A Hilbert Transform for Matrix Functions on Fractal Domains

Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

Clifford analysis is a higher-dimensional function theory offering a refinement of classical harmonic analysis. It is centred around monogenic functions, i.e. null solutions of the rotation invariant vector-valued Dirac operator. In this context, generalizations of classical orthogonal polynomials on the real line have been introduced in order to use them as building blocks for wavelets, as has been the case, e.g. for the generalized Clifford–Hermite polynomials. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focuses on simultaneous null solutions of two Hermitean Dirac operators which are invariant under the action of the unitary group. In this Hermitean setting, generalized Hermitean Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Hermitean Dirac operators. In this paper, we establish their mutual orthogonality relations w.r.t. a well-chosen weight function.

We study time-harmonic scattering in [Formula: see text] ([Formula: see text]) by a planar screen (a “crack” in the context of linear elasticity), assumed to be a non-empty bounded relatively open subset [Formula: see text] of the hyperplane [Formula: see text], on which impedance (Robin) boundary conditions are imposed. In contrast to previous studies, [Formula: see text] can have arbitrarily rough (possibly fractal) boundary. To obtain well-posedness for such [Formula: see text] we show how the standard impedance boundary value problem and its associated system of boundary integral equations must be supplemented with additional solution regularity conditions, which hold automatically when [Formula: see text] is smooth. We show that the associated system of boundary integral operators is compactly perturbed coercive in an appropriate function space setting, strengthening previous results. This permits the use of Mosco convergence to prove convergence of boundary element approximations on smoother “prefractal” screens to the limiting solution on a fractal screen. We present accompanying numerical results, validating our theoretical convergence results, for three-dimensional scattering by a Koch snowflake and a square snowflake.

Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary

Options have been developed for constructing circulant matrices of any M-sequence (MS) based on automorphic multiplicative groups of the extended Galois field, constructed using an irreducible primitive polynomial, on the basis of which the original MS is formed. The result of this approach is the identification of new methods for transforming MS circulant matrices to a matrix of Walsh functions, ordered by the powers of the antiderivative element of the field. It is shown for the first time that, depending on the initial conditions of the transformation, a set of any number of any cyclic shifts of the MP, shifted relative to each other by one symbol, can be transformed to any rows of the ordered matrix of Walsh functions, following one another. This circumstance makes it possible to simplify the MS synchronization algorithm for a known range of its cyclic shifts, especially in the case of large periods of its repetition, and also to reduce the computational complexity of the processing algorithm when working in a truncated basis of Walsh–Hadamard functions.

This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary. Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous Neumann conditions, the emphasis is placed on transparent boundary conditions, which allow the computation of the solutions in the subdomains obtained by stopping the geometric construction after a finite number of steps. The proposed methods and algorithms will be used numerically in forecoming papers.

UDC 517.9 At present, there are many articles devoted to the solvability of the boundary value problems describing deformation of static and quasistatic ideally elastoplastic bodies. Namely, solvability of variational inequalities resulting from the original (exact) statements of boundary value problems is proven in the case of quasistatic deformation for the flow-type models and in the case of static deformation for the Hencky-type models [1-7]. A peculiarity of solutions to the variational inequalities is the fact that some boundary conditions are lost. The smoothness assumption on a solution to the variational inequality does not allow us to establish the presence of the lost boundary conditions, since the set of feasible stresses is not a vector space. It was demonstrated in [8-11] that in the case of one-dimensional models of elastoplastic deformation (beam, rod, etc.) a solution to the variational inequality does not loose the boundary conditions that are involved in the original statement of the boundary value problem; i.e., all boundary conditions hold. Until recently, there were no such results for three-dimensional (two-dimensional) models of elastoplasticity. In the present article, we prove an existence theorem for the equilibrium problem for an elastoplastic plate which guarantees validity of all boundary conditions. Moreover, we consider domains with smooth boundaries as well as domains with cuts. The proof is based on a special combination of the regularization method and the penalty method. Namely, we apply elliptic regularization for solving the penalized boundary value problem that approximates the original elastoplastic problem. We obtain an existence theorem for the original problem by passing to the limit over the regularization and penalty parameters. The proposed way of regularization turns out to be rather effective and useful for analysis of other elastoplastic problems. The essence of the method consists in the fact that the regularization of the penalized equations is accompanied by special regularization of the boundary conditions. 1. A domain with smooth boundary. Suppose that fl C R 2 is a bounded domain with smooth boundary F and �9 : R 3 ---* R is a continuous convex function. The equilibrium problem for an elastoplastic plate is stated as follows [12]: Find functions w, m = {mij}, and ~ii, i,j = 1, 2, in satisfying the following equations and inequalities: -rnii,ij = f,

