On the complexity of parabolic initial-value problems with variable drift

We study the intrinsic difficulty of solving linear parabolic initial-value problems numerically at a single point. We present a worst-case analysis for deterministic as well as for randomized (or Monte Carlo) algorithms, assuming that the drift coefficients and the potential vary in given function spaces. We use fundamental solutions (parametrix method) for equations with unbounded coefficients to relate the initial-value problem to multivariate integration and weighted approximation problems. Hereby we derive lower and upper bounds for the minimal errors. The upper bounds are achieved by algorithms that use Smolyak formulas and, in the randomized case, variance reduction. We apply our general results to equations with coefficients from Hölder classes, and here, in many cases, the upper and lower bounds almost coincide and our algorithms are almost optimal.     

A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations

We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations.     

Two Simple Derivations of Universal Bounds for the C.B.S. Inequality Constant

Universal bounds for the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for piecewise linear-linear and piecewise quadratic-linear finite element spaces in 2 space dimensions are derived. The bounds hold for arbitrary shaped triangles, or equivalently, arbitrary matrix coefficients for both the scalar diffusion problems and the elasticity theory equations.     

The Rate of Convergence of Finite-Difference Approximations for Bellman Equations with Lipschitz Coefficients

We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order h^1/2 for certain types of finite-difference schemes are obtained.     

Unique continuation for second-order parabolic operators at the initial time

We consider second-order parabolic equations with time independent coefficients. Under reasonable assumptions, it is known that the fundamental solution satisfies certain Gaussian bounds related to the associated geodesic distance. In this article we prove a sharp unique continuation property at the initial time which matches exactly the above-mentioned kernel bounds.

     

On the optimum choice of weight functions in a class of variational calculations

Summary We consider the variational solution of a particular class of second order differential equations and show that expansions in terms of Chebychev and a range of ultraspherical polynomials lead to operator matrices that are asymptotically diagonal, and that hence their convergence properties can be completely characterised using a previously developed analysis. For a given class of weight functions bounds are given on the convergence of the coefficients and of the weighted mean square error, in terms of the analyticity properties of the coefficients in the differential equation. These bounds are used to discuss the optimum choice of weight function for such a calculation.     

Positivity and stabilisation for nonlinear stochastic delay differential equations

We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero.

We do not require that any component of the coefficients of the equations satisfy Lipschitz conditions. Instead, we require that the functional part of each coefficient which feeds back the present state of the process admit to bounds imposed by a member of a particular class of concave functions. Lipschitz conditions are included as a special case of these bounds.

We generalise these results to the finite dimensional case, also constructing perturbations that can destabilise the otherwise stable solutions of a deterministic system of equations.     

Explicit bounds for second-order difference equations and a solution to a question of Stevi?

This note gives explicit, applicable bounds for solutions of a wide class of second-order difference equations with nonconstant coefficients. Among the applications is an affirmative answer to a recent question of Stevi?.     

Estimates for the wandering rate of solutions of a linear differential equation via its coefficients

For nontrivial solutions of a linear nonautonomous differential equation with integrally small coefficients, we improve earlier-known upper bounds for the wandering rate. In particular, our estimates imply that the upper bound of the range of the wandering rate for equations of arbitrary order tends to zero as all of their coefficients uniformly (on the time half-line) tend to zero at infinity.     

Improved bounds for some nonstandard problems in generalized heat conduction

We consider the Maxwell-Cattaneo system of equations for generalized heat conduction where the temperature and heat flux satisfy a nonstandard auxiliary condition which prescribes a combination of their values initially and at a later time. We obtain L₂ bounds for the temperature and heat flux by means of Lagrange identities. These bounds extend the range of validity for the parameter in the nonstandard condition under a constraint on the coefficients in the differential equations.     

The Rate of Convergence of Finite-Difference Approximations for Parabolic Bellman Equations with Lipschitz Coefficients in Cylindrical Domains

We consider degenerate parabolic and elliptic fully nonlinear Bellman equations with Lipschitz coefficients in domains. Error bounds of order h^1/2 in the sup norm for certain types of finite-difference schemes are obtained.     

On solutions of algebraic differential equations whose coefficients are entire functions of finite order

Summary We determine bounds for the growth of entire solutions of first order equations whose coefficients are entire functions of finite order. Entrata in Redazione il giorno 8 gennaio 1969. This research was supported in part by the National Science Foundation (GP 7374).     

Elliptic operators and maximal regularity on periodic little-H?lder spaces

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-H?lder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions.     

Comparison theorems for diffusion processes

We prove comparison theorems for diffusion processes onR ^d. From these theorems we derive lower and upper bounds for the transition probabilities of a diffusion process. In contrast to the known estimates for fundamental solutions of parabolic equations our bounds do not depend on the moduli of continuity of the coefficients of the differential operator.     

An improvement of the complexity bound for solving systems of polynomial equations

In 1984, the author suggested an algorithm for solving systems of polynomial equations. Now we modify it and improve the bounds on its complexity as well as the degrees and lengths of coefficients from the ground filed of the elements constructed by this algorithm. Bibliography: 4 titles.     

The convergence rate of continued fractions representing solutions of a Riccati equation

The aim of this note is to generalize and apply results on matrix continued fractions representing the solution of discrete matrix Riccