An oscillation theory for second-order integral differential equations

The primary purpose of this paper is to give an oscillation theory for second-order integral differential equations. It is shown that this theory follows in a natural way as “a corollary” from the more abstract approximation theory of quadratic forms given previously by the author. Thus, our ideas are primarily constructive and quantitative as opposed to the usual qualitative methods. We also note that the usual oscillation theory for second-order differential equations follows directly by our methods. Furthermore, our methods provide a unified theory for eigenvalue problems, optimization problems, and numerical approximation problems within this setting.In Section 1 we give the preliminaries for the remainder of the paper. In Section 2 we define the basic quadratic form and integral differential equation and give the relationships between them. These relationships are used (in Section 3) to give a theory of oscillation in our setting and some basic oscillation results. Finally, in Section 4 we give some deeper oscillation results.To emphasize the unifying methods of our ideas, this paper is presented as a companion paper to “A Numerical Approximation Theory for Second Order Integral Differential Equations.”     

Elliptic quadratic forms,focal points,and a generalized theory of oscillation

The purpose of this paper is to generalize the theory, methods, and results for oscillation of second-order normal ordinary differential equations. This purpose is obtained by use of a theory of quadratic forms on Hilbert spaces given by Hestenes and the author.In particular, the ideas of this paper may be applied to second-order abnormal problems of differential equations, higher-order control problems, integral and partial differential equations, abstract approximation problems, and to finite dimensional approximations which lead to meaningful computer algorithms.For expository purposes some examples are included. Finally we show that specific existence and comparison theorems for the second-order case may be generalized to the 2nth-order case.     

Approximate inverses of multidiagonal matrices and application to the block PCG method

A fast method to compute high-order approximate inverses based on truncated elimination is constructed for multidiagonal matrices of diagonal dominance. Together with the block preconditioned conjugate gradient method, it can be used for the numerical solution of elliptic partial differential equations and related problems. Through numerical experiments it is shown that the method is robust and efficient.     

A review of infinite matrices and their applications

Infinite matrices, the forerunner and a main constituent of many branches of classical mathematics (infinite quadratic forms, integral equations, differential equations, etc.) and of the modern operator theory, is revisited to demonstrate its deep influence on the development of many branches of mathematics, classical and modern, replete with applications. This review does not claim to be exhaustive, but attempts to present research by the authors in a variety of applications. These include the theory of infinite and related finite matrices, such as sections or truncations and their relationship to the linear operator theory on separable and sequence spaces. Matrices considered here have special structures like diagonal dominance, tridiagonal, sign distributions, etc. and are frequently nonsingular. Moreover, diagonally dominant finite and infinite matrices occur largely in numerical solutions of elliptic partial differential equations.The main focus is the theoretical and computational aspects concerning infinite linear algebraic and differential systems, using techniques like conformal mapping, iterations, truncations etc. to derive estimates based solutions. Particular attention is paid to computable precise error estimates, and explicit lower and upper bounds. Topics include Bessel’s, Mathieu equations, viscous fluid flow, simply and doubly connected regions, digital dynamics, eigenvalues of the Laplacian, etc. Also presented are results in generalized inverses and semi-infinite linear programming.     

An efficient method for solving difference systems for elliptic differential equations

A method for solving systems of linear algebraic equations arising in connection with the approximation of boundary value problems for elliptic partial differential equations is proposed. This method belongs to the class of conjugate directions method applied to a preliminary transformed system of equations. A model example is used to explain the idea underlying this method and to investigate it. Results of numerical experiments that confirm the method’s efficiency are discussed.     

Coiflet interpolation and approximate solutions of elliptic partial differential equations

In this article, we prove a higher order interpolation result for square–integrable functions by using generalized coiflets. Convergence of approximation by using generalized coiflets is shown. Applications to wavelet–Galerkin approximation of elliptic partial differential equations and some numerical examples are also given. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13:303–320, 1997.     

A numerical approximation theory for second-order integral differential equations

This paper is a companion paper to “An Oscillation Theory for Second-Order Integral Differential Equations.” The underlying theme is that both topic (oscillation theory and numerical oscillation theory) follow as a corollary to an approximation theory of quadratic forms given previously by the author. In Section I we give the mathematical preliminaries; some of which are not included in the earlier paper. This includes the relationship between the fundamental quadratic form and its integral differential equation (the Euler-Lagrange equations). In Section II the approximating quadratic forms are defined on the approximating Hilbert space. In Section III we show that our approximating hypothesis are satisfied and give the fundamental inequality relationships [Eqs. (12) and (13)]. We also show that the mth oscillation point is a continuous function of our approximating parameter. Finally in Section IV we show how that the approximating indices may be easily obtained by computer algorithms.     

