Boundary-value problem for equations with variable coefficients unsolvable with respect to the higher time derivative

In a cylindrical domain, we study the unique solvability of a boundary-value problem with data given on the whole boundary of the domain for a certain class of linear equations with partial higher-order derivatives that are unsolvable with respect to the higher time derivative with variable coefficients depending on spatial coordinates. The obtained results are transferred to the case where the equation is perturbed by a nonlinear term in the linear part.     

Spatial behaviour of solutions of the dual‐phase‐lag heat equation

In this paper we study the spatial behaviour of solutions of some problems for the dual‐phase‐lag heat equation on a semi‐infinite cylinder. The theory of dual‐phase‐lag heat conduction leads to a hyperbolic partial differential equation with a third derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary‐value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial‐time lines. A class of non‐standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T are assumed proportional to their initial values. Copyright © 2004 John Wiley & Sons, Ltd.     

Variational problems in the theory of optimum processes

This paper presents a review of the optimization problems for control processes described by ordinary differential equations and of the variational methods for solving these problems. The following cases are studied: problems with constraints on the controls or the coordinates, problems described by equations with discontinuous right-hand sides, problems with functionals depending on intermediate coordinates, and problems with given discontinuities in the coordinates. Variational problems of synthesis of optimal systems are also discussed. The method of solution is based on the multiplier rule and the Weierstrass necessary condition for the strong minimum of a functional. In some cases, the Legendre-Clebsch necessary condition for the weak minimum of a functional is used.     

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which in the case of the Benjamin–Ono equation, are the mean, a spatial phase, a temporal phase, and the real part of one of the Fourier modes at t=0.     

Time-Variable Extension of the Solution of a Nonlocal Multipoint Problem for Partial Differential Equations with Constant Coefficients

A problem with nonlocal multipoint conditions for the nth-order partial differential equation with constant coefficients is considered. In the case where conditions of strict averaging of time intervals are specified, the existence of a solution of the problem in a cylinder that is the Cartesian product of a time interval and a p-dimensional spatial torus is discussed. It is found that under certain conditions of separability of the roots of the characteristic equation for almost all (in the sense of the Lebesgue measure) coefficients of the equation and parameters of the conditions, the solution of the problem cannot be extended in the time variable beyond the extreme points at which the conditions are given.     

Existence and stability of travelling waves in fixed-bed reactors

The model equations of the catalytic fixed-bed reactor often possess solutions in the form of travelling wave fronts similar to the well-known case of Fisher's equation. The mathematical investigation of these waves requires searching for solutions of singular boundary value problems in the phase plane or in the three-dimensional phase space. In this paper necesary and sufficient conditions are derived which are to be satisfied by the model parameters and the propagation velocity of the wave front if wave solutions exist. Moreover, sufficient conditions for the asymptotic stability of these solutions are proved where the perturbations are supposed to belong to a certain weighted L²-space. Finally, the connection between the initial distribution of the state variable and the velocity of the wave is discussed.     

On an algorithm for tracking the motion of the reference system with aftereffect when only part of the coordinates is measured

We consider the problem of tracking a given trajectory by a system described by an equation with aftereffect. We suggest an algorithm, stable to information noise and numerical errors, for solving this problem in the case of incomplete information on the phase trajectory (measurement of part of the coordinates). The algorithm is based on the dynamic inversion and guaranteed control method.     

Single‐cell fourth‐order difference approximations for $${\font\twelvesl=cmsl10\partial {\sl\bf u}\over\partial {\sl\bf x}},{\partial \sl\bf u\over\partial\sl\bf y}$$ , and $${\font\twelvesl=cmsl10\partial {\sl\bf u}\over\partial{\sl\bf z}}$$ of the three‐dimensional quasi‐linear elliptic equation

Finite difference methods of O(h⁴) are proposed for obtaining estimates of first‐order partial derivatives of the solution of three‐dimensional quasi‐linear elliptic equation with mixed derivative terms subject to Dirichlet boundary conditions on a uniform cubic grid. In all the cases, we use a single computational cell and the methods are applicable to the problems both in cartesian and polar coordinates. The utility of the new methods is shown by testing the methods on three‐dimensional poisson solvers in polar coordinates. Some numerical examples are provided to demonstrate the accuracy and efficiency of the methods discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 417–425, 2000     

