Unperturbed Problem Research Articles

Weak long‐range diffusion

AbstractWe study the spreading solutions of the nonlinear diffusion equation when the far‐field diffusivity is small. The method of strained coordinates is used to construct a uniform asymptotic correction to the similarity solution of the unperturbed problem. The equation provides a possible analogue to similar models of fluid jets and plumes.

Asymptotic method for solving the problem of transition process optimization in a three-tempo singularly perturbed system

The problem of constructing a transition process with minimal energy costs for a linear singularly perturbed system containing three groups of variables with significantly different rates of change is considered. Asymptotic approximations to solving this problem are constructed in the form of an open-loop and feedback controls. The main advantage of the proposed computational procedures is that the original problem is split into three unperturbed optimal control problems of lower dimension.

Periodic perturbations of central force problems and an application to a restricted 3-body problem

We consider a perturbation of a central force problem of the formx¨=V′(|x|)x|x|+ε∇xU(t,x),x∈R2∖{0}, where ε∈R is a small parameter, V:(0,+∞)→R and U:R×(R2∖{0})→R are smooth functions, and U is τ-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem (ε=0) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular τ-periodic solutions bifurcating from invariant tori at ε=0. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential V(r)=κ/rα for α∈(−∞,2)∖{−2,0,1}). Finally, an application is given to a restricted 3-body problem with a non-Newtonian interaction.

On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property

This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function px, which belongs to the space L20,1. The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when px≡0) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions px, for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in L20,1, is dense within the space L20,1. Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.

Nontrivial solutions for an asymptotically linear Delta alpha-Laplace equation

In this paper, we study a class of degenerate unperturbed problems. We first investigate some properties of eigenvalues and eigenfunctions for the strongly degenerate elliptic operator and then obtain two existence theorems of nontrivial solutions when the nonlinearity is a function with an asymptotically condition.

Locally Hölder Continuity of the Solution Map to a Boundary Control Problem with Finite Mixed Control-State Constraints

The local stability of the solution map to a parametric boundary control problem governed by semilinear elliptic equations with finite mixed pointwise constraints is considered in this paper. We prove that the solution map is locally Hölder continuous in -norm of control variable when the strictly nonnegative second-order optimality conditions are satisfied for the unperturbed problem.

On the antimaximum principle for the p-Laplacian and its sublinear perturbations

We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $$-\Delta _p u = \lambda \,m(x)|u|^{p-2}u+ \eta \,a(x)|u|^{q-2}u+ f(x)$$ in a bounded domain $$\Omega \subset \mathbb {R}^N$$ , where $$q<p$$ . Under certain regularity and qualitative assumptions on the weights m, a and the source function f, we identify ranges of the parameters $$\lambda $$ and $$\eta $$ for which solutions satisfy maximum and antimaximum principles in weak and strong forms. Some of our results, especially on the validity of the antimaximum principle under low regularity assumptions, are new for the unperturbed problem with $$\eta =0$$ , and among them there are results providing new information even in the linear case $$p=2$$ . In particular, we show that for any $$p>1$$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the p-Laplacian provided $$m,f \in L^\gamma (\Omega )$$ with $$\gamma >N$$ . For completeness, we also investigate the existence of solutions.

Wavenumber-Explicit Parametric Holomorphy of Helmholtz Solutions in the Context of Uncertainty Quantification

A crucial role in the theory of uncertainty quantification (UQ) of PDEs is played by the regularity of the solution with respect to the stochastic parameters; indeed, a key property one seeks to establish is that the solution is holomorphic with respect to (the complex extensions of) the parameters. In the context of UQ for the high-frequency Helmholtz equation, a natural question is therefore: how does this parametric holomorphy depend on the wavenumber ? The recent paper [35] showed for a particular nontrapping variable-coefficient Helmholtz problem with affine dependence of the coefficients on the stochastic parameters that the solution operator can be analytically continued a distance into the complex plane. In this paper, we generalize the result in [35] about -explicit parametric holomorphy to a much wider class of Helmholtz problems with arbitrary (holomorphic) dependence on the stochastic parameters; we show that in all cases the region of parametric holomorphy decreases with and show how the rate of decrease with is dictated by whether the unperturbed Helmholtz problem is trapping or nontrapping. We then give examples of both trapping and nontrapping problems where these bounds on the rate of decrease with of the region of parametric holomorphy are sharp, with the trapping examples coming from the recent results of [31]. An immediate implication of these results is that the -dependent restrictions imposed on the randomness in the analysis of quasi-Monte Carlo methods in [35] arise from a genuine feature of the Helmholtz equation with large (and not, for example, a suboptimal bound).

