Asymptotic Representation Of Solutions Research Articles

On some asymptotic representations of solutions to elliptic equations and their applications

The question on asymptotic representations of Green's function for classical boundary value problems for second-order elliptic equations depending on a complex parameter is considered. The equations are defined either in a domain with a compact boundary or in the whole space or the half-space. The main asymptotic term is written out. The results can be applied for the study of some inverse problems of recovering point sources in heat and mass transfer problems.

Asymptotic Integration of Certain Differential Equations in Banach Space

In this work, we investigate the problem of constructing asymptotic representations for weak solutions of a certain class of linear differential equations in the Banach space as an independent variable tends to infinity. We consider the class of equations that represent a perturbation of a linear autonomous equation, in general, with an unbounded operator. The perturbation takes the form of a family of bounded operators that, in a sense, oscillatorally decreases at infinity. It is assumed that the unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence of a center-like manifold (a critical manifold) for the initial equation. This manifold is positively invariant with respect to the initial equation and attracts all trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional system of ordinary differential equations. The asymptotics of the fundamental matrix of this system can be constructed by using the method developed by P.N. Nesterov for asymptotic integration of systems with oscillatory decreasing coefficients. We illustrate the proposed technique by constructing the asymptotic representations for solutions of the perturbed heat equation.

Asymptotic representations of solutions with slowly varying derivatives

Second-order differential equations with power and exponential nonlinearities on the right hand side play an important role in the development of a qualitative theory of differential equations. The authors of most works devoted to the establishment of asymptotic representations of solutions investigate equations with power and with regularly varying nonlinearities. Recently, the consideration of differential equations with exponential and a wider class than exponential functions - rapidly varying functions - has begun. In this paper, the asymptotic representations of solutions with slowly varying first-order derivatives of some new class of second-order differential equations with rapidly and regularly varying nonlinearities are established.

Asymptotic representations of solutions of nonlinear two term difference equations

In this paper we establish the necessary and sufficient conditions for existence of solutions of the nonlinear two term difference equation where and are regularly varying functions, Also we obtain their asymptotic representations as Since and could be more complicated than power functions and for proving the main result we formulate and prove an additional statement about existence of vanishing solutions of quasilinear system of difference equations.

Singular Initial-Value and Boundary-Value Problems for Integrodifferential Equations in Dynamical Insurance Models with Investments

We investigate two insurance mathematical models of the following behavior of an insurance company in the insurance market: the company invests a constant part of the capital in a risk asset (shares) and invests the remaining part in a risk-free asset (a bank account). Changing parameters (characteristics of shares), this strategy is reduced to the case where all the capital is invested in a risk asset. The first model is based on the classical Cramer–Lundberg risk process for the exponential distribution of values of insurance demands (claims). The second one is based on a modification of the classical risk process (the so-called stochastic premium risk process) where both demand values and insurance premium values are assumed to be exponentially distributed. For the infinite-time nonruin probability of an insurance company as a function of its initial capital, singular problems for linear second-order integrodifferential equations arise. These equations are defined on a semiinfinite interval and they have nonintegrable singularities at the origin and at infinity. The first model yields a singular initial-value problem for integrodifferential equations with a Volterra integral operator with constraints. The second one yields more complicated problem for integrodifferential equations with a non-Volterra integral operator with constraints and a nonlocal condition at the origin. We reduce the problems for integrodifferential equations to equivalent singular problems for ordinary differential equations, provide existence and uniqueness theorems for the solutions, describe their properties and long-time behavior, and provide asymptotic representation of solutions in neighborhoods of singular points. We propose efficient algorithms to find numerical solutions and provide the computational results and their economics interpretation.

Asymptotic representations of solutions of k-th order Emden–Fowler difference equation

In this paper we establish the asymptotical representations of solutions of k-th order Emden–Fowler difference equation where p(n) is close to a power sequence. We obtained the necessary conditions for the existence of solutions and we also found the sufficient conditions of their existence for non-critical classes. These results are generalizations of corresponding theorems for second-order Emden–Fowler equation.

Wiman’s formula for a second order dynamic equation

Abstract We derive Wiman’s asymptotic formula for the number of generalized zeros of (nontrivial) solutions of a second order dynamic equation on a time scale. The proof is based on the asymptotic representation of solutions via exponential functions on a time scale. By using the Jeffreys et al. approximation we prove Wiman’s formula for a dynamic equation on a time scale. Further we show that using the Hartman-Wintner approximation one can derive another version of Wiman’s formula. We also prove some new oscillation theorems and discuss the results by means of several examples. MSC:34E20, 34N05.