On a new method for the study of the spatial behavior in a homogeneous elastic arch-like region

We consider a two-dimensional homogeneous elastic state in the arch-like region a?≤?r?≤?b, 0?≤?θ?≤?α, where (r,θ) denotes plane polar coordinates. We assume that three of the edges are traction-free, while the fourth edge is subjected to a (in plane) self-equilibrated load. The Airy stress function ‘?’ satisfies a fourth-order differential equation in the plane polar coordinates with appropriate boundary conditions. We develop a method which allows us to treat in a unitary way the two problems corresponding to the self-equilibrated loads distributed on the straight and curved edges of the region. In fact, we introduce an appropriate change for the variable r and for the Airy stress functions to reduce the corresponding boundary value problem to a simpler one which allows us to indicate an appropriate measure of the solution valuable for both the types of boundary value problems. In terms of such measures we are able to establish some spatial estimates describing the spatial behavior of the Airy stress function. In particular, our spatial decay estimates prove a clear relationship with the Saint-Venant's principle on such regions.     

Global stress regularity of convex and some nonconvex variational problems

We derive a global regularity theorem for stress fields which correspond to minimizers of convex and some special nonconvex variational problems with mixed boundary conditions on admissible domains. These are Lipschitz domains satisfying additional geometric conditions near those points, where the type of the boundary conditions changes. In the first part it is assumed that the energy densities defining the variational problem are convex but not necessarily strictly convex and satisfy a convexity inequality. The regularity result for this case is derived with a difference quotient technique. In the second part the regularity results are carried over from the convex case to special nonconvex variational problems taking advantage of the relation between nonconvex variational problems and the corresponding (quasi-) convexified problems. The results are applied amongst others to the variational problems for linear elasticity, the p-Laplace operator, Hencky elasto-plasticity with linear hardening and for scalar and vectorial two-well potentials (compatible case).      

Local existence and uniqueness theorems for a free boundary problem

Free boundary problems are considered, where the tangential and normal components u_t and u_n of an otherwise unknown plane harmonic vector field are prescribed along the unknown boundary curve as a function of the coordinates x, y and the tangent angle θ. The vector field is required to exist either in the interior region G⁺ or in the exterior G^?. In each case the free boundary is characterized by a nonlinear integral equation. A linearised version of this equation is a one-dimensional singular integral equation. Under rather general hypotheses which are easy to check, the properties of the linear equation are described by Noether's theorems. The regularity of the solution is studied and the effect of the nonlinear terms is estimated. A variant of the Nash-Moser implicit-function theorem can be applied. This yields local existence and uniqueness theorems for the free boundary problem in Hölder-classes H^2+μ. The boundary curve depends continuously on the defining data. Finally some examples are given, where the linearised equation can be completely discussed.     

Single cell finite difference approximations of O(kh2 + h4) for ∂u/∂x for one space dimensional nonlinear parabolic equation

We report a new two‐level explicit finite difference method of O(kh² + h⁴) using three spatial grid points for the numerical solution of for the solution of one‐space dimensional nonlinear parabolic partial differential equation subject to appropriate initial and Dirichlet boundary conditions. The method is shown to be unconditionally stable when applied to a linear equation. The proposed method is applicable to the problems both in cartesian and polar coordinates. Numerical examples are provided to demonstrate the efficiency and accuracy of the method discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 408–415, 2000     

Inverse problems for elliptic equations in the space: I

We consider inverse problems of finding the right-hand side of a linear secondorder elliptic equation of general form. We study the first boundary value problem. Two ways of representation of the additional information (over-determination) are considered: specification of the trace of the solution of the boundary value problem inside the domain on some manifold of smaller dimension and specification of values of the normal derivative on a part of the boundary. The Fredholm alternative is proved for the considered problems. Stability estimates are given for the case of unique solvability. The analysis is performed in classes of continuous functions whose derivatives satisfy the Hölder condition.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

A Fast Algorithm for Two-Dimensional Elliptic Problems

In this paper, we extend the work of Daripa et al. [14–16,7] to a larger class of elliptic problems in a variety of domains. In particular, analysis-based fast algorithms to solve inhomogeneous elliptic equations of three different types in three different two-dimensional domains are derived. Dirichlet, Neumann and mixed boundary value problems are treated in all these cases. Three different domains considered are: (i) interior of a circle, (ii) exterior of a circle, and (iii) circular annulus. Three different types of elliptic problems considered are: (i) Poisson equation, (ii) Helmholtz equation (oscillatory case), and (iii) Helmholtz equation (monotone case). These algorithms are derived from an exact formula for the solution of a large class of elliptic equations (where the coefficients of the equation do not depend on the polar angle when written in polar coordinates) based on Fourier series expansion and a one-dimensional ordinary differential equation. The performance of these algorithms is illustrated for several of these problems. Numerical results are presented.     

