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1.
ON THE EQUIVALENCE OF THE ABEL EQUATION 总被引:2,自引:0,他引:2
This article uses the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Abel equation. The results are applied to discuss the behavior of solutions of these complicated differential equations. 相似文献
2.
Matrix polynomials satisfying first order differential equations and three term recurrence relations
Mirta M. Castro 《Journal of Computational and Applied Mathematics》2010,233(6):1491-1498
We describe families of matrix valued polynomials satisfying simultaneously a first order differential equation and a three term recurrence relation. Our goal is to address the classification of the matrix valued polynomials satisfying first order differential equations through the solutions of the so-called bispectral problem. At the heart of this lies the need to solve some complicated nonlinear equations with matrix coefficients called ad-conditions. The solutions of these equations are studied under a variety of sufficient conditions on its coefficients. 相似文献
3.
Summary In this paper Lie series are presented in Chebyshev form and applied to the iterative solution of initial value problems in differential equations. The resulting method, though algebraically complicated, is of theoretical interest as a generalisation of Taylor series methods and iterative Chebyshev methods. The theory of the method is discussed and the solutions of some simple scalar equations are analysed to illustrate the behaviour of the process. 相似文献
4.
Norbert Gorenflo 《Integral Equations and Operator Theory》1999,35(3):366-377
In some earlier publications it has been shown that the solutions of the boundary integral equations for some mixed boundary value problems for the Helmholtz equation permit integral representations in terms of solutions of associated complicated singular algebraic ordinary differential equations. The solutions of these differential equations, however, are required to be known on some infinite interval on the real line, which is unsatisfactory from a practical point of view. In this paper, for the example of one specific boundary integral equation, the relevant solutions of the associated differential equation are expressed by integrals which contain only one unknown generalized function, the support of this generalized function is no longer unbounded but a compact subset of the real line. This generalized function is a distributional solution of the homogeneous boundary integral equation. By this null space distribution the boundary integral equation can be solved for arbitrary right-hand sides, this solution method can be considered of being analogous to the method of variation of parameters in the theory of ordinary differential equations. The nature of the singularities of the null space distribution is worked out and it is shown that the null space distribution itself can be expressed by solutions of the associated ordinary differential equation. 相似文献
5.
Jin Liang Gaston M. N’Guérékata Ti-Jun Xiao Jun Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2009
This paper is concerned with some properties of pseudo-almost automorphic functions, which are more general and complicated than pseudo-almost periodic functions. Using these properties, we establish an existence and uniqueness theorem for pseudo-almost automorphic mild solutions to semilinear differential equations in a Banach space. 相似文献
6.
7.
Generic operator equations allowing multidimensional operators are solved by a decomposition method allowing solution of nonlinear and/or stochastic partial differential equations by accurate and convenient approximation. Green's functions for complicated ordinary or partial differential linear equations are similarly determinable. 相似文献
8.
9.
Models of tumor growth and treatment based on one or two ordinary differential equations are heavily used in practice because they are simple but can often still capture the essence of complicated interactions. Currently relevant examples of such models are given here: some classic growth equations, an ODE pair for the interplay between tumor and neovascularization during cancer growth or therapy, and an ODE pair for response to ionizing radiation. Mathematically more sophisticated generalizations of various kinds, usually more realistic but less practical, are mentioned very briefly. 相似文献
10.
Zhou Zhengxin 《Journal of Applied Mathematics and Computing》2004,16(1-2):265-277
This article deals with the number of periodic solutions of the second order polynomial differential equation using the Riccati equation, and applies the property of the solutions of the Riccati equation to study the property of the solutions of the more complicated differential equations. Many valuable criterions are obtained to determine the number of the periodic solutions of these complex differential equations. 相似文献
11.
T. Global in time solutions of the equations Ou = H(u, u') on Minkowski space-time are considered. Results available so far involve complicated decay and energy estimates and also careful choice of Banach spaces and associated ordinary differential inequalities. This work tries to simplify some of the existing arguments and to develop a new technique for other nonlinear evolution equations. The method is motivated by the work of Christodoulou and Baez, Segal, and Zhou, on nonlinear wave equations. The key idea is to use the Penrose conformal compactification that transforms the equations from Minkowski space to the Einstein universe in order to change the global existence question to the local one. 相似文献
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13.
