共查询到20条相似文献,搜索用时 15 毫秒
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In this survey, results on the existence, growth, uniqueness, and value distribution of meromorphic (or entire) solutions of linear partial differential equations of the second order with polynomial coefficients that are similar or different from that of meromorphic solutions of linear ordinary differential equations have been obtained. We have characterized those entire solutions of a special partial differential equation that relate to Jacobian polynomials. We prove a uniqueness theorem of meromorphic functions of several complex variables sharing three values taking into account multiplicity such that one of the meromorphic functions satisfies a nonlinear partial differential equations of the first order with meromorphic coefficients, which extends the Brosch??s uniqueness theorem related to meromorphic solutions of nonlinear ordinary differential equations of the first order. 相似文献
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In this paper, we study the growth, in terms of the Nevanlinna characteristic function, of meromorphic solutions of three types of second-order nonlinear algebraic ordinary differential equations (ODEs). We give all their meromorphic solutions explicitly, and hence show that all of these ODEs satisfy the classical conjecture proposed by Hayman in 1996. 相似文献
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This paper is concerned with entire and meromorphic solutions of linear partial differential equations of second order with polynomial coefficients. We will characterize entire solutions for a class of partial differential equations associated with the Jacobi differential equations, and give a uniqueness theorem for their meromorphic solutions in the sense of the value distribution theory, which also applies to general linear partial differential equations of second order. The results are complemented by various examples for completeness. 相似文献
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The celebrated Malmquist theorem states that a differential equation, which admits a transcendental meromorphic solution, reduces into a Riccati differential equation. Motivated by the integrability of difference equations, this paper investigates
the delay differential equations of form $w(z+1)-w(z-1)+a(z)\frac{w''(z)}{w(z)}=R(z, w(z))(*),$ where $R(z, w(z))$ is an irreducible rational function in $w(z)$ with rational coefficients and $a(z)$ is a rational function. We characterize all reduced forms when the equation $(*)$ admits a transcendental entire solution with hyper-order less than one. When we compare with the results obtained by Halburd and Korhonen[Proc. Amer. Math. Soc. 145, no.6 (2017)], we obtain the reduced forms without the assumptions that the denominator of rational function $R(z,w(z))$ has roots that are nonzero rational functions in $z$. The value distribution and forms of transcendental entire solutions for the reduced delay differential equations are studied. The existence of finite iterated order entire solutions of the Kac-van Moerbeke delay differential equation is also detected. 相似文献
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Zifeng Huang Liming Zhang Qiuhui Chen Wenjun Yuan 《Mathematical Methods in the Applied Sciences》2014,37(10):1553-1560
In this paper, we employ the Nevanlinna's value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for a class of odd order algebraic differential equations with the weak ?p,q?and dominant conditions. Moreover, we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of the Kuramoto–Sivashinsky equation by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
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1.IntroductionandResultsConsidernon-homogeneouslineardifferentialequationsoftheform1.Lameprovedin[7]TheoremA.LetB(z),PO(z),PI(z)*06epolynomialssuchthatdegB=n21,degPO=p<(n k)/kandH=PI(z)epo('),then(a)IfdegPI相似文献
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《数学物理学报(B辑英文版)》2015,(5)
In this paper, we first employ the complex method to deritive all meromorphic solutions of an auxiliary ordinary differential equation, and then find all meromorphic exact solutions of the modified ZK equation, modified Kd V equation, nonlinear Klein-Gordon equation and modified BBM equation. Our work shows that there exist some classes of rational solutionswr,2(z) and simple periodic solutionsws,1(z) which are new and are not degenerated successively to by the elliptic function solutions. 相似文献
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We investigate whether certain Diophantine equations have or have not solutions in entire or meromorphic functions defined on a non-Archimedean algebraically closed field of characteristic zero. We prove that there are no non-constant meromorphic functions solving the Erdös–Selfridge equation except when the corresponding curve is a conic. We also show that there are infinitely many non-constant entire solutions to the Markoff–Hurwitz equation. 相似文献
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Wenjun Yuan Fanning Meng Jianming Lin Yonghong Wu 《Mathematical Methods in the Applied Sciences》2016,39(8):2083-2092
Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions wr,2(z) and simply periodic solutions ws,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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Wenjun Yuan Yezhou Li Jianming Lin 《Mathematical Methods in the Applied Sciences》2013,36(13):1776-1782
In this paper, we employ the complex method to obtain first all meromorphic solutions of an auxiliary ordinary differential equation and then find all meromorphic exact solutions of the classical Korteweg–de Vries equation, Boussinesq equation, ( 3 + 1)‐dimensional Jimbo–Miwa equation, and Benjamin–Bona–Mahony equation. Our results show that the method is more simple than other methods. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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In this paper,we shall use Nevanlinna theory of meromorphic functions to investigate the complex oscillation theory of solutions of some higher order linear differential equation.Suppose that A is a transcendental entire function with ρ(A)<1/2.Suppose that k≥2 and f(k)+A(z)f=0 has a solution f with λ(f)<ρ(A),and suppose that A1=A+h,where h≡0 is an entire function with ρ(h)<ρ(A).Then g(k)+A1(z)g=0 does not have a solution g with λ(g)<∞. 相似文献
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利用亚纯函数的Nevanlinna值分布理论,研究了一类复高阶微分方程的亚纯允许解的存在性问题.证明了在适当条件的假设下,该类复微分方程的亚纯解不是允许解的结果,推广了以前一些文献的结论,并且文中有例子表明结果是精确的. 相似文献
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By meas of the Nevanlinna theory of the value distribution of meromorphic functions,this paper discusses the orders of growth of meromorphic solutions of differential equation and proves that the form of the solution is determined if the order are sufficiently large. 相似文献
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高凌云 《纯粹数学与应用数学》2005,21(4):305-309
利用亚纯函数的Nevanlinna值分布理论和微分代数知识,研究了一类高阶代数微分方程解的解析式问题,该类高阶微分方程解的解析式被得到. 相似文献
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研究了一类亚纯函数为系数的二阶非齐次线性微分方程的解及其微分多项式和小函数的关系,并得到了这类微分方程解以及解的一阶,二阶导数与微分多项式的不动点性质. 相似文献
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We consider differential polynomials of Fermat–Waring type, const-ructed using polynomials of Yi’s type for meromorphic functions in a non-Archimedean field. Similarly to the Hayman Conjecture, we prove that the considered differential polynomials assume all values. We established also a uniqueness theorem for these differential polynomials. 相似文献
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研究了一类高阶齐次线性微分方程解的零点收敛指数,并得到当方程的系数A_0为整函数,其泰勒展式为缺项级数,并且A_0起控制作用时,方程f~((k))+A_(k-2)f~((k-2))+…+A_1f′+A_0f=0的任意两个线性无关解f_1,f_2满足max{λ(f_1),λ(f_2)}=∞,其中λ(f)表示亚纯函数.f的零点收敛指数. 相似文献