonlinear case.We overcome these di?culties by using the Banach fixed point theorem in subintervals,adding terms to move u(t) to the left-hand side, and using the iterated solution to prove thesuper-convergence at the collocation points which is di?erent from [6], respectively.
In this paper, we firstly gave the definition of Condition A, then we got the existenceand uniqueness of the solution of nonstandard Volterra integral equation. And we got theregularity properties of the solutions under certain conditions. At last, we applied collocationmethod to solve nonstandard Volterra integral equations, and obtained the convergence andsuperconvergence of collocation method.
Firstly, we gave the Condition A as followsDefinition. The function K is said to satisfy Condition A, if(?) where D := {(t, s) : 0≤s≤t≤T}, 0 L(t, s) M0 and(?)
Under Condition A, we obtained the theorem of existence and uniquenessTheorem. (Existence and Uniqueness) Assume(i) g∈C(I), K∈C(D×R×R),(ii) the function K satisfies Condition A.
Then there exists a unique function u∈C(I) of the NVIE (0.1) on I.Under certain conditions we got the theorem of regularity as followsTheorem. (Regularity)Assume(i) g∈Cm(I) and K∈Cm(D×R×R) for some integer m 1;(ii) the function K satisfies Condition A.Then the solution u of (0.1) is Cm smooth.Then we applied collocation method to solve nonstandard Volterra integral equation andobtained the theorem of convergenceTheorem. (Convergence) Assume that(i) the functions g∈Cm(I) and K∈Cm(D×R×R) for some integer m 1;(ii) the function K satisfies Condition A;(iii) uh∈S m(??11)(Ih) is the collocation solution to (0.1) defined by (3.4) with h∈(0, hˉ).Then(?)where C depends on {ci} but not on h.
At last, we got the theorem of local superconvergenceTheorem. (Local super-convergence on Xh) Assume that(i) the functions satisfy g∈Cm+1(I) and K∈Cm+2(D×R×R) for some integer 1 m 2;(ii) function K satisfies Condition A; (iii) uh∈S m(??11)(Ih)(h∈(0, hˉ)) is the collocation solution for (0.1), with collocationparameters {ci} satisfying the orthogonality condition(?)
Then with uniform mesh the collocation solution is superconvergent on Xh , withmax(?)where C depends on the {ci} and on u(m+1)∞but not on h.
引文
[1] Atkinson, K., Flores, J.(1993) The discrete collocation method for nonlinear integralequations, IMA J. Numer. Anal., 13(2), 195–213.
[2] Atkinson, K.E.(1992) A survey of numerical methods for solving nonlinear integralequations, J. Integral Equations Appl., 4(1),15–46.
[3] Blom, J.G. & Brunner, H.(1987) The numerical solution of nonlinear Volterra integralequations of the second kind by collocation and iterated collocation methods, SIAM J.Sci. Statist. Comput., 8(5),806–830.
[4] Bohner, M. & Peterson, A. editors.(2003) Advances in dynamic equations on timescales. Birkha¨user Boston Inc., Boston, MA, 2003.
[5] Brauer, F. & Castillo-Cha′vez, C.(2001) Mathematical models in population biology andepidemiology, volume 40 of Texts in Applied Mathematics. Springer-Verlag, New York.
[6] Brunner, H.(1980) Superconvergence in collocation and implicit Runge-Kutta methodsfor Volterra-type integral equations of the second kind. In Numerical treatment of inte-gral equations (Workshop, Math. Res. Inst., Oberwolfach, 1979), volume 53 of Internat.Ser. Numer. Math., pages 54–72. Birkha¨user, Basel.
[7] Brunner, H.(1992) On discrete superconvergence properties of spline collocation meth-ods for nonlinear Volterra integral equations, J. Comput. Math., 10(4),348–357.
[8] Brunner, H.(2004) Collocation methods for Volterra integral and related functionaldi?erential equations, volume 15 of Cambridge Monographs on Applied and Computa-tional Mathematics. Cambridge University Press, Cambridge.
[9] Brunner, H. & Yan, N.(1996) On global superconvergence of iterated collocationsolutions to linear second-kind Volterra integral equations, J. Comput. Appl. Math.,67(1),185–189.
[10] Burton, T.A.(1983) Volterra integral and di?erential equations, volume 167 of Math-ematics in Science and Engineering. Academic Press Inc., Orlando, FL, 1983.
[11] Davis, P.J. & Rabinowitz, P.(1984) Methods of numerical integration. Computer Sci-ence and Applied Mathematics. Academic Press Inc., Orlando, FL, second edition.
[12] Elnagar, G.N. & Kazemi, M.(1996) Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations, J. Comput. Appl. Math., 76(1-2),147–158.
[13] Han, G.Q(1993) Asymptotic error expansion of a collocation-type method for Volterra-Hammerstein integral equations, Appl. Numer. Math., 13(5),357–369.
