A note concerning the integral equations for magnetostatic fields

Zeitschrift für angewandte Mathematik und Physik -     

Optimal integrability of some system of integral equations

We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in R^n ：{u（x）=1/｜x｜^α｜∫R^n v（y）^q｜y｜^β｜x-y｜^λdy,v（x）=1/｜x｜^β∫R^n u（y）^p｜y｜^α｜x-y｜^λdy.C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26： 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q 〉 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q ≥1 and pq ≠1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when ｜x｜ →0 and when ｜x｜→∞.     

Optimal control of certain hyperbolic and integral stochastic equations

Using Gateaux differentiation of the quality functional we obtain necessary conditions for optimality of a control for stochastic differential equations of hyperbolic type containing two-parameter white noise and for stochastic integral equations.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 110–116, 1987.     

Optimal control of Volterra integral equations via triangular functions

This paper presented an approximate method for solving optimal control problem of Volterra integral equations. The method is based upon orthogonal triangular functions. The error estimates and associated theorems have been proved for optimal control and cost functionals. Some numerical examples illustrate the efficiency of the proposed method.     

Variational-iterative method for integral equations

We study applications of the variational-iterative method to nonlinear integral equations with potential strongly monotone and Lipschitz-continuous operators and investigate its rate of convergence for special systems of coordinate functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1626–1635, December, 1990.     

Boundary arcs for integral equations

Behavior of boundary arcs for control systems is investigated when the systems are governed by integral equations of the Volterra type. The main result is in the form of a maximum principle. This result is then used to obtain necessary conditions for a minimum control problem.     

Optimal methods for specifying information in the solution of integral equations with analytic kernels

We determine the exact order of the minimum radius of information in the logarithmic scale for Fredholm integral equations of the second kind with periodic analytic kernels and free terms. We show that the information complexity of the solution of Fredholm equations with analytic kernels is greater in order than the complexity of the approximation of analytic functions. This distinguishes the analytic case from the case of finite smoothness.     

Liouville type theorems for integral equations and integral systems

In this paper, we establish some Liouville type theorems for positive solutions of some integral equations and integral systems in R ^N . The main technique we use is the method of moving planes in an integral form.     

An integral basis for algebraic fields

Let A be a principal ideal domain, K be the quotient field of A, and let L be a cubic extension of K. In this paper we establish the existence of a special type of integral basis of the field L over K which is a generalization of the integral basis of Voronoi for cubic extensions of the field Q of rational numbers.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 229–234, February, 1973.     

Volterra integral equations and evolution equations with an integral condition

We suggest a simple method for reducing problems with an integral condition for evolution equations to a Volterra integral equation of the first kind. For Volterra equations of the convolution type, we indicate necessary and sufficient solvability conditions for the case in which the right-hand side lies in some classes of functions of finite smoothness. We use these conditions to construct examples of nonexistence of a local solution for the heat equation with an integral condition.     

Smoothed integral equations

For a linear integral equation there is a resolvent equation and a variation of parameters formula . It is assumed that B is a perturbed convex function and that a(t) may be badly behaved in several ways. When the first two equations are treated separately by means of a Liapunov functional, restrictive conditions are required separately on a(t) and B(t,s). Here, we treat them as a single equation where S is an integral combination of a(t) and B(t,s). There are two distinct advantages. First, possibly bad behavior of a(t) is smoothed. Next, properties of S needed in the Liapunov functional can be obtained from an array of properties of a(t) and B(t,s) yielding considerable flexibility not seen in standard treatment. The results are used to treat nonlinear perturbation problems. Moreover, the function is shown to converge pointwise and in L²[0,∞) to x(t).     

Influence coefficients for variational integral equations

We compute exact formulas for the influence coefficients deriving from the finite element discretization of integral equation methods. We consider the case of the Newtonian potential and plane triangles of the lower degree. To cite this article: M. Lenoir, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

An embedding theorem for integral equations

In this paper, an embedding theorem is established for a system of nonlinear integral equations of the Volterra type. The main result is basic in the development of a maximum principle for an optimal control problem in which the state variables are determined as solutions to integral equations.     

Numerical solutions for functional integral equations

Convergence criteria are given for a family of numerical methodsfor functional integral equations with delay. Several numericalmethods are compared.     

Volterra integral equations

One presents a survey of the investigations in the theory of Volterra integral equations, reviewed in Ref. Zh. Mat. between 1966–1976.Translated from Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 15, pp. 131–198, 1977.