Power Singularities Research Articles

General Boundary-Value Problem for Nonuniformly Parabolic Equations with Power Singularities

General Boundary-Value Problem for Nonuniformly Parabolic Equations with Power Singularities

Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number μ

We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number μ, approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a τ preconditioning when the variable coefficient wave number μ is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented.

Boundary-Value Problem for Nonuniformly Elliptic Equations with Power Singularities

Boundary-Value Problem for Nonuniformly Elliptic Equations with Power Singularities

Matrix‐less methods for the spectral approximation of large non‐Hermitian Toeplitz matrices: A concise theoretical analysis and a numerical study

AbstractIt is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non‐Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non‐Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix‐size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non‐Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high‐precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented.

Cauchy Problem for Nonuniformly Parabolic Equations with Power Singularities

Cauchy Problem for Nonuniformly Parabolic Equations with Power Singularities

Plane and axisymmetric problems of an interfacial crack in a power-law graded material

Plane and axisymmetric problems of interfacial cracks in a power-law graded infinite medium are examined. It is shown that the models are governed by integral equations of Mellin’s convolution type whose kernels are expressed through the hypergeometric function. Exact solutions are derived by the method of orthogonal Jacobi polynomials in a series form and by the Wiener–Hopf method by quadratures. Mode-I stress intensity factors and the associated weight functions for power-law graded materials in the plane and axisymmetric cases are introduced and evaluated. It is shown that, although the displacement jump and the normal traction component have power singularities at the crack tip different from 1/2, the strain energy variation is proportional to the crack length change as in the case of homogeneous materials. A Griffith-type criterion of crack propagation in power-law graded materials is proposed and results of numerical tests are reported.

On rational approximations of Poisson integrals on the interval by Fejer sums of Fourier – Chebyshev integral operators

Approximations of the Fejér sums of the Fourier – Chebyshev rational integral operators with restrictions on numerical geometrically different poles are herein studied. The object of research is the class of functions defined by Poisson integrals on the segment [–1, 1]. Integral representations of approximations and upper estimates of uniform approximations are established. In the case when the boundary function has a power singularity on the segment [–1, 1], upper estimates of pointwise and uniform approximations are found, and the asymptotic representation of the majorant of uniform approximations is found. As a separate problem, approximations of Poisson integrals for two geometrically different poles of the approximating rational function are considered. In this case, the optimal values of the parameters at which the highest rate of uniform approximations by the studied method is achieved are found. If the function |x|s, s ∈ (0, 1], is approximated, then this rate is higher than the corresponding polynomial analogues. Consequently, asymptotic expressions of the exact upper bounds of the deviations of Fejer sums of polynomial Fourier – Chebyshev series on classes of Poisson integrals on a segment are obtained. Estimates of uniform approximations by Fejer sums of polynomial Fourier – Chebyshev series of functions given by Poisson integrals on a segment with a boundary function having a power singularity are also obtained.

Internal and Startup Controls of the Solutions of Boundary-Value Problem for Parabolic Equations with Degenerations

For a second-order parabolic equation with degenerations, we construct the solution of the problem of optimal control for systems described by the first boundary-value problem with internal and startup controls. The coefficients of parabolic equation have power singularities of any order in time and space variables on a certain set of points.

Optimal control in the boundary value problem for elliptic equations with degeneration

The problem of optimal control of the system described by the oblique derivative problem forthe elliptic equation of the second order is studied. Cases of internal and boundary managementare considered. The quality criterion is given by the sum of volume and surface integrals.The coefficients of the equation and the boundary condition allow power singularities of arbitraryorder in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special H\"{o}lder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will bethe solution to the given problem.
 The necessary and sufficient conditions for the existence of the optimal solution of the systemdescribed by the boundary value problem for the elliptic equation with degeneracy have been established.

Asymptotics for the eigenvalues of Toeplitz matrices with a symbol having a power singularity

AbstractThe present work is devoted to the construction of an asymptotic expansion for the eigenvalues of a Toeplitz matrix as goes to infinity, with a continuous and real‐valued symbol having a power singularity of degree with , at one point. The resulting matrix is dense and its entries decrease slowly to zero when moving away from the main diagonal, we apply the so called simple‐loop (SL) method for constructing and justifying a uniform asymptotic expansion for all the eigenvalues. Note however, that the considered symbol does not fully satisfy the conditions imposed in previous works, but only in a small neighborhood of the singularity point. In the present work: (i) We construct and justify the asymptotic formulas of the SL method for the eigenvalues with , where the eigenvalues are arranged in nondecreasing order and is a sufficiently small fixed number. (ii) We show, with the help of numerical calculations, that the obtained formulas give good approximations in the case . (iii) We numerically show that the main term of the asymptotics for eigenvalues with , formally obtained from the formulas of the SL method, coincides with the main term of the asymptotics constructed and justified in the classical works of Widom and Parter.

Crisis and Literature: Future Imperfect, or the Case of Don DeLillo’s Cosmopolis

In his afterword to The Cambridge Companion to Don DeLillo (2008), Joseph M. Comte makes a strong case for positioning the author as a writer of historical liminality, and citing DeLillo himself, he claims that Cosmopolis is a text “poised liminally ‘between the end of the Cold War and the beginning of the Age of Terror’” (183). Not yet aware of the shift taking place in the author’s interest from all matters historical to his previous preoccupation with corporeality and writing, Comte goes on to argue that DeLillo’s novel of 2003 stages how “[c]yber-capital and terrorism contend within the singularity of global power” (185), inasmuch as the text is preoccupied with what commentators usually identify as “the technological sublime” (186) in DeLillo’s oeuvre, in this case representing the “interaction between technology and capital, the inseparability” of the two (23). Comte and other scholarly commentators praise Cosmopolis exactly for what it was criticized for at the time of its publication, its witty handling of academically embedded ideas, thereby somewhat downplaying how the text, as I will argue, discusses, or indeed embodies, some of these ideas in relation to the white male body and terrorism in a curious temporal structure: written after 9/11, but presenting what one may call reverse déjà vu of the terror attacks. Comte’s estimation is, therefore, in line with the contemporary reviews of the book at the time of its publication, and stresses the intellectual achievement and poetic qualities of the text.