Analyticity of solutions of analytic non-linear general elliptic boundary value problems, and some results about linear problems

The main aim of this paper is to discuss the problem concerning the analyticity of the solutions of analytic non-linear elliptic boundary value problems. It is proved that if the corresponding first variation is regular in Lopatinskiĭ sense, then the solution is analytic up to the boundary. The method of proof really covers the case that the corresponding first variation is regularly elliptic in the sense of Douglis-Nirenberg-Volevich, and hence completely generalize the previous result of C. B. Morrey. The author also discusses linear elliptic boundary value problems for systems of elliptic partial differential equations where the boundary operators are allowed to have singular integral operators as their coefficients. Combining the standard Fourier transform technique with analytic continuation argument, the author constructs the Poisson and Green’s kernel matrices related to the problems discussed and hence obtain some representation formulae to the solutions. Some a priori estimates of Schauder type and L _p type are obtained. __________ Translated from Acta Sci. Natur. Univ. Jilin, 1963, (2): 403–447 by GAO Wenjie.     

The use of second degree normalized implicit conjugate gradient methods for solving large sparse systems of linear equations

Second degree normalized implicit conjugate gradient methods for the numerical solution of self-adjoint elliptic partial differential equations are developed. A proposal for the selection of certain values of the iteration parameters ?_i, γ_i involved in solving two and three-dimensional elliptic boundary-value problems leading to substantial savings in computational work is presented. Experimental results for model problems are given.     

Quasi-periodic solutions of nonlinear elliptic partial differential equations

In a recent paper [9] the KAM theory has been extended to non-linear partial differential equations, to construct quasi-periodic solutions. In this article this theory is illustrated with three typical examples: an elliptic partial differential equation, an ordinary differential equation and a difference equation related to monotone twist mappings.     

Order invariant properties of some matrices

Certain important Toeplitz and composite Toeplitz matrices have order invariant properties. A class of matrices is defined, and a notation for sign patterns of vector and matrix elements is developed which enables some order invariant properties of the matrices to be described. Several examples are given to show how these ideas apply to some important matrices, including those commonly arising from the numerical solution of elliptic partial differential equations. The following information about the inverse of these matrices is obtained: the signs of the elements, the row in which the maximum row sum occurs, and the signs of the elements of the eigenvector corresponding to its dominant eigenvalue.     

Exact solutions of the Euler equations for some two-dimensional incompressible flows

Two-dimensional (plane and axisymmetric) steady flows of an ideal incompressible fluid are considered in a potential field of external forces. An elliptic partial differential equation is obtained such that each of its solutions is a stream function of a flow described by a certain solution of the Euler equations. Examples of such new exact solutions are given. These solutions can be used, in particular, for testing numerical algorithms and computer programs.     

NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

In this article, we describe on a state of the art of validated numerical computations for solutions of differential equations. A brief overview of the main techniques for self-validating numerics for initial and boundary value problems in ordinary and partial differential equations including eigenvalue problems will be presented. A fairly detailed introductions are given for the author's own method related to second-order elliptic boundary value problems. Many references which seem to be useful for readers are supplied at the end of the article.     

The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass

Some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications. In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. The properties of Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

A superconvergent linear FE approximation for the solution of an elliptic system of PDEs

The aim of this work is to study a nonstandard piecewise linear finite element method for elliptic systems of partial differential equations. This nonstandard method was considered by the authors for scalar elliptic equations and for a planar elasticity problem. The method enables us to compute a superconvergent numerical approximation to the solution of the system of partial differential equations.     

Discretizations of Linear Elliptic Partial Differential Inclusions

Differential inclusions provide a suitable framework for modelling choice and uncertainty. In finite dimensions, the theory of ordinary differential inclusions and their numerical approximations is well-developed, whereas little is known for partial differential inclusions, which are the deterministic counterparts of stochastic partial differential equations.

The aim of this article is to analyze strategies for the numerical approximation of the solution set of a linear elliptic partial differential inclusion. The geometry of its solution set is studied, numerical methods are proposed, and error estimates are provided.     

A generalized conjugate gradient algorithm for solving a class of quadratic programming problems

In this paper we apply matrix splitting techniques and a conjugate gradient algorithm to the problem of minimizing a convex quadratic form subject to upper and lower bounds on the variables. This method exploits sparsity structure in the matrix of the quadratic form. Choices of the splitting operator are discussed, and convergence results are established. We present the results of numerical experiments showing the effectiveness of the algorithm on free boundary problems for elliptic partial differential equations, and we give comparisons with other algorithms.     

Singularities of multivalued solutions of nonlinear differential equations,and nonlinear phenomena

The purpose of this paper is to use the geometrical theory of nonlinear partial differential equations and the theory of singularities of maps in order to obtain the general scheme for constructing shock waves from multivalued solutions, given by smooth integral manifolds. This scheme is illustrated by some examples from gas dynamics, mechanics, acoustics and thermodynamics.     

Asymptotics in a neighborhood of infinity of solutions with finite Dirichlet integral of second-order elliptic equations

In this paper we answer a question posed by E. Sanchez-Palencia which arose in the theory of homogenization of differential operators. The asymptotic behavior of solutions at infinity which have finite Dirichlet integral is studied and uniqueness theorems are also proved for exterior boundary problems for second-order elliptic equations in divergent form.Translated from Trudy Seminara im. I. G. Petrovskogo, No. 12, pp. 149–163, 1987.