Some differential and integral equations with applications to Toeplitz operators

One of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex place C is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators defined on the harmonic Bergman space.

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

A Toeplitz operator with respect to a contractive representation of an abelian semigroup in a Hilbert space H is an operator such that (Error rendering LaTeX formula) for all . We show that if has a minimal isometric dilation , then Toeplitz operators can be obtained in a unique way as compressions of operators , called Toeplitz symbols, such that (Error rendering LaTeX formula) . This approach to lifting the Toeplitz equation (Error rendering LaTeX formula) is shown to be unitarily equivalent to the one proposed by Muhly in 1972. We use our approach to extend to this case a number of theorems about classical Toeplitz operators and, finally, we show that some classes of well-known operators like Wiener-Hopf operators Toeplitz operators in or Toeplitz operators in the sense of Murphy fall within the class studied in this paper. The main objective of this paper is to provide a general framework that, we hope, will be useful in order to extend to wider classes of operators some of the more recent and deep advances in the theory of Toeplitz operators.

may be reduced to a single linear integral equation whose kernel is defined on a?x ? a+n(b-a), a_s <a+n(b-a). Greggi? considered a system of the form (1) and by use of the transformation introduced by Fredholm showed the form of the resolvent matrix for the system; for the symmetric system where KKj(x; s) = Ksi(s; x) (i, j = 1, 2, , n) he also stated theorems analogous to those proved by Schmidtll for a single integral equation with symmetric kernel. System (1) also comes under the class of systems treated by Platrier.? Weatherburn** has treated the system (1) without using the transformation introduced by Fredholm, but by vector method throughout. He states all results for the case n = 3, but his method of procedure is equally applicable to the general case. In the present paper a special system of the form (1), to which is applied the term self-adjoint, is considered and the existence of a countable infinity of real characteristic numbers is established, together with expansion theorems in terms of the characteristic solutions of the system of integral equations. A definitely self-adjoint system of integral equations includes as a special case the symmetric system with closed matrix kernel. It also includes the system of integral equations to which a boundary value problem for a system of ordinary linear differential equations of the first order which

In a recent paper (L. Greengard and V. Rokhlin, On the Numerical Solution of Two‐Point Boundary Value Problems. in Communications on Pure and Applied Mathematics, Volume XLIV, 1991, pages 419‐452). L. Greengard and V. Rokhlin introduce a numerical technique for the rapid solution of integral equations resulting from linear two‐point boundary value problems for second‐order ordinary differential equations. In this paper, we extend the method to systems of ordinary differential equations. After reducing the system of differential equations to a system of second kind integral equations, we discretize the latter via a high‐order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O(N.p2.n3) operations with N the number of nodes on the interval, p the desired order of convergence, and n the number of equations in the system. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to endpoint singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods.We in addition present a Newton method for solving boundary value problems for nonlinear first‐order systems in which each Newton iterate is the solution of a second kind integral equation; the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems. © 1994 John Wiley &amp; Sons. Inc.

One of the methods for studying the solvability of infinite systems of integral or differential equations is the application of various fixed-point theorems to operators acting in appropriate functional Banach spaces. This method is fairly well developed, frequently used, and effective in many situations. However, there are cases in which certain infinite systems of differential equations arise—linked to the modeling of significant real-world phenomena—where this method, based on situating considerations within Banach spaces, fails and cannot be applied. In this paper, we propose a slightly different approach, which involves conducting the analysis within appropriate functional Fréchet spaces. We discuss the fundamental properties of these spaces and formulate compactness criteria. The main result of this paper is a positive answer, using the proposed method, to an open problem concerning the modeling of a stochastic birth-and-death process, as formulated in one of the cited publications. The most important conclusion is that the presented computational technique, based on functional Fréchet spaces, can be regarded as a more effective alternative to methods based on Banach spaces.

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda &amp;amp; Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda &amp;amp; Erdogan (Konda &amp;amp; Erdogan 1994).

