Riemann-Hilbert problems for poly-Hardy space on the unit ball

In this paper, we focus on a Riemann–Hilbert boundary value problem (BVP) with a constant coefficients for the poly-Hardy space on the real unit ball in higher dimensions. We first discuss the boundary behaviour of functions in the poly-Hardy class. Then we construct the Schwarz kernel and the higher order Schwarz operator to study Riemann–Hilbert BVPs over the unit ball for the poly-Hardy class. Finally, we obtain explicit integral expressions for their solutions. As a special case, monogenic signals as elements in the Hardy space over the unit sphere will be reconstructed in the case of boundary data given in terms of functions having values in a Clifford subalgebra. Such monogenic signals represent the generalization of analytic signals as elements of the Hardy space over the unit circle of the complex plane.     

Decompositions of functions and Dirichlet problems in the unit disc

In this paper, we establish a decomposition theorem for polyharmonic functions and consider its applications to some Dirichlet problems in the unit disc. By the decomposition, we get the unique solution of the Dirichlet problem for polyharmonic functions (PHD problem) and give a unified expression for a class of kernel functions associated with the solution in the case of the unit disc introduced by Begehr, Du and Wang. In addition, we also discuss some quasi-Dirichlet problems for homogeneous mixed-partial differential equations of higher order. It is worthy to note that the decomposition theorem in the present paper is a natural extension of the Goursat decomposition theorem for biharmonic functions.     

Non-self-adjoint elliptic problems with a polynomial property in domains possessing cylindrical outlets to infinity

Solvability of mixed boundary-value problems in domains with cylindrical outlets to infinity is investigated for a certain class of non-self-adjoint differential operator-matrices., The structure of these matrices is such that the corresponding asymmetric quadratic forms possess a polynomial property, i.e., they degenerate only on finite-dimensional lineals of vector polynomials. With the help of elementary algebraic operations, this property enables one to indicate some attributes of boundary-value problems, namely, to calculate the operator index, to describe the kernel and co-kernel of an operator, to find an asymptotic behavior of solutions at infinity, etc. Bibliography: 16 titles. Dedicated to O. A. Ladyzhenskaya on the occasion of her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997, pp. 212–230. Translated by S. A. Nazarov.     

Spectral factorization in the disk algebra

Every strictly positive function f, given on the unit circle of the complex plane, defines an outer function. This article investigates the behavior of these outer functions on the boundary of the unit disk. It is shown that even if the given function f on the boundary is continuous, the corresponding outer function is generally not continuous on the closure of the unit disk. Moreover, any subset E∈ [-π ,π) of Lebesgue measure zero is a valid divergence set for outer functions of some continuous functions f. These results are applied to study the solutions of non-linear boundary-value problems and the factorization of spectral density functions.     

The Complexity of Two-Point Boundary-Value Problems with Piecewise Analytic Data

Previous work on the ε-complexity of elliptic boundary-value problems Lu = f assumed that the class F of problem elements f was the unit ball of a Sobolev space. In a recent paper, we considered the case of a model two-point boundary-value problem, with F being a class of analytic functions. In this paper, we ask what happens if F is a class of piecewise analytic functions. We find that the complexity depends strongly on how much a priori information we have about the breakpoints. If the location of the breakpoints is known, then the ε-complexity is proportional to ln (ε⁻¹), and there is a finite element p-method (in the sense of Babu ka) whose cost is optimal to within a constant factor. If we know neither the location nor the number of breakpoints, then the problem is unsolvable for ε < √2. If we know only that there are b ≥ 2 breakpoints, but we de not know their location, then the ε-complexity is proportional to bε⁻¹, and a finite element h-method is nearly optimal. In short, knowing the location of the breakpoints is as good as knowing that the problem elements are analytic, whereas only knowing the number of breakpoints is no better than knowing that the problem elements have a bounded derivative in the L₂ sense.     