[14] Kulik, T. & Tisdell, C.C.(2008) Volterra integral equations on time scales: basic qual-itative and quantitative results with applications to initial value problems on unboundeddomains, Int. J. Di?erence Equ., 3(1),103–133.
[15] Ma, J. & Brunner, H.(2006) A posteriori error estimates of discontinuous Galerkinmethods for non-standard Volterra integro-di?erential equations, IMA J. Numer. Anal.,26(1),78–95.
[16] Miller, R.K.(1971) Nonlinear Volterra integral equations. W. A. Benjamin, Inc.,Menlo Park, Calif., 1971. Mathematics Lecture Note Series.
[17] Pachpatte, D.B.(2008) Properties of solutions to nonlinear dynamic integral equationson time scales, Electron. J. Di?erential Equations, pages No. 136, 8.
[18] Pachpatte, D.B.(2009) On a nonstandard Volterra type dynamic integral equation ontime scales. Electron. J. Qual. Theory Di?er. Equ., pages No. 72, 14, 2009.
[19] Tisdell, C.C. & Zaidi, A.(2008) Basic qualitative and quantitative results for solutionsto nonlinear, dynamic equations on time scales with an application to economic mod-elling, Nonlinear Anal., 68(11),3504–3524.
[2] Atkinson, K.E.(1992) A survey of numerical methods for solving nonlinear integralequations, J. Integral Equations Appl., 4(1),15–46.
[3] Blom, J.G. & Brunner, H.(1987) The numerical solution of nonlinear Volterra integralequations of the second kind by collocation and iterated collocation methods, SIAM J.Sci. Statist. Comput., 8(5),806–830.
[4] Bohner, M. & Peterson, A. editors.(2003) Advances in dynamic equations on timescales. Birkha¨user Boston Inc., Boston, MA, 2003.
[5] Brauer, F. & Castillo-Cha′vez, C.(2001) Mathematical models in population biology andepidemiology, volume 40 of Texts in Applied Mathematics. Springer-Verlag, New York.
[6] Brunner, H.(1980) Superconvergence in collocation and implicit Runge-Kutta methodsfor Volterra-type integral equations of the second kind. In Numerical treatment of inte-gral equations (Workshop, Math. Res. Inst., Oberwolfach, 1979), volume 53 of Internat.Ser. Numer. Math., pages 54–72. Birkha¨user, Basel.
[7] Brunner, H.(1992) On discrete superconvergence properties of spline collocation meth-ods for nonlinear Volterra integral equations, J. Comput. Math., 10(4),348–357.
[8] Brunner, H.(2004) Collocation methods for Volterra integral and related functionaldi?erential equations, volume 15 of Cambridge Monographs on Applied and Computa-tional Mathematics. Cambridge University Press, Cambridge.
[9] Brunner, H. & Yan, N.(1996) On global superconvergence of iterated collocationsolutions to linear second-kind Volterra integral equations, J. Comput. Appl. Math.,67(1),185–189.
[10] Burton, T.A.(1983) Volterra integral and di?erential equations, volume 167 of Math-ematics in Science and Engineering. Academic Press Inc., Orlando, FL, 1983.
[11] Davis, P.J. & Rabinowitz, P.(1984) Methods of numerical integration. Computer Sci-ence and Applied Mathematics. Academic Press Inc., Orlando, FL, second edition.
[12] Elnagar, G.N. & Kazemi, M.(1996) Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations, J. Comput. Appl. Math., 76(1-2),147–158.
[13] Han, G.Q(1993) Asymptotic error expansion of a collocation-type method for Volterra-Hammerstein integral equations, Appl. Numer. Math., 13(5),357–369.
[14] Kulik, T. & Tisdell, C.C.(2008) Volterra integral equations on time scales: basic qual-itative and quantitative results with applications to initial value problems on unboundeddomains, Int. J. Di?erence Equ., 3(1),103–133.
[15] Ma, J. & Brunner, H.(2006) A posteriori error estimates of discontinuous Galerkinmethods for non-standard Volterra integro-di?erential equations, IMA J. Numer. Anal.,26(1),78–95.
[16] Miller, R.K.(1971) Nonlinear Volterra integral equations. W. A. Benjamin, Inc.,Menlo Park, Calif., 1971. Mathematics Lecture Note Series.
[17] Pachpatte, D.B.(2008) Properties of solutions to nonlinear dynamic integral equationson time scales, Electron. J. Di?erential Equations, pages No. 136, 8.
[18] Pachpatte, D.B.(2009) On a nonstandard Volterra type dynamic integral equation ontime scales. Electron. J. Qual. Theory Di?er. Equ., pages No. 72, 14, 2009.
[19] Tisdell, C.C. & Zaidi, A.(2008) Basic qualitative and quantitative results for solutionsto nonlinear, dynamic equations on time scales with an application to economic mod-elling, Nonlinear Anal., 68(11),3504–3524.