Integral equations as evolution equations

We study a class of time-dependent linear integrodifferential equations (VE) with the evolution equation approach. We determine the generators of a time-dependent evolution equation (DE) which is equivalent to the given integrodifferential equation. Under very general assumptions we prove the well-posedness and continuity of (VE) from the stability of (DE). The related question of convergence of a family of approximate solutions is examined. As an application, we include an example of hyperbolic integro-partial-differential equation to illustrate the theory.     

Convolution equations and nonlinear functional equations

The survey is devoted to applications of nonlinear integral equations to linear convolution equations, their discrete analogues, and also the connection of these equations with problems of radiative transfer, in particular, with the Ambartsumyan equations.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 175–244, 1984.     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.     

Some nonlinear equations reducible to diffusion-type equations

The problem of the reducibility of a system of second-order quasi-linear parabolic differential equations to diffusion-type equations is considered. An effective solution algorithm is suggested for this problem in the nondegenerate case. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 26–37, April, 2000.     

Second-order type-changing evolution equations with first-order intermediate equations

This paper presents a partial classification for C^∞ type-changing symplectic Monge-Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems.     

Some differences between difference equations and differential equations

In this expository paper we discuss some instances in which analogues which might be expected between the behavior of solutions of differential equations and difference equations fail to hold, focussing particularly on questions related to boundedness and oscillation.     

Dendriform equations

We investigate solutions for a particular class of linear equations in dendriform algebras. Motivations as well as several applications are provided. The latter follow naturally from the intimate link between dendriform algebras and Rota–Baxter operators, e.g. the Riemann integral map or Jackson's q-integral.     

Transport equations

We study questions of the general theory and solution methods for linear transport problems in a homogenous half-space and in a plane layer of finite thickness. We propose developing a factorization interpretation of the operator approach and consider the nonlinear factorization equations and generalized Ambartsumian equation in combination with the possibilities of two analytic semigroups.     

Set equations

This note discusses some set equations, using set matrices, i.e. matrices whose entries are sets.     

Differential equations

Book Reviews

Differential equationsC. M. Dafermos, G. Ladas, and G. Papanicolaou (editors): Lecture Notes in Pure and Applied Mathematics 118, Marcel Dekker, New York, Basel, 1989, xiv + 787 pp.     

Transforming algebraic Riccati equations into unilateral quadratic matrix equations

The problem of reducing an algebraic Riccati equation XCX − AX − XD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind PX ² + QX + R = 0 is analyzed. New transformations are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm (SDA) of Anderson (Int J Control 28(2):295–306, 1978) is in fact the cyclic reduction algorithm of Hockney (J Assoc Comput Mach 12:95–113, 1965) and Buzbee et al. (SIAM J Numer Anal 7:627–656, 1970), applied to a suitable UQME. A new algorithm obtained by complementing our transformations with the shrink-and-shift technique of Ramaswami is presented. The new algorithm is accurate and much faster than SDA when applied to some examples concerning fluid queue models.     

Backward stochastic differential equations associated with the vorticity equations

In this paper we derive a non-linear version of the Feynman–Kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions. We prove the existence (global in time) and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2. A particular class of terminal values provide, via these probabilistic methods, solutions for the vorticity equation.     

Integrability of Riccati equations and stationary Korteweg-de vries equations

By using the Lie infinitesimal method, we establish the correspondence between the integrability of a one-parameter family of Riccati equations and the hierarchy of the higher Korteweg-de Vries equations. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal. Vol. 51, No. 6, pp. 856–860, June, 1999.