On the Effect of Irregularity of the Domain Boundary on the Solution of a Boundary Value Problem for the Laplace Equation

We consider an inhomogeneous boundary value problem with mixed boundary conditions for the Laplace equation in a domain representing a perturbationof a rectangle where one of its sides is replaced by some curve of minimal smoothness. An estimate is obtained for the difference between the solutions of the perturbed and unperturbed problems in the norm of the Sobolev space on their common domain.

Investigating non-reciprocity in time-periodic media using a perturbative approach.

Lorentz famous theorem leads to clear reciprocity conditions for linear, time-invariant media based on their constitutive parameters. By contrast, reciprocity conditions for linear time-varying media are not fully explored. In this paper, we investigate whether, and how a structure containing a time-periodic medium can be truly identified as reciprocal or not. To that end, a necessary and sufficient condition is derived which requires both the constitutive parameters and the electromagnetic fields inside the dynamic structure. As solving for the fields for such problems is challenging, a perturbative approach is proposed which expresses the aforementioned non-reciprocity condition in terms of the electromagnetic fields and the Green's functions of the unperturbed static problem and is particularly applicable for the case of structures with weak time modulation. Reciprocity of two famous canonical time-varying structures are then studied using the proposed approach and their reciprocity/non-reciprocity is investigated. In the case of one-dimensional propagation in a static medium with two point-wise modulations, our proposed theory clearly explains the often observed maximization of non-reciprocity when the modulation phase difference between the two points is 90 degrees. In order to validate the perturbative approach, analytical and Finite-Difference Time-Domain (FDTD) methods are employed. Then, solutions are compared and considerable agreement between them is observed.

On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case

We study the asymptotic behavior of the viscosity solutions of the Hamilton-Jacobi (HJ) equation as the positive discount factor λ tends to 0, where is the perturbation of a Hamiltonian –periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential The constant c(G) appearing above is defined as the infimum of values for which the HJ equation in admits bounded viscosity subsolutions. We prove that the functions locally uniformly converge, for to a specific solution of the critical equation We identify in terms of projected Mather measures for G and of the limit to the unperturbed periodic problem. Our work also includes a qualitative analysis of the critical equation with a weak KAM theoretic flavor.

Local stability of solutions to a parametric multi-objective optimal control problem

This paper studies local stability of solutions to a parametric multi-objective optimal control problem with constraints. By combining the scalarization method and non-scalarization methods, we show that if the unperturbed problem has a locally strong Pareto solution, then the Pareto solution set is locally Hölder continuous at a reference parameter.

Chaos in the vicinity of a singularity in the three-body problem: The equilateral triangle experiment in the zero angular momentum limit

We present numerical simulations of the gravitational three-body problem, in which three particles lie at rest close to the vertices of an equilateral triangle. In the unperturbed problem, the three particles fall towards the center of mass of the system to form a three-body collision, or singularity, where the particles overlap in space and time. By perturbing the initial positions of the particles, we are able to study chaos in the vicinity of the singularity. Here we cover both the singular region close to the unperturbed configuration and the binary-single scattering regime where one side of the triangle is very short compared to the other two. We make phase space plots to study the regular and ergodic subsets of our simulations and compare them with the outcomes expected from the statistical escape theory of the three-body problem. We further provide fits to the ergodic subset to characterize the properties of the left-over binaries. We identify the discrepancy between the statistical theory and the simulations in the regular subset of interactions, which only exhibits weak chaos. As we decrease the scale of the perturbations in the initial positions, the phase space becomes entirely dominated by regular interactions, according to our metric for chaos. Finally, we show the effect of general relativity corrections by simulating the same scenario with the inclusion of post-Newtonian corrections to the equations of motion.