Fuchsian systems of ordinary differential equations that are irreducible and free from accessory parameters are classified into eight classes. Canonical systems of these classes were determined by Haraoka [SIAM J. Math. Anal., 25 (1994), pp. 1203--1226]. Among them the canonical systems of exactly four classes are extended to the completely integrable systems of total differential equations of two variables that are introduced by Yokoyama [Funkcial. Ekvac., 35 (1992), pp. 65--93]. This paper presents monodromy groups of the four systems of total differential equations. The theory for computing monodromy groups of the systems developed by Yokoyama is applied.

The paper deals with a numerical algorithm to reduce the overall system of integral equations describing the heat transfer process at any geometrically complex area (both twodimensional and three-dimensional), to the iterative solution of a system of independent onedimensional integral equations. This approach has been called "string method" and has been used to solve a number of applications, including the problem of the detonation wave front for the calculation of heat loads in pulse detonation engines. In this approach "the strings" are a set of limited segments parallel to the coordinate axes, into which the whole solving area is divided (similar to the way the strings are arranged in a tennis racket). Unlike other grid methods where often for finding solutions, the values of the desired function in the region located around a specific central point here in each iteration step is determined by the solution throughout the length of the one-dimensional "string", which connects the two end points and set them values and determine the temperature distribution along all the strings in the first step of an iterative procedure.

Applications of the Lanczos method

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.

The purpose of this paper is to derive a means of scaling the dependent variables of a system of linear differential equations so that the normalized system accurately reflects the dependence of the system on initial data. The theory is developed for the special case of a system of first order hyperbolic partial differential equations in two independent variables. Several other systems of differential equations, to which the same normalization could be applied, are also indicated. The scaling is equivalent to row and column scaling commonly used in linear algebra. For the case of positive matrices (that is, for the case where the diagonalizing matrix and its inverse have no zero elements), the theory for the “1” and “$\infty $ ” norms has been thoroughly developed by Bauer [1]. The assumption of positive matrices is too stringent for some applications. Thus, the theory is generalized to show that for irreducible matrices a best scaling exists (for both the “1” and “ $\infty $” norms), while for simply reducible matrices a best scaling exists for at least one of these two norms. Several examples are given.

The basic system of integral and differential equations is obtained that describes the evolution of a liquid-liquid interface in inhomogeneous ground layers such that the square root of their conductivity is described by a metaharmonic function. A discrete scheme for the basic system of equations is constructed using the method of discrete vortex pairs.

In the previous chapters we explored varieties of periodic traits of linear difference equations, piece-wise difference equations, rational difference equations, and Max-Type difference equations. First of all, we observed how the periodicity nature varied from the existence of unique periodic cycles to every solution being periodic. Second, we discovered the necessary and sufficient conditions for the existence of periodic cycles. Furthermore, we perceived the disparity between patterns of the even ordered periodic cycles and odd ordered periodic cycles with uniqueness of solutions and alternating patterns. Our next aim is to broaden our developed knowledge to ascertain the periodicity essence of third and higher order difference equations, systems of difference equations, when the periodic sequence \(\{A_{n}\}_{n=0}^{\infty }\) is periodic with period-k, (k ≥ 3) and study of new systems of piecewise difference equations and neural networking models. How will these criteria for the existence of periodic cycles be similar or different compared to our current mastery in previous chapters? What new periodic traits can we expect that will be quite different? How will the delay m ≥ 3 affect the new periodic traits? Will even and odd values of delay m ≥ 3 influence the new periodic traits? We will pose conjectures that will lead to new theorems and results based on our previous studies and will pose new research questions.

The initial attached cell layer in multispecies biofilm growth is studied in this paper. The corresponding mathematical model leads to discuss a free boundary problem for a system of nonlinear hyperbolic partial differential equations, where the initial biofilm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biofilm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass flux from the bulk liquid. The differential equations are converted into an equivalent system of Volterra integral equations. We use Newton-Raphson method to solve the nonlinear system of Volterra integral equations (SVIEs) of the second kind. This method converts the nonlinear system of integral equations into a linear integral equation at each step.