再生核空间中积分和微分方程的求解方法
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摘要
许多自然现象都是借助于线性、非线性方程来描述的,而对这些现象的分析一般可归结为微分方程、积分方程和微分-积分方程的求解问题.因此,如何求解这些有实际意义的方程也就变得越来越重要了.本文运用再生核空间的技巧,给出了几类非线性积分和微分方程的求解方法.
首先,在再生核空间中给出了第一类Fredholm积分方程的最小范数解的表达式.进一步地,如果方程的解不唯一,给出了它的所有解的表达形式.并给出了求解第一类Fredholm积分方程逼近解的稳定性分析.
其次,讨论了非线性Volterra-Fredholm积分方程的求解.利用再生核表达式得到了一组标准正交基,将方程的未知函数在这组基上进行Fourier级数展开,构造了一种收敛的迭代序列,从而得到了方程解的一种级数表达形式.通过截断级数得到方程的逼近解.同时,将此迭代法推广到求解非线性Fredholm积分方程组.
第三,在再生核空间中给出了带有积分边界条件的抛物型微分方程的求解方法.通过构造一组满足积分边界条件的标准正交基,利用再生核的技巧给出了方程解的表达式.
第四,在再生核空间中给出了求解非线性非局部边界条件的二阶偏微分方程的方法.通过构造一组满足非线性非局部边界条件的标准正交基,给出了一个有界的并且收敛的迭代序列,从而得到了方程解的级数表达形式.
Some natural phenomena may be described by linear or nonlinear equations.Analysis on these phenomena would generally come down to solve solutions ofdi?erential equations, integral equations or di?erential-integral equations. Thus,how to solve these equations with actual prospecting becomes more and moreimportant problems. In this paper, some methods of solving nonlinear di?erentialand integral equations are proposed using skills in reproducing kernel space(RKS).
Firstly, a representation of minimal norm solutions on Fredholm integralequations of the first kind is obtained in RKS. Representations of the solutionsare further given if it has solutions. At the same time, a analysis on stability ofapproximate solutions of Fredholm integral equations of the first kind is provided.
Secondly, a method of solving nonlinear Volterra-Fredholm integral equationsis proposed. An orthnormal basis is obtained using representation of reproducingkernel and unknown functions of nonlinear Volterra-Fredholm integral equationsare expressed as Fourier series using the basis. A convergent iterative sequence isconstructed. A series representation of solutions for the equations is obtained andapproximate solutions are given by truncating the series. The iterative methodis further generalized to solve systems of nonlinear Fredholm integral equation.
Thirdly, a method of solving parabolic di?erential equations with integralboundary conditions is proposed in RKS. A representation of the solutions isprovided by constructing an orthnormal basis satisfying integral boundary con-ditions and applying skills of reproducing kernel.
Finally, a method of solving partial di?erential equations with nonlinearnonlocal boundary conditions is proposed. A bounded and convergent iterativesequence is provided by constructing an orthnormal basis satisfying nonlinearnonlocal boundary conditions. Hence, a representation of solutions for the equa-tions is obtained.
首先,在再生核空间中给出了第一类Fredholm积分方程的最小范数解的表达式.进一步地,如果方程的解不唯一,给出了它的所有解的表达形式.并给出了求解第一类Fredholm积分方程逼近解的稳定性分析.
其次,讨论了非线性Volterra-Fredholm积分方程的求解.利用再生核表达式得到了一组标准正交基,将方程的未知函数在这组基上进行Fourier级数展开,构造了一种收敛的迭代序列,从而得到了方程解的一种级数表达形式.通过截断级数得到方程的逼近解.同时,将此迭代法推广到求解非线性Fredholm积分方程组.
第三,在再生核空间中给出了带有积分边界条件的抛物型微分方程的求解方法.通过构造一组满足积分边界条件的标准正交基,利用再生核的技巧给出了方程解的表达式.
第四,在再生核空间中给出了求解非线性非局部边界条件的二阶偏微分方程的方法.通过构造一组满足非线性非局部边界条件的标准正交基,给出了一个有界的并且收敛的迭代序列,从而得到了方程解的级数表达形式.
Some natural phenomena may be described by linear or nonlinear equations.Analysis on these phenomena would generally come down to solve solutions ofdi?erential equations, integral equations or di?erential-integral equations. Thus,how to solve these equations with actual prospecting becomes more and moreimportant problems. In this paper, some methods of solving nonlinear di?erentialand integral equations are proposed using skills in reproducing kernel space(RKS).
Firstly, a representation of minimal norm solutions on Fredholm integralequations of the first kind is obtained in RKS. Representations of the solutionsare further given if it has solutions. At the same time, a analysis on stability ofapproximate solutions of Fredholm integral equations of the first kind is provided.
