Hilbert boundary value problem for generalized analytic functions with a singular line

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on a circle for a generalized Cauchy-Riemann equation with a singular coefficient. To solve this problem, we conducted a complete study of the solvability of the Hilbert boundary value problem of the theory of analytic functions with an infinite index due to a finite number of points of a special type of vorticity. Based on these results, we have derived a formula for the general solution and studied the existence and number of solutions to the boundary value problem of the theory of generalized analytic functions.

Of one over determined system differential equations at private derivative second order with one singular lines in general case

In this paper, an over determined system of second-order partial differential equations with a single singular line in the General case is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with a single singular line in the General case. If the compatibility condition is met, integral representations of the variety of solutions are found explicitly in terms of three arbitrary constants, when the singular line is in the boundaries of the domain for which initial data problems (Cauchy-type Problems) can be set.

Integral representation of solution manifolds for over determined systems with one singular line

In this paper, an over determined system of second-order partial differential equations with one singular line is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with one singular line. Under the condition of compatibility, introducing a new function, we come to a over determined system of partial differential equations of the second order with one singular line of a simpler form. The integral representation of the manifold of solutions of the redefined second-order partial differential system with one singular line is found explicitly through three arbitrary constants, for which initial data problems (Cauchy type problems) can be posed.

Of one over determined system differential equation at private derivative second order with one singular point and one singular line

In this paper, a over determined system of second-order partial differential equations with one singular point and one singular line is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with one singular point and one singular line. If the compatibility condition is met, integral representations of the variety of solutions are found explicitly in terms of three arbitrary constants, when the singular line is in the boundaries of the domain for which initial data problems (Cauchy-type Problems) can be set. In this paper considers a redefined system of second-order partial differential equations, when the coefficients and right parts have one singular point and one singular line. Obtaining a variety of solutions and studying boundary value problems for linear differential equations of the hyperbolic type of the second order, some linear redefined systems of the first and second order with one and two supersingular lines and supersingular points is devoted to the monograph of academician of the National Academy of Sciences of the Republic of Tatarstan Rajabov N. - 1992 "Introduction to the theory of partial differential equations with supersingular coefficients" [6, p.126]. Using the obtained results of The monograph of Rajabov N., a variety of solutions of redefined systems of partial differential equations of the second order with one singular point and one singular line in an explicit form, through three arbitrary constants, was found.

Bifurcations of Solitary Waves of a Simple Equation

In this paper, we consider a simple equation which involves a parameter [Formula: see text], and its traveling wave system has a singular line. Firstly, using the qualitative theory of differential equations and the bifurcation method for dynamical systems, we show the existence and bifurcations of peak-solitary waves and valley-solitary waves. Specially, we discover the following novel properties: (i) In the traveling wave system, there exist infinitely many periodic orbits intersecting at a point, or two points and passing through the singular line, and there is no singular point inside a homoclinic orbit. (ii) When [Formula: see text], in the equation there exist three types of bifurcations of valley-solitary waves including periodic wave, blow-up wave and double solitary wave. (iii) When [Formula: see text], in the equation there exist two types of bifurcations of valley-solitary wave including periodic wave and blow-up wave, but there is no double solitary wave bifurcation. Secondly, we perform numerical simulations to visualize the above properties. Finally, when [Formula: see text] and the constant wave speed equals [Formula: see text], we give exact expressions to the above phenomena.

Schwarz boundary-value problems for solutions of a generalized Cauchy-Riemann system with a singular line

We consider a generalized Cauchy-Riemann system with a rectilinear singular interval of the real axis. Schwarz boundary value problems for generalized analytic functions which satisfy the mentioned system are reduced to the Fredholm integral equations of the second kind under natural assumptions relating to the boundary of a domain and the given boundary functions.

On rational solutions of two differential equations with а moving singular line

The study is devoted to the analytical theory of ordinary differential equations. In the introduction, it is said that the object of investigation is nonlinear third-order autonomous differential equations with a moving singular line, and whose two-parameter rational solutions are known. The study aims to clarify how to obtain two-parameter rational solutions from the general solutions of these equations. In the analytical theory of differential equations, as a rule, nonlinear differential equations have negative resonances. Among these resonances is the resonance equal to –1 (trivial case). However, it is asserted in some of the researchers’ papers that the nature of these negative resonances has not been found until now. The equations only with non-trivial negative resonance arose the interest of researchers. And it appears that the rational solutions of the equations can be constructed by their negative resonances. In this paper, a necessary and sufficient condition is indicated, at which the two-parameter rational solution of the equation with a moving singular line can be obtained from its general solution.