D Schley R Shail S. A Gourley 《International Journal of Mathematical Education in Science & Technology》2013,44(3):359-375
Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay differential equations arising in population dynamics may serve as good teaching examples for these methods. The determination of linear stability properties for an ordinary differential equation with a varying time delay is carried out through discrete point analysis, either by seeking explicit solutions or leading to the consideration of a difference equation and the roots of a characteristic polynomial. Numerical simulations carried out using MATLAB Simulink are compared to the analytical solutions, and computation is also used to suggest extensions to some results. 相似文献
14.
Master equations of different types describe the evolution (reduced dynamics) of a subsystem of a larger system generated
by the dynamic of the latter system. Since, in some cases, the (exact) master equations are relatively complicated, there
exist numerous approximations for such equations, which are also called master equations.
In the paper, we develop an exact master equation describing the reduced dynamics of the Wigner function for quantum systems
obtained by a quantization of a Hamiltonian system with a quadratic Hamilton function. First, we consider an exact master
equation for first integrals of ordinary differential equations in infinite-dimensional locally convex spaces. After this,
we apply the results obtained to develop an exact master equation corresponding to a Liouville-type equation (which is the
equation for first integrals of the (system of) Hamilton equation(s)); the latter master equation is called the master Liouville
equation; it is a linear first-order differential equation with respect to a function of real variables taking values in a
space of functions on the phase space. If the Hamilton equation generating the Liouville equation is linear, then the vector
fields that define the first-order linear differential operators in the master Liouville equations are also linear, which
in turn implies that for a Gaussian reference state the Fourier transform of a solution of the master Liouville equation also
satisfies a linear differential equation.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 203–219, 2005. 相似文献
15.
The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This article employs an elementary method to derive the Milstein scheme and its first order strong rate of convergence for stochastic delay differential equations. 相似文献
16.
A. V. Kitaev 《Journal of Mathematical Sciences》1995,73(4):446-459
On the basis of the connection between the theories of linear and nonlinear special functions, we present a method which makes
it possible to consider the well known formal limits from complicated Painlevé equations to simpler ones as the double asymptotics
of specific solutions of these equations with respect to the parameter and the argument under some special relation between
them. The hierarchies of the first and second Painlevé equations are interpreted as special functions that describe the isomonodromic
collision of turning points for linear systems of ordinary differential equations. Bibliography: 28 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 187, pp. 53–74, 1990.
Translated by B. M. Bekker. 相似文献
17.
《Communications in Nonlinear Science & Numerical Simulation》2011,16(6):2421-2437
Using the solutions of an auxiliary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some Wick-type nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained. In addition, the links between Wick-type partial differential equations and variable coefficient partial differential equations are also clarified generally. 相似文献
18.
In this paper, a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled Riccati differential equations arising in connection with control of some linear stochastic systems is considered.The system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite-dimensional Hilbert space adequately choosen.This allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of Lyapunov type.Our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions.Conditions providing the existence of a maximal solution (minimal solution respectively) with respect to some classes of global solutions are presented. It is shown that if the coefficients of the equations are periodic functions all these special solutions (stabilizing, maximal, minimal) are periodic functions, too.Whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves. 相似文献
19.
Yuexin Yan Yingxin Pan Feifei Lu Zhengxin Zhou 《Journal of Nonlinear Modeling and Analysis》2019,1(2):207-220
In this paper, first of all we give the necessary and sufficient con-
ditions of the center of a class of planar quintic differential systems by using
reflecting function method, and provide a simple proof of this results. Sec-
ondly, We use the reflecting integral to research the equivalence of the Abel
equation and some complicated equations and derive their center conditions
and discuss their integrability. 相似文献
20.
In this article, differential transform method (DTM) has been successfully applied to obtain the approximate analytical solutions of the nonlinear homogeneous and non-homogeneous gas dynamic equations, shock wave equation and shallow water equations with fractional order time derivatives. The true beauty of the article is manifested in its emphatic application of Caputo fractional order time derivative on the classical equations with the achievement of the highly accurate solutions by the known series solutions and even for more complicated nonlinear fractional partial differential equations (PDEs). The method is really capable of reducing the size of the computational work besides being effective and convenient for solving fractional nonlinear equations. Numerical results for different particular cases of the equations are depicted through graphs. 相似文献