ON THE STRUCTURE OF SOLUTIONS OF LINEAR PARTIAL DIFFERENTIAL EQUATION sum form i+j≤n aijpiqjφ=0 WITH TWO INDEPENDENT VARIABLES AND CONSTANT COEFFICIENTS

This paper is a continuation of [1]. In this paper, the solutions of the more general linear partial differential equation form i+j≤n a_ijpⁱq^jφ=0 with two independent variables and constant coefficients are discussed in detail. The general solution which can be used in the approximation to the conditions of the definite solution of the practical problems is presented. To illustrate the use of the results obtained in this paper, some practical examples in mechanics are given.     

The global structural stability and application of the toroidal differntial equations\left\{ {\begin{array}{*{20}c} {\dot x = a_1 [\sin (x - y) - \sin x] - \Delta \omega } \\ {\dot y = a_2 [\sin (y - x) - \sin y] - \Delta \omega } \\ \end{array} } \right.

In this paper, the global structural stability of the toroidal differential equationshas been obtained, and applied to the cross-coupled phase - locked loop, where Δω≥0     

Necessary and Sufficient Conditions for the Invertibility of the Nonlinear Difference Operator \left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1}\right) - f\left( {x\left( t \right)} \right) in the Space of Functions Bounded and Continuous on the Axis

We find necessary and sufficient conditions for the nonlinear difference operator $\left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1} \right) - f\left( {x\left( t \right)} \right)$ $t \in \mathbb{R}$ , where $f:\mathbb{R} \to \mathbb{R}$ is a continuous function, to have the inverse in the space of functions bounded and continuous on $\mathbb{R}$ .     

Topological Conjugation Classes of Tightly Transitive Subgroups of $$\text {Homeo}_{+}(\mathbb {S}^1)$$ Homeo + ( S 1 )

Journal of Dynamics and Differential Equations - Let $$\text {Homeo}_{+}(\mathbb {S}^1)$$ denote the group of orientation preserving homeomorphisms of the circle $$\mathbb {S}^1$$ . A subgroup G of...     

The Riemann problem for the system $$u_1 + \sigma _x = 0{\text{ and }}(\sigma - f(u))_t + (\sigma - uf(u))$$

Archive for Rational Mechanics and Analysis -     

New nonautonomous combined multi-wave solutions for ( $$\varvec{2+1}$$ 2 + 1 )-dimensional variable coefficients KdV equation

Nonlinear Dynamics - A variety of new types of nonautonomous combined multi-wave solutions of the ( $$2+1$$ )-dimensional variable coefficients KdV equation is derived by means of the generalized...     

Residual symmetry,Bäcklund transformation and CRE solvability of a ( $$\mathbf{2}{\varvec{+}}{} \mathbf{1}$$ 2 + 1 )-dimensional nonlinear system

Nonlinear Dynamics - In this paper, the truncated Painlevé expansion is employed to derive a Bäcklund transformation of a ( $$2+1$$ )-dimensional nonlinear system. This system can be...     

Bilinear form,soliton, breather,lump and hybrid solutions for a ( $$\varvec{2+1}$$ 2 + 1 )-dimensional Sawada–Kotera equation

Nonlinear Dynamics - In this paper, we investigate a ( $$2+1$$ )-dimensional Sawada–Kotera (SK) equation for the atmosphere, rivers, lakes, oceans, as well as the conformal field and...