OPTIMAL CONTROL IN THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS WITH DEGENERATION

The theory of optimal control of systems, which is described by partial differential equations, is rich in results and is actively developing nowadays. The popularity of this kind of research is connected with its active use in solving problems of natural science, in particular hydro and gas dynamics, heat physics, diffusion, and the theory of biological populations. The problem of optimal control of the system described by the Dirichlet problem for the elliptic equation of the second order is studied. Cases of internal control are considered. The quality criterion is given by the volumetric integral. The coefficients of the equation admit power singularities of arbitrary order in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will be the solution to the given problem. The necessary and sufficient conditions for the existence of the optimal solution of the system described by the Dirichlet problem for the elliptic equation with degeneracy have been established.

Asymptotic existence theorem for formal solutions with singularities of nonlinear partial differential equations via multisummability

In this paper, we consider the summability of formal solutions with singularities (such as logarithmic singularities, functional power singularities, etc.) of nonlinear partial differential equations in the complex domain. The main result is as follows: when a formal solution with singularities is given, under appropriate assumptions related to the formal solution, the equation has a true solution that admits the given formal solution as an asymptotic expansion. The proof is done by constructing a new formal solution that is equivalent to the given formal solution in the asymptotic sense and is multisummable in a suitable direction. The assumptions are stated in terms of the Newton polygon associated with the given formal solution.

Bifurcation Analysis of a Piecewise Smooth Map with Two Asymptotes

In this paper, we consider a discontinuous piecewise smooth system involving four parameters and two asymptotes, recently introduced as a model in engineering sciences. We classify and investigate its bifurcation behaviour. A local bifurcation analysis of the system in the range of parameters which has not been studied so far is undertaken and then supported by numerical computations. This reveals the existence of a flip bifurcation depends on the power singularity. Moreover, we state that a set of positive measure of points with divergent dynamic behaviour exists.

General boundary value problem for nonuniformly parabolic equations with power singularities

General boundary value problem for nonuniformly parabolic equations with power singularities

Boundedness and compactness of a class of integral operators with power and logarithmic singularity when pleq q

In this paper, necessary and sufficient conditions for the boundedness and compactness of one class of integral operators with power and logarithmic singularities in weighted Lebesgue spaces are obtained.

Коаналитическая задача продолжения периодической функции в пространствах с весом, имеющим особенность на границе

It is known that any periodic square-summable function with a power-law weight in a half-strip is uniquely represented as an orthogonal sum of analytic and coanalytic components; therefore, it is natural to regard the coanalytic component as a certain nonanalyticity characteristic of the function. The article considers the problem of finding a periodic function extension so that it would deviate from the weighted Sobolev subspace of analytic functions to a minimal extent (the problem of minimizing the coanalytic deviation). In my previous publication, I considered the problem of extending a function to inside a circle in spaces with a weight that has a singularity at the boundary. Other researchers studied similar coanalytic problems in the weightless case for the unit circle, half-strip, and an arbitrary bounded simply connected domain with a smooth boundary. The coanalytic problem is formulated in the spaces of periodic functions with a weight having a power singularity at the boundary, and the boundary values of the functions are taken from the corresponding Besov space. The known properties of weight spaces, including the direct and inverse theorems about the traces of functions from the classes under consideration have been formulated. Using these properties, a theorem about the existence and uniqueness of the coanalytic problem solution is proved within the principles of the monotone operator theory.

A MULTIPOINT IN-TIME PROBLEM FOR THE 2b-PARABOLIC EQUATION WITH DEGENERATION

In recent decades, special attention has been paid to problems with nonlocal conditions for partial differential equations. Such interest in such problems is due to both the needs of the general therapy of boundary value problems and their rich practical application (the process of diffusion, oscillations, salt and moisture transport in soils, plasma physics, mathematical biology, etc.). A multipoint in-time problem for a nonuniformly 2b-parabolic equation with degeneracy is studied. The coefficients of the parabolic equation of order 2b allow for power singularities of arbitrary order both in the time and spatial variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions of auxiliary problems, the limiting value of which will be the solution of the given problem. Estimates of the solution of the multipoint time problem for the 2b-parabolic equation are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of degeneracy of the coefficients of the groups of higher terms and the power features of the coefficients of the lower terms of the parabolic equation. With certain restrictions on the right-hand side of the equation, an integral image of the solution to the given problem is obtained.

Boundary value problem for non-uniformly elliptic equations with power singularities

Boundary value problem for non-uniformly elliptic equations with power singularities

On Qualitative Properties of Sign-Constant Solutions of Some Quasilinear Parabolic Problems

We study the Cauchy problem for quasilinear parabolic inequalities containing squares of the first derivatives of an unknown function (the so-called nonlinearities of the KPZ type). The coefficients of the leading nonlinear terms of the inequalities considered either can be continuous functions (the regular case) or can admit power singularities (the singular case) of degree no greater than 1. For the regular case, we prove the damping of global nonnegative solutions to the problem studied. By damping, we mean the boundedness of the support of a solution for each positive t, uniform (with respect to t) convergence to zero as |x| → ∞, and vanishing (for any x) starting with a certain sufficiently large t. For the singular case, we proved that the problem considered has no global positive solutions.