Secondly, a method of solving nonlinear Volterra-Fredholm integral equationsis proposed. An orthnormal basis is obtained using representation of reproducingkernel and unknown functions of nonlinear Volterra-Fredholm integral equationsare expressed as Fourier series using the basis. A convergent iterative sequence isconstructed. A series representation of solutions for the equations is obtained andapproximate solutions are given by truncating the series. The iterative methodis further generalized to solve systems of nonlinear Fredholm integral equation.
Thirdly, a method of solving parabolic di?erential equations with integralboundary conditions is proposed in RKS. A representation of the solutions isprovided by constructing an orthnormal basis satisfying integral boundary con-ditions and applying skills of reproducing kernel.
Finally, a method of solving partial di?erential equations with nonlinearnonlocal boundary conditions is proposed. A bounded and convergent iterativesequence is provided by constructing an orthnormal basis satisfying nonlinearnonlocal boundary conditions. Hence, a representation of solutions for the equa-tions is obtained.
引文
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3 S. Zaremba. Sur Le Calcul Numerique Des Founctions Demandness Dans LeProblems De Dirichlet Et Le Problems Hydrodynamique. Bulletin Intera-tional de I Academie des Sciences de Cracovie. 1908,196:125-195
4 S. Bergman. U¨ber Die Entwicklung Der Harmonischen Funcktionen DerEbeneunddes Raumes Nach Orthogonal-funkionen. Math. Ann.. 1922,86:238-271
5 S. Bergman. U¨ber Ein Verfahren Zur Konstruktion Der NaherungslO¨-SungslO¨Sungen Der Gleichung u+τ2u = 0 . Prikl. Math. Meh.. 1936,5:97-107
6 S. Bergman. Zur Theorie Der Funktionen Die EineLlinear Partielle Di?fer-entialgleichung Berriedigen. Soviet. Math. Dokl.. 1937,15:227-230
7 S. Bergman. Sur Un Lien Entre Lal Th′eorie Des Equations Auxderives Par-tielles Elliptiques Et Celle Des Functions D’une Variable Complex. C. R.Acad. Sci. Paris. 1937,205:1198-1200 1360-1362
8 S. Bergman. The Approximation of Functions Satisfying A Linear PartialDi?erential Equation. Duke Math. J.. 1940,6:537-561
9 S. Bergman. A Remark on The Mapping of Multiply-Connected Domains.Amer. J. Math.. 1946,68:20-28
10 S. Bergman. Partial Di?erential Equations:Advanced Topics. Brown Univer-sity Providence. 1941:211-214
11 N. Aronszajn. Theory of Reproducing Kernels. Trans. Amer. Math. Soc..1950,68:337-404
12 F. M. Larkin. Optimal Approximation in Hilbert Space with ReproducingKernel Function. Math. Comp.. 1970,22(4):911-921
13 M. M. Chawla. Optimal Rules with Pilynomial Precision for Hilbert SpacesPossessing Reproducing Kernel Functions. Numer. Math.. 1974,22:207-218
14 J. S. Chen, V. Kotta, H. Lu, D. Wang, D. Moldovan , D. Wolf. A VariationalFormulation and A Double-grid Method for Meso-scale Modeling of StressedGrain Growth in Polycrystalline Materials. Comput. Methods Appl. Mech.Engrg.. 2004,193:1277-1303
15 S. Shimorin. Complete Nevanlinna-Pick Property of Dirichlet-Type Spaces.Journal of Functional Analysis. 2002,191:276-296
16 D. Alpay. A Structure Theorem for Reproducing Kernel Pontryagin Spaces.Journal of Computational and Applied Mathematics. 1998,99:413-422
17 D. Alpay, C. Dubi. Carath′eodory Fej′er Interpolation in The Ball with MixedDerivatives. Linear Algebra and its Applications. 2004,382:117-133
18 N. Askour, A. Intissar,Z. Mouayn. Explicit Formulas for Reproducing Kernelsof Generalized Bargmann Spaces. C. R. Acad. Sci. Paris. 1997,325(1):707-712
19 B. Raneea. Geometric Quantization of Real Minimal Nilpotent Orbits. Dif-ferential Geometry and its Applications. 1998,9(1-2):5-58
20 J. K. Misiewicz. Infinite Divisibility of Sub-stable Processes. Part I. Geome-try of Sub-spaces of Lα - Space. Stochastic Processes and their Applications.1995,56:101-116
21 A. Daniela, B. Vladimira, D. Aadb, R. Amesc, S. Cora. Hilbert Spaces Con-tractively Included in The Hardy Space of The Bidisk. Comptes Rendus del’Academie des Sciences Series I Mathematics. 1998,326(12):1365-1370
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