Multipoint solution of two-point boundary-value problems

The purpose of this paper is to report on the application of multipoint methods to the solution of two-point boundary-value problems with special reference to the continuation technique of Roberts and Shipman. The power of the multipoint approach to solve sensitive two-point boundary-value problems with linear and nonlinear ordinary differential equations is exhibited. Practical numerical experience with the method is given.Since employment of the multipoint method requires some judgment on the part of the user, several important questions are raised and resolved. These include the questions of how many multipoints to select, where to specify the multipoints in the interval, and how to assign initial values to the multipoints.Three sensitive numerical examples, which cannot be solved by conventional shooting methods, are solved by the multipoint method and continuation. The examples include (1) a system of two linear, ordinary differential equations with a boundary condition at infinity, (2) a system of five nonlinear ordinary differential equations, and (3) a system of four linear ordinary equations, which isstiff.The principal results are that multipoint methods applied to two-point boundary-value problems (a) permit continuation to be used over a larger interval than the two-point boundary-value technique, (b) permit continuation to be made with larger interval extensions, (c) converge in fewer iterations than the two-point boundary-value methods, and (d) solve problems that two-point boundary-value methods cannot solve.     

Exact L 2-Small Ball Asymptotics of Gaussian Processes and the Spectrum of Boundary-Value Problems

We sharpen a classical result on the spectral asymptotics of boundary-value problems for self-adjoint ordinary differential operator. Using this result, we obtain the exact L ₂-small ball asymptotics for a new class of zero-mean Gaussian processes. This class includes, in particular, the integrated generalized Slepian process, integrated centered Wiener process, and integrated centered Brownian bridge. Partially supported by RFFR grant No.07-01-00159 and by grant NSh-227.2008.1.     

Determination of the stress-strain state of anisotropic thermosensitive cylinders

In the three-dimensional formulation we study a class of problems involving the stressed state of an axisymmetrically heated anisotropic cylinder arbitrarily inhomogeneous over the thickness taking account of the dependence of mechanical characteristics on the temperature. The solution of the boundary-value problems is carried out numerically. We study the temperature and mechanical fields in composite cylinders. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 73–77.     

Optimization of a Class of Nonlinear Dynamic Systems: New Efficient Method without Lagrange Multipliers

This paper deals with optimization of a class of nonlinear dynamic systems with n states and m control inputs commanded to move between two fixed states in a prescribed time. Using conventional procedures with Lagrange multipliers, it is well known that the optimal trajectory is the solution of a two-point boundary-value problem. In this paper, a new procedure for dynamic optimization is presented which relies on tools of feedback linearization to transform nonlinear dynamic systems into linear systems. In this new form, the states and controls can be written as higher derivatives of a subset of the states. Using this new form, it is possible to change constrained dynamic optimization problems into unconstrained problems. The necessary conditions for optimality are then solved efficiently using weighted residual methods.     

Total ordering for intuitionistic fuzzy numbers

Intuitionistic fuzzy set plays a vital role in decision making, data analysis, and artificial intelligence. Many decision‐making problems consist of different types of datum, where fuzzy set theoretical approaches may fail to obtain the optimal decision. Numerous approaches for intuitionistic fuzzy decision‐making problem have been introduced in the literature to overcome these short comings. But there is no single approach that can be used to solve all kinds of problems because of the partial ordering defined on the collection of intuitionistic fuzzy numbers (IFNs). Even though ranking of fuzzy numbers have been studied from early seventies in the last century, a total order on the entire class of fuzzy numbers has been introduced by Wang and Wang (Fuzzy Sets Syst 2014, 243, 131–141) only on 2014. A total order on the collection of all IFN is an open problem till today. In this article, a total order on the entire class of IFN using upper lower dense sequence in the interval [0, 1] is proposed and compared with existing techniques using illustrative examples, further an algorithm (which is problem independent) for solving any intuitionistic fuzzy multicriteria decision‐making problem (Intuitionistic fuzzy MCDM) is introduced. This new total ordering on IFNs generalizes the total ordering defined in Wang and Wang ( 22 ) for fuzzy numbers. © 2016 Wiley Periodicals, Inc. Complexity 21: 54–66, 2016     

Fictitious region method for one class of nonlinear boundary-value problems

We consider the application of the fictitious region method to solve one class of nonlinear boundary-value problems with nonlinearity only in the boundary condition. A rate of convergence bound is obtained.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 16–19, 1986.     

