Vera V. Dmitrieva
Point transformations of the 3-rd order ordinary differential equations are considered. Special classes of ordinary differential equations that are invariant under the general point transformations are constructed.
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The Riccati equation method is used to establish Kamenev type conditions for the existence of oscillatory solutions to third order linear ordinary differential equations. Three oscillatory theorems are proved, which generalize the Lazer's oscillation criterion.
We provide linearizability criteria for a class of systems of third-order ordinary differential equations (ODEs) that is cubically semi-linear in the first derivative, by differentiating a system of second-order quadratically semi-linear ODEs and using the original system to replace the second derivative. The procedure developed splits into two cases, those where the coefficients are constant and those where they are variables. Both cases are discussed and examples given.
We establish in this paper the equivalence between a Volterra integral equation of second kind and a singular ordinary differential equation of third order with two initial conditions and an integral boundary condition, with a real parameter. This equivalence allow us to obtain the solution to some problems for nonclassical heat equation, the continuous dependence of the solution with respect to the parameter and the corresponding explicit solution to the considered problem.
We study symmetry and holomorphy of the third-order ordinary differential equation satisfied by the third Painlevé Hamiltonian.
Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order or systems of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define new type of…
Bagderina \cite{Bagderina2008} solved the equivalence problem for scalar third-order ordinary differential equations (ODEs), quadratic in the second-order derivative, via point transformations. However, the question is open for the general class $y'''=f(x,y,y',y'')$ which is not quadratic in the second-order derivative. We utilize Lie's infinitesimal method to study the differential invariants of this general class under pseudo-group of fiber preserving equivalence transformations $\bar{x}=ϕ(x), \bar{y}=ψ(x,y)$. As a result, all third-order differential invariants of this group and the invaria…
We calculate in detail the conditions which allow the most general third order ordinary differential equation to be linearised in X'''(T)=0 under the transformation X(T)=F(x,t), dT=G(x,t)dt. Further generalisations are considered.
Admissible point transformations of classes of $r$th order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational form, the Laguerre-Forsyth form, the first and second Arnold forms) are exhaustively described. Using these results, the group classification of such equations is revisited within the algebraic approach in three different ways.
Meleshko presented a new method for reducing third order autonomous ordinary differential equations (ODEs) to Lie linearizable second order ODEs. We extended his work by reducing fourth order autonomous ODEs to second and third order linearizable ODEs and then applying the Ibragimov and Meleshko linearization test for the obtained ODEs. The application of the algorithm to several ODEs is also presented.
Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations and third order quintically semi-linear ordinary differential equations, we extend to the fourth order by differentiating the third order equation. This yields criteria for linearizability of a class of fourth order semi-linear ordinary differential equations, which have not been discussed in the literature previously. It is shown that the procedure can be extended to higher order. Though the results for the higher orders are complicated, they are doable by algebraic computing.…
Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations, we extend to the third order by differentiating the second order equation. This yields criteria for linearizability of a class of third order semi-linear ordinary differential equations, which is distinct from the classes available in the literature. Some examples are given and discussed.
The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler equation and to the Van der Pol type equation.
The linearization problem by use of the Cartan equivalence method for scalar third-order ODEs via point transformations was solved partially in [1,2]. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third-order ordinary differential equation u'''=f(x,u,u',u'') which admits a four-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained invariant is…
In this paper, we show that the Dickson polynomials of the third kind satisfy a nonhomogeneous second order linear ordinary differential equation whose general solution contains Legendre functions.
We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-for…
A six-stage third order additive method for stiff ordinary differential equations
本文构造了一种用于求解加性分裂自治刚性常微分方程组的六阶段三阶加性方法，该方法对隐式部分L稳定且允许使用任意雅可比矩阵近似，并通过局部误差和稳定性控制实现自动步长选择。
Five equivalence classes had been found for systems of two second-order ordinary differential equations, transformable to linear equations (linearizable systems) by a change of variables. An "optimal (or simplest) canonical form" of linear systems had been established to obtain the symmetry structure, namely with 5, 6, 7, 8 and 15 dimensional Lie algebras. For those systems that arise from a scalar complex second-order ordinary differential equation, treated as a pair of real ordinary differential equations, a "reduced optimal canonical form" is obtained. This form yields three of the five equ…
A non-homogeneous method of third order for additive stiff systems of ordinary differential equations
针对加性分裂自治刚性常微分方程组，构造了一种三阶L-稳定方法，其第四阶段显式且允许任意雅可比矩阵近似，通过局部误差和稳定性控制实现自动步长选择，数值实验验证了算法的可靠性和效率。
We present here the solution of the problem on linearization of fourth-order equations by means of point transformations. We show that all fourth-order equations that are linearizable by point transformations are contained in the class of equations which is linear in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation.
In this paper we use the Riccati equation method with other ones to establish global solvability, stability and oscillation criteria for a class of two dimensional nonlinear systems of ordinary differential equations, which is a generalization of wide classes of second order nonlinear ordinary differential equations, studied by many authors. The applicability of some of these criteria are illustrated by examples.
