Comment on “Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients”

The present paper corrects the way of using Jordan canonical forms for studying the symmetry structures of systems of linear second-order ordinary differential equations with constant coefficients applied in [1]. The approach is demonstrated for a system consisting of two equations.     

The a posteriori error estimates of Chebyshev–Petrov–Galerkin methods for second-order equations

In this paper, the a posteriori error estimates of Chebyshev–Petrov–Galerkin approximations are investigated. For simplicity, we choose the Poisson equation with Dirichlet boundary conditions to discuss the a posteriori error estimators, and deduce their efficient and reliable properties. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error estimators.     

Solvability of a system of equations for the Fourier–Chebyshev coefficients when solving ordinary differential equations by the Chebyshev series method

A solvability theorem for a system of equations with respect to approximate values of Fourier–Chebyshev coefficients is formulated. This theorem is a theoretical justification for numerical solution of ordinary differential equations using Chebyshev series.     

Oscillation criteria for second-order Emden–Fowler delay differential equations with a sublinear neutral term

New oscillation criteria for the second-order Emden–Fowler delay differential equation with a sublinear neutral term are presented. An essential feature of our results is that oscillation of the studied equation is ensured via only one condition. Furthermore, as opposed to the results by Agarwal et al. (Ann. Mat. Pura Appl. (4) 193 (2014), no. 6, 1861–1875), Li and Rogovchenko (Math. Nachr. 288 (2015), no. 10, 1150–1162; Monatsh. Math. 184 (2017), no. 3, 489–500), and Xu (Monatsh. Math. 150 (2007), no. 2, 157–171), new criteria can be applied to Emden–Fowler delay differential equations with noncanonical operators and a sublinear neutral term. Our results essentially improve, extend, and simplify some known ones reported in the literature. The results are illustrated with examples.     

Point equivalence of second-order ordinary differential equations to the fifth Painlevé equation with one and two nonzero parameters

Theoretical and Mathematical Physics - We consider the problem of the equivalence of scalar second-order ordinary differential equations under invertible point transformations. To solve this...     

On the second-order approximate symmetry classification and optimal systems of subalgebras for a forced Korteweg–de Vries equation

We investigate a forced Korteweg–de Vries (fKdV) equation, $u_{\text{,}t}+{\mathit{cu}}_{\text{,}x}+\alpha {\mathit{uu}}_{\text{,}x}+\beta u_{\text{,}\mathit{xxx}}=\beta F(t)$, which arises in the modelling of tsunami generation by submarine landslides. Approximate symmetries are found for the fKdV equation using the method as proposed by Fushchich and Shtelen [6]. Symmetries are found to second order in the perturbation parameter using the MAPLE symmetry package ASP [11], an add-on to the symmetry package DESOLVII [18]. ASP also allows particular forms of the arbitrary function $F(t)$ to be found which extend the symmetry algebra and hence a full approximate symmetry classification of the fKdV equation is obtained. Optimal systems of one-dimensional subalgebras are also determined. Corresponding approximate invariant solutions to the fKdV equation are then constructed for particular $F(t)$ using DESOLVII routines.     

A canonical system of two differential equations with periodic coefficients and the Poincaré-Denjoy theory of differential equations on a torus

The passage from Cartesian to polar coordinates in a canonical system with periodic coefficients gives rise to a nonlinear differential equation whose right-hand side is periodic in time and the polar angle and thus this equation can be regarded as a differential equation on a torus. In accord with Poincaré-Denjoy theory, the behavior of a solution to a differential equation on a torus is characterized by the rotation number and some homeomorphic mapping of a circle onto itself. We study connections of strong stability (instability) of a canonical system, including the membership in the nth stability (instability) domain, with the rotation number and fixed points of this mapping.     

A DIRECT METHOD OF OPERATIONAL CALCULUS(Ⅰ)——Research for the solution of second-order linear ordinary differential equations with variable coefficients

According to Mikusinski's fundamental view, we reconstruct a direct method of operation acalbulus which may be adapted to search for the solution of differential equations with variablecoefficients.The present paper is devoted to solutions of two types——indirect (undetermined) method and direct (series of integral) method-for solving second-order linear differential equations: The latter displays forthright feature while the former can solve second-order linear differential equation and interrelate first-order Riccati's non-linear differential equation at the same time.     