Asymptotics of the density matrix for a system of a large number of identical particles

The Wigner equation is considered for a system of a large numberN of identical particles with interaction factor of the order of 1/N. In both the Bose and the Fermi cases, we construct the asymptotics of the solution of the Cauchy problem for this equation with regard to the exchange effect for the case in which the Planck constant is of the order ofN ^−1/d , whered is the space dimension. This asymptotics is interpreted in terms of the theory of the complex germ on a curved phase space equivalent to the space of functions with values on the Riemann sphere in the Fermi case and on the Lobachevskii plane in the Bose case. The classical equations of motion in both cases are reduced to the Vlasov equation; since the phase space is infinite-dimensional, the complex germ is subjected to additional conditions depending on the type of statistics. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 84–106, January, 1999.     

Analytic invariant curves for a planar mapping

This paper is concerned with the existence of analytic invariant curves for a planar mapping. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. In this paper, we discuss not only the general case, but also the critical cases as well, in particular, the case where β is a unit root is discussed.     

Some Properties of Minimal Splines

S. G. Mikhlin was the first to construct systematically coordinate functions on an equidistant grid solving a system of approximate equations (called “fundamental relations”, see [5]; Goel discussed some special cases earlier in 1969; see also [1, 4, 6]). Further, the idea was developed in the case of irregular grids (which may have finite accumulation points, see [1] ). This paper is devoted to the investigation of A-minimal splines, introduced by the author; they include polynomial minimal splines which have been discussed earlier. Using the idea mentioned above, we give necessary and sufficient conditions for existence, uniqueness and g-continuity of these splines. The application of these results to polynomial splines of m-th degree on an equidistant grid leads us, in particular, to necessary and sufficient conditions for the continuity of their i-th derivative (i = 1, ?, m). These conditions do not exclude discontinuities of other derivatives (e.g. of order less than i). This allows us to give a certain classification of minimal spline spaces. It turns out that the spline classes are in one-to-one-correspondence with certain planes contained in a hyperplane.     

Lax pairs and a new spectral method for linear and integrable nonlinear PDEs

A new transform method for solving initial-boundary value problems for linear and integrable nonlinear PDEs in two independent variables has been recently introduced in [1]. For linear PDEs this method involves: (a) formulating the given PDE as the compatibility condition of two linear equations which, by analogy with the nonlinear theory, we call a Lax pair; (b) formulating a classical mathematical problem, the so-called Riemann-Hilbert problem, by performing a simultaneous spectral analysis of both equations defining the Lax pair; (c) deriving certain global relations satisfied by the boundary values of the solution of the given PDE. Here this method is used to solve certain problems for the heat equation, the linearized Korteweg-deVries equation and the Laplace equation. Some of these problems illustrate that the new method can be effectively used for problems with complicated boundary conditions such as changing type as well as nonseparable boundary conditions. It is shown that for simple boundary conditions the global relations (c) can be analyzed using only algebraic manipulations, while for complicated boundary conditions, one needs to solve an additional Riemann-Hilbert problem. The relationship of this problem with the classical Wiener-Hopf technique is pointed out. The extension of the above results to integrable nonlinear equations is also discussed. In particular, the Korteweg-deVries equation in the quarter plane is linearized.     

Inverse problems for a quasilinear hyperbolic equation in the case of moving observation point

We consider two inverse coefficient problems for a quasilinear hyperbolic equation, where the additional information used for finding the coefficients is the values of the solution on some curve. (This corresponds to measurements performed at a moving observation point.) The unknown coefficient depends on the space variable in the first inverse problem and on the solution of the equation in the second inverse problem. We prove theorems of uniqueness of solution to the inverse problems.