Using the only admissible rank-two realisations of the Lie algebra of the affine group in one dimension in terms of the Lie algebra of Lie symmetries of the Ermakov-Pinney (EP) equation, some classes of second order nonlinear ordinary differential equations solvable by reduction method are constructed. One class includes the standard EP equation as a special case. A new EP equation with a perturbed potential but admitting the same solution formula as EP itself arises. The solution of the dissipative EP equation is also discussed.
We study second order and third order linear differential equations with analytic coefficients under the viewpoint of finding formal solutions and studying their convergence. We address some untouched aspects of Frobenius methods for second order as the convergence of formal solutions and the existence of Liouvillian solutions. A characterization of regular singularities is given in terms of the space of solutions. An analytic classification of such linear homogeneous ODEs is obtained. This is done by associating to such an ODE a Riccati differential equation and therefore a global holonomy gr…
It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation $\mathbf{y}^{(n)}$=0. For arbitrary systems of ordinary differential equations of order $n \geq 3$ reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras…
Invariant characterization of scalar third-order ODEs that admit the maximal contact symmetry Lie algebra
利用Cartan等价法对具有最大接触对称李代数的标量三阶常微分方程进行不变表征，提供辅助函数以确定接触变换，将其约化为最简线性方程，且给出大量示例说明该方法。
The problem of Painleve classification of ordinary differential equations lasting since the end of XIX century saw significant advances for the limited equation order, however not that much for the equations of higher orders. In this work we propose the complete Painleve classification for ordinary differential equations of the arbitrary order with right-hand side being a quadratic form on the dependent variable and all of its derivatives. The total of seven classes of the equations with Painleve property have been found. Five of them having the order up to four are already known. Sixth one of…
Considering a differential operator of third order that does not increase the degree of polynomials, we analyse some properties of elements of the dual space of 2-orthogonal polynomial eigenfunctions. In two classes of such generic operator, we prove that a 2-orthogonal polynomial solution fulfils Hahn's property.
The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'"=f(x,u,u',u")$ which admits the maximal seven-dimensional point symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation $\bar{u}'"=0$. Moreover, examples are given to illustrate the method.
We characterize stability under composition, inversion, and solution of ordinary differential equations for ultradifferentiable classes, and prove that all these stability properties are equivalent.
Admissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
该研究针对含任意因变量的正规二阶常微分方程线性系统，在多规范下分析容许变换，划分奇异与正则子类，刻画等价群胚，确立李对称性质与维数界，给出降阶积分方法，并完成两因变量情形的完全群分类。
The Cartan equivalence method is applied to provide an invariant characterization of the third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits a five-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function $f$ in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained constant invariant is also presented. We also show how one obtains the point transformation that does the reduction to linear form. Moreover, some applications are provided.
The outcomes of this paper are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class PTIME of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous elegant and simple characterization of PTIME. We believe it is the first time complexity classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the re…
We survey recent work of Burghelea, Malta and both authors on the topology of critical sets of nonlinear ordinary differential operators. For a generic nonlinearity $f$, the critical set of the first order nonlinear operator $F_1(u)(t) = u'(t) + f(u(t))$ acting on the Sobolev space $H^1_p$ of periodic functions is either empty or ambient diffeomorphic to a hyperplane. For the second order operator $F_2(u)(t) = -u''(t) + f(u(t))$ on $H^2_D$ (Dirichlet boundary conditions), the critical set is ambient diffeomorphic to a union of isolated parallel hyperplanes. For second order operators on $H^2_p…
We investigate the connection between the linear harmonic oscillator equation and some classes of second order nonlinear ordinary differential equations of Liénard and generalized Liénard type, which physically describe important oscillator systems. By using a method inspired by quantum mechanics, and which consist on the deformation of the phase space coordinates of the harmonic oscillator, we generalize the equation of motion of the classical linear harmonic oscillator to several classes of strongly non-linear differential equations. The first integrals, and a number of exact solutions of th…
We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain non\-auto\-nomous ordinary differential equations have such a property. The first criterion applies to a semilinear differential equation provided that its linear part is hyperbolic and the nonlinearity is small in a neighborhood of the prescribed set. The second criterion requires that the logarithmic norm of the derivative of the right-hand side with respect to …
Second order ordinary differential equations that possesses the constant invariant are investigated. Four basic types of these equations were found. For every type the complete list of nonequivalent equations is issued. As the exampes the equivalence problem for the Painleve II equation, Painleve III equation with three zero parameters, Emden equations and for some other equations is solved.
The purpose of this paper is to make a few connections among specific concepts occurring in differential geometry and the theory of differential equations with the aim of identifying an intriguing class of undetermined nonlinear ordinary differential equations whose solutions satisfy a specific side condition consisting in a homogeneous third-order linear ordinary differential equation. A method for solving this class of Wronskian-involving differential equations based on the proposed side condition is presented. The Tzitzeica curve equation arising in the theory of space curves is considered …
This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable $x$ and one dependent variable $y$. As opposed to ODEs the variable $x$ figures in more than one point (we consider the case of two points, $x$ and $x_-$). The dependent variable $y$ and its derivatives figure in both $x$ and $x_-$. Two previous articles were devoted to {\it first}-order DODSs, here we concentrate on a large class of {\it second}-order ones.…
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