Related papers: Riccati-parameter solutions of nonlinear second-or…

This paper gives out the general solutions of variable coefficients ODE and Riccati equation by way of integral series E(X) and F(X). Such kinds of integral series are the generalized form of exponential function, and keep the properties of…

An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ xi = F(x), eta = P(x) y + Q(x) ], where xi d/dx + eta d/dy is the symmetry generator - is presented. To these {\it linear} symmetries one can…

We study the dynamics of the positive solutions of a second-order, Ricker-type exponential difference equation with periodic parameters. We find that qualitatively different dynamics occur depending on whether the period p of the main…

Some properties of global solution of scalar Riccati equation are studied. On the basis of these properties using the Whiburn's and Leighton - Nehary's theorems some oscillatory and criteria are proved for second order linear systems of…

The Riccati equation method is used to establish some oscillatory criteria for the second order linear functional - differential equations of multiple terms with locally integrable coefficients. An interval oscillation criterion for the…

General solutions of nonlinear ordinary differential equations (ODEs) are in general difficult to find although powerful integrability techniques exist in the literature for this purpose. It has been shown that in some scalar cases…

The Riccati differential equation is examined in light of its connection to second order linear time varying systems. In that light it becomes the clear generalization for the characteristic equation of linear time invariant systems, and is…

We present particular solutions for the following important nonlinear second order differential equations: modified Emden, generalized Lienard, convective Fisher, and generalized Burgers-Huxley. For the latter two equations these solutions…

We present two algorithms for computing what we call the absolute factorization of a difference operator. We also give an algorithm to solve third order difference equations in terms of second order equations, together with applications to…

The logistic function is shown to be solution of the Riccati equation, some second-order nonlinear ordinary differential equations and many third-order nonlinear ordinary differential equations. The list of the differential equations having…

A recent problem [B. Gardas, J. Math. Phys. 52, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead…

The Riccati equation method is used to establish three new oscillatory criteria for the second order linear ordinary differential equations in the marginal, sub extremal and extremal cases.We show that the first of these criteria implies…

A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…

In this paper, we first propose a new parameterized definition of comparison matrix of a given complex matrix, which generalizes the definition proposed by \cite {Axe1}. Based on this, we propose a new class of complex nonsymmetric…

The scalar Riccati equation is a prototypical nonlinear ODE having diverse mathematical connections. In the centuries since its initial formulation, a standard textbook theory has emerged according to which the general solution may be…

Parametric factorizations of linear partial operators on the plane are considered for operators of orders two, three and four. The operators are assumed to have a completely factorable symbol. It is proved that ``irreducible'' parametric…

An algebraic approach for factorizing nonlinear partial differential equations (PDEs) and systems of PDEs is provided. In the particular case of second order linear and nonlinear PDEs and systems of PDEs, necessary and sufficient conditions…

We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic…

We analyse the behaviour of the Dirac equation in $d=1+1$ with Lorentz scalar potential. As the system is known to provide a physical realization of supersymmetric quantum mechanics, we take advantage of the factorization method in order to…

There was proposed the method of a factorization of PDE. The method is based on reduction of complicated systems to more easy ones (for example, due to dimension decrease). This concept is proposed in general case for the arbitrary PDE…