Lie symmetry,nonlocal symmetry analysis,and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation

Theoretical and Mathematical Physics - We use the method of Lie symmetry analysis to investigate the properties of a (2+1)-dimensional KdV–mKdV equation. Using the Ibragimov method, which...     

Approximations of Euler–Maruyama type for stochastic differential equations with Markovian switching,under non-Lipschitz conditions

We develop the Euler–Maruyama scheme for a class of stochastic differential equations with Markovian switching (SDEwMSs) under non-Lipschitz conditions  . Both L¹$L^1$ and L²$L^2$-convergence are discussed under different non-Lipschitz conditions. To overcome the mathematical difficulties arisen from the Markovian switching as well as the non-Lipschitz coefficients, several new analytical techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems.     

Generalization of the lions theory for first-order evolution differential equations with discontinuous operators and with α ∈ [1/2, 1]

We prove existence, uniqueness, and smoothness theorems for weak solutions of the problem $$ du(t)/dt + A(t)u(t) = f(t), t \in ]0,T[; u(0) = u_0 \in H, $$ where, for almost all t, the linear unbounded operators A(t) with domains D(A(t)) depending on t are closed and maximal accretive and have bounded inverses A ^?1(t) discontinuous with respect to t in the Hilbert space H. There exists an α ∈ [1/2, 1] such that the following is true in H for almost all t: the power A ^α (t) is subordinate to the power A* ^α (t) of the adjoint operators A*(t), the operators A ^α (t) and A* ^α (t) do not form an obtuse angle, and the domains D(A* ^α (t)) of the operators A* ^α (t) are not increasing with respect to t. This paper is the first to prove the well-posedness of the mixed problem for the multidimensional linearized Korteweg-de Vries equation smooth in time with boundary conditions piecewise constant in time.     

On a majorant–minorant criterion for the total preservation of global solvability of distributed controlled systems

For distributed controlled systems that can be represented by a functional-operator equation of the Hammerstein type with an additional term on the right-hand side in the form of a linear operator (an extended equation of the Hammerstein type), we prove a criterion for the total (over the entire set of admissible controls) preservation of global solvability. In this connection, we develop an earlier-proved majorant–minorant criterion for the total preservation of global solvability. As an example, we study a controlled semilinear integro-differential equation describing radiation transport with Compton scattering diagram. The choice of this example is explained, in particular, by the fact that the preceding version of the majorant–minorant criterion is not applicable to it.     

Solvability of a periodic boundary-value problem for a system of first-order ordinary differential equations with (β,γ,δ)-comparison pairs

Existence theorems are proved for an incomplete set of upper and lower functions. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 231–241. Translated by L. Yu. Kolotilina.     

An efficient Levenberg–Marquardt method with a new LM parameter for systems of nonlinear equations

A modified Levenberg–Marquardt method for solving singular systems of nonlinear equations was proposed by Fan [J Comput Appl Math. 2003;21;625–636]. Using trust region techniques, the global and quadratic convergence of the method were proved. In this paper, to improve this method, we decide to introduce a new Levenberg–Marquardt parameter while also incorporate a new nonmonotone technique to this method. The global and quadratic convergence of the new method is proved under the local error bound condition. Numerical results show the new algorithm is efficient and promising.     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

Dulac–Cherkas criterion for exact estimation of the number of limit cycles of autonomous systems on a plane

The problem of exact nonlocal estimation of the number of limit cycles surrounding one point of rest in a simply connected domain of the real phase space is considered for autonomous systems of differential equations with continuously differentiable right-hand sides. Three approaches to solving this problem are proposed that are based on sequential two-step usage of the Dulac–Cherkas criterion, which makes it possible to find closed transversal curves dividing the connected domain in doubly connected subdomains that surround the point of rest, with the system having precisely one limit cycle in each of them. The effectiveness of these approaches is exemplified with polynomial Liènard systems, a generalized van der Pol system, and a perturbed Hamiltonian system. For some systems, the derived estimate holds true in the entire phase space.     

Asymptotic behavior of solutions for nonlinear wave equations of Kirchhoff type with a positive–negative damping

The initial–boundary value problem of Kirchhoff type with an intermittent damping is considered. Under some appropriate assumptions, we give some sufficient conditions for the asymptotic stability of the solutions.