Constructing autonomous boundary-value linear systems from acausal input-output functions

Afrequency-domain realization theory is developed for the class of autonomous, but not necessarily stationary, boundary-value linear systems. It is shown that this realization problem, which consists of constructing autonomous boundary-value linear systems from prescribed input-output functions (weighting patterns), reduces to the factorization of several rational matrices in two variables having separable denominators. This factorization problem is examined and a method is given for constructing minimal factorizations for such rational matrices.     

An approach to numerical solution of multipoint boundary-value problems

An approach is proposed to solving multipoint boundary-value problems for linear differential equation of w-th order, based on reduction to two-point boundary-value problems. The two-point problems are solved by the stable discrete orthogonalization method. Some numerical examples are considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 36–45, 1986.     

Application of fractional calculus operators for a new class of univalent functions with negative coefficients defined by Hohlov operator

In this paper, we introduce a new class W(a, b, c, γ, β) which consists of analytic and univalent functions with negative coefficients in the unit disc defined by Hohlov operator, we obtain distortion theorem using fractional calculus techniques for this class. Also coefficient inequalities and some results for this class are obtained.     

Mutually-adjoint boundary-value problems with deviation from the characteristic for a multidimensional Gellerstedt equation

In this paper, we study mutually-adjoint boundary-value problems with a deviation from the characteristic for multidimensional Gellerstedt equation. In [3, 4], for the equation of the vibration of a string, the boundary-value problem with a deviation from the characteristic was studied, where the main attention was paid to the study of such problems for hyperbolic equations. For hyperbolic equations on the plane, this problem was studied in [5, 9].     

A mathematical model of a class of problems for the transport of a mixture in air

An analytic solution of a class of boundary-value problems of mathematical physics describing the transport of a mixture in the atmosphere is considered. To solve these problems we apply the substitution method and the Fourier method. The solution of a boundary problem describing the process of contamination of the atmosphere by various substances is presented in the form of a series. The result obtained is useful for the solution of problems concerning the protection of the atmosphere.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 87–90, 1989.     

Introduction to the theory of functional differential equations and their applications. Group approach

In this work, we give an introduction to the theory of nonlinear functional differential equations of pointwise type on a finite interval, semi-axis, or axis. This approach is based on the formalism using group peculiarities of such differential equations. For the main boundary-value problem and the Euler-Lagrange boundary-value problem, we consider the existence and uniqueness of the solution, the continuous dependence of the solution on boundary-value and initial-value conditions, and the “roughness” of functional differential equations in the considered boundary-value problems. For functional differential equations of pointwise type we also investigate the pointwise completeness of the space of solutions for given boundary-value conditions, give an estimate of the rank for the space of solutions, describe types of degeneration for the space of solutions, and establish conditions for the “smoothness” of the solution. We propose the method of regular extension of the class of ordinary differential equations in the class of functional differential equations of pointwise type. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 8, Functional Differential Equations, 2004.     

On the Univalence of an Integral on Subclasses of Meromorphic Functions

We study the integral operator $P_\lambda |f|(\zeta ) = \int {_{\zeta _0 }^\zeta } \left( {f\prime \left( t \right)} \right)^\lambda dt,{\text{ }}|\zeta |{\text{ }} > 1$ , acting on the class ∑ of functions meromorphic and univalent in the exterior of the unit disk. We refine the ranges of the parameter λ for which the operator preserves univalence either on ∑ or on its subclasses consisting of convex functions. As a consequence, a two-sided estimate is deduced for the separating constant in the sufficient condition for the univalent solvability of exterior inverse boundary-value problems.     

Maximum Principle of Optimal Control of the Primitive Equations of the Ocean With State Constraint

We investigate in this paper Pontryagin's maximum principle for a class of control problems associated with the primitive equations (PEs) of the ocean. These optimal problems involve a state constraint similar to that considered in Wang and Wang (Nonlinear Analysis 2003; 52:1911–1931) for the three-dimensional Navier–Stokes (NS) equations. The main difference between this work and Wang and Wang (Nonlinear Analysis 2003; 52:1911–1931) is that the nonlinearity in the PEs is stronger than in the three-dimensional NS systems.