АЛГОРИТМЫ ВЫЧИСЛЕНИЯ СОБСТВЕННЫХ ЗНАЧЕНИЙ ДИСКРЕТНЫХ ПОЛУОГРАНИЧЕННЫХ ОПЕРАТОРОВ, ЗАДАННЫХ НА КВАНТОВЫХ ГРАФАХ

Spectral problems for differential operators defined on quantum graphs are of great scientific interest related to problems in quantum mechanics, computer network modeling, image processing, ranking algorithms, modeling of electrical, and mechanical and acoustic processes, in networks of a diverse nature, in designing nano systems with prescribed properties and in other areas. Theoretical solutions of direct and inverse spectral problems on quantum graphs have been developed, but computational algorithms based on these methods are computationally inefficient. We have not seen any published works that consider examples of numerical solutions of spectral problems on finite connected graphs with a large number of vertices and edges. Therefore, the development of new computationally effective algorithms for numerical solution of spectral problems given on finite connected graphs is urgent. This paper develops a technique for finding the eigenvalues of boundary value problems on finite connected graphs with a required number of vertices and edges. To use this technique, it is necessary to know the eigenvalues and vectors of the eigenfunctions of corresponding unperturbed vector operators which are usually self-adjoint. Finding them manually, if the graph has a large number of vertices and edges, is difficult. This led to writing a package of programs in the mathematical environment Maple to find transcendental equations in the symbolic mode to calculate eigenvalues and find the eigenfunctions of unperturbed boundary value problems. Examples of calculating eigenvalues for a quantum graph which models an anthracene aromatic compound molecule are presented.

Limits of solutions to the semilinear plate equation with small parameter

We study the existence of the limits of solutions to the semilinear plate equation with boundary Dirichlet condition with a small parameter coefficient of the second order derivative in time. We establish the convergence of solutions to the perturbed problem and their derivatives in spacial variables to the corresponding solutions to the unperturbed problem as the small parameter tends to zero.

Applied Problems of Perturbation Theory

In this paper, we study solutions to nonlinear singularly perturbed differential equations with initial conditions. Proving the asymptotic closeness of the solutions of the perturbed and unperturbed problems on the real axis is a top-priority task. But it doesn't always work out. For the first time in works in this direction, the concept of bistability of solutions was introduced. The definition of stability to the right and to the left is given. As well as definitions of bistability of solutions. Examples are given. If the solution is bistable, then it is always possible to show the asymptotic closeness of the solutions of the perturbed and unperturbed problems on the real domain.

Solution of the nonregular problem for a parabolic equation with the time derivative in the boundary condition

There is studied the Hölder space solution u ε of the problem for parabolic equation with the time derivative ε ∂ t u ε | Σ in the boundary condition, where ε &gt; 0 is a small parameter. The unique solvability of the perturbed problem and estimates of it’s solution are obtained. The convergence of u ε as ε → 0 to the solution of the unperturbed problem is proved. Boundary layer is not appeared.

Conditions for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result, specific for ideal efficient solutions to unperturbed problem and enhanced by related error bounds, is discussed. Some connections with the concept of robustness in multi-objective optimization are also sketched.

Stability for Semilinear Parabolic Optimal Control Problems with Respect to Initial Data

A distributed optimal control problem for a semilinear parabolic partial differential equation is investigated. The stability of locally optimal solutions with respect to perturbations of the initial data is studied. Based on different types of sufficient optimality conditions for a local solution of the unperturbed problem, Lipschitz or Hölder stability with respect to perturbations are proved. Moreover, a particular example with semilinear equation, constant initial data, and standard quadratic tracking type objective functional is constructed that has at least two different locally optimal solutions. By the perturbation analysis, the existence of a problem with non-constant initial data is shown that also has at least two different locally optimal solutions.

Исследование решений системы сингулярно возмущенных дифференциальных уравнений

Solutions of linear systems of singularly perturbed differential equations are investigated in the work, in the case when the matrix function had multiple eigenvalues. And also in the study of solutions to a system of singularly perturbed differential equations, we apply the level line method. We define a stable and unstable interval. We take the starting point in stable intervals. Passing to the complex domain, we define the domain that we study for solutions of the problem under consideration. We divide the defined areas near the singular point into several areas. In each area, we estimate the solutions of the problem. To do this, we choose the integration path and prove the lemma and theorem. As a result, we will prove the asymptotic proximity of the solutions of the perturbed and unperturbed problems.