Far fields of internal gravity waves in a stratified liquid of varying depth

Asymptotic representations of solutions describing the far fields of internal gravity waves in a stratified medium of varying depth are constructed. The effect of space-frequency cutoff of the wave field for a real oceanic shelf is revealed. Depending on frequency characteristics of the wave field and bottom topography, far fields of internal waves either are located in a certain confined space domain (trapped waves) or propagate in the absence of turning points over sufficiently large distances when compared with the sea depth (progressive waves). The space domain where the progressive waves penetrate is fully determined by the presence of turning points whose locations depend on the medium stratification and inhomogeneities of bottom topography.

Asymptotic representations of solutions of essentially nonlinear systems of ordinary differential equations with regularly and rapidly varying nonlinearities

1. Statement of the Problem and Auxiliary Notation Consider a system of differential equations y 0 i D ipi.t/’iC1.yiC1/; iD1;n; (1.1) in which 2 i 2 f1;1 g; i D 1;n; and piWaa;!a ! c0;C1a; i D 1;n; are continuous functions, 1 < a < ! C1; 3 and ’iWA.Y 0 i /!c0IC1a; i D 1;n; where A.Y 0

Asymptotic representations of solutions of one class of nonlinear differential equations of the second order

We establish asymptotic properties of some types of solutions of one class of essentially nonlinear nonautonomous differential equations of the second order and find necessary and sufficient conditions for the existence of these solutions.

Linear second-order differential equations with degenerating coefficient of the second derivative

For an ordinary linear second-order differential equation with degenerating coefficient of the higher derivative, we obtain exact asymptotic representations of solutions near the degeneration point, which are convergent uniformly and absolutely in some neighborhood of that point. The results permit one to obtain well-posed statements of initial and boundary value problems for degenerating equations and establish oscillation properties of solutions.

Asymptotic representations of solutions of one class of second-order ordinary differential equations

We establish asymptotic representations for some classes of solutions of nonautonomous second-order differential equations close, in a certain sense, to linear equations.

Far fields of internal gravity waves in stratified media of variable depth

The present paper is devoted to presenting the solution of the problem on the far field of internal gravity waves in a stratified medium whose depth linearly depends on the space variable. Uniform asymptotic representations of solutions are constructed; these representations enable us to describe the far fields of internal gravity waves excited by a moving source and characterize the main specific features of the wave fields near caustics and wave fronts.

Asymptotic representations of solutions of essentially nonlinear two-dimensional systems of ordinary differential equations

We establish asymptotic representations for one class of solutions of two-dimensional systems of ordinary differential equations that are more general than systems of the Emden–Fowler type.

Asymptotic representations of solutions of nonautonomous second-order differential equations with nonlinearities close to power-type nonlinearities

Asymptotic representations are found for solutions of nonautonomous second-order differential equations with nonlinearities close to power-type nonlinearities.

FRACTIONAL LANDAU–GINZBURG EQUATIONS ON A SEGMENT

We study the initial-boundary value problem for the fractional Landau–Ginzburg equations on a segment. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem and to find the main term of the asymptotic representation of solutions.

Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order

We establish asymptotic representations for unbounded solutions of nonlinear nonautonomous differential equations of the third order that are close, in a certain sense, to equations of the Emden-Fowler type.

Asymptotic analysis of solutions to parabolic systems

We study asymptotics as $$t\to\infty$$ of solutions to a linear, parabolic system of equations with time-dependent coefficients in $$\Omega \times (0,\infty)$$ , where $$\Omega$$ is a bounded domain. On $$\partial \Omega\times(0,\infty)$$ we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function $$\kappa(t)$$ . This includes in particular situations when the coefficients may take different values on different parts of $$\Omega$$ and the boundaries between them can move with t but stabilize as $$t\to\infty$$ . The main result is an asymptotic representation of solutions for large t. A consequence is that for $$\kappa \in L^1(0,\infty)$$ , the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.

Asymptotics for critical pseudodifferential equations on a half-line

We study the initial boundary value problem for a general class of nonlinear pseudodifferential equations on a half-line Pseudodifferential operator is defined by the inverse Laplace transform. The aim of this article is to prove the global existence of solutions to the inital-boundary value problem (1) in the critical case and to find the main term of the asymptotic representation of solutions taking into account the influence of no homogeneous boundary data. Also we give general theory of the study of initial boundary value problem with no homogeneous boundary data and critical convective type of the nonlinearity.

Asymptotics for critical Rosenau type equations with nonconvective nonlinearites on a half-line

We study nonlinear pseudodifferential equations on a half-line with nonanalytic symbol where where Qm and Pm are polynomials of order m ∈ N such that ReK(p)>0 for all Rep = 0 and The aim of this article is to prove the global existence of solutions to the initial boundary value problem and to find the main term of the asymptotic representation of solutions in critical case, when the nonlinear term of equation has the time decay rate less than that of the linear terms. Also in this study for the first time we give some general approaches to obtain the global existence of solution of initial boundary value problem in critical nonconvective case and elaborate on the general sufficient conditions to obtain asymptotic expansion of solution.