On the basis of A. N. Tikhonov’s regularization theory, a method is developed for solving inverse heat conduction problems of identifying a smooth outer boundary of a two-dimensional region with a known boundary condition. For this, the smooth boundary to be identified is approximated by Schoenberg’s cubic splines, as a result of which its identification is reduced to determining the unknown approximation coefficients. With known boundary and initial conditions, the body temperature will depend only on these coefficients. With the temperature expressed using the Taylor formula for two series terms and substituted into the Tikhonov functional, the problem of determining the increments of the coefficients can be reduced to solving a system of linear equations with respect to these increments. Having chosen a certain regularization parameter and a certain function describing the shape of the outer boundary as an initial approximation, one can implement an iterative process. In this process, the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of coefficients in the previous iteration and the vector of the increments of these coefficients, obtained as a result of solving a system of linear equations. Having obtained a vector of coefficients as a result of a converging iterative process, it is possible to determine the root-mean-square discrepancy between the temperature obtained and the temperature measured as a result of the experiment. It remains to select the regularization parameter in such a way that this discrepancy is within the measurement error. The method itself and the ways of its implementation are the novelty of the material presented in this paper in comparison with other authors’ approaches to the solution of geometric inverse heat conduction problems. When checking the effectiveness of using the method proposed, a number of two-dimensional test problems for bodies with a known location of the outer boundary were solved. An analysis of the influence of random measurement errors on the error in identifying the outer boundary shape is carried out.

Quasi-linear Venttsel’ problems with nonlocal boundary conditions on fractal domains

The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries

Pseudodifferential operator theory was apparently developed no later than half a century ago (Vishik and Eskin, ; Eskin, ; Mikhlin, ; Mikhlin and Prössdorf, ; Malgrange, ; Calderon and Zygmund, ; Hörmander, ; Treves, ; Taylor, ; Rempel and Schulze, ; Sternin, ), but it is not so young because its basic achievements were invented in 1960s or 1970s. The main point of this theory is a symbolic calculus for pseudodifferential operators and boundary value problems for pseudodifferential equations on manifolds with a smooth boundary. This is not so good for manifolds with a non-smooth boundary although there are many studies on this problem (see, for example, the papers of B.-W. Schulze and his colleagues). The first paper in this direction was Kondratiev’s paper of 1967, in which the general boundary value problem for a partial differential operator in a cone was studied. But in this paper, and almost all of he following studies, the conical singularity is geometrically treated as direct product of a manifold with a smooth boundary and a half-line. The author suggests another point of view (for elliptic equations in this time) related to the concept of wave factorization of elliptic symbol (Vasil’ev, ; Vasilyev, ). From this point of view the conical singularity is something indivisible, and it cannot be reduced to other cases which were studied earlier. “Curiously enough,” V.A. Kondratiev said to me (in answer to my question) “there is nothing interesting any more in the theory of boundary value problems in domains with non-smooth boundary” (December 2004). This urged the author to collect his own outlines into one but still unfinished fragment. The author hopes that in the future it will be possible to obtain a “non-smooth” version of Sternin’s results (Sternin, ) by the wave factorization method.

A two-stage fourth order time-accurate discretization for Lax–Wendroff type flow solvers II. High order numerical boundary conditions

Part 1 Chaos before chaos: frequency demultiplication, B. Van der Pol and J. Van der Mark. Part 2 Description and quantification of chaotic behaviour: geometry from time series, N.H. Packard et al. Part 3 Analytical methods: a partial differential equation with infinitely many periodic orbits - chaotic oscillations of forced beam, P. Holmes and J. Marsden. Part 4 Classical non-linear oscillators - Duffing, Van der Pol and Pendulum: universal scaling property in bifurcation structure of Duffing's and generalized Duffing's equations, S. Sato et al. Part 5 Other oscillatory systems: complex dynamics of compliant off-shore structures, J.M.T. Thompson. Part 6 Chaos in noisy systems: fluctuations and onset of chaos, B.A. Huberman and J.P. Crutchfield. Part 7 Strange non-chaotic attractors: dimensions of strange nonchaotic attractors, M. Ding et al. Part 8 Spatial chaos: chaos as a limit in a boundary value problem, C. Kahlert and O.E. Rossler. Part 9 Fractal basin boundaries: fractal basin boundaries and homoclinic orbit for periodic motion in a two-well potential, F.C. Moon and G.H. Li.

The second part of this book is organized analogously to the first one. We begin with boundary value problems in domains with isolated singularities of the boundaries, after that we study the asymptotics of the solutions of boundary value problems in domains that are perturbed near such singularities. However, in contrast to the first part we consider here general elliptic boundary value problems. The reader who is interested only in concrete problems of mathematical physics may restrict himself to a superficial reading of Chapters 3–5.KeywordsElliptic OperatorElliptic BoundaryHomogeneous ProblemCanonical SystemOperator PencilThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On an elliptic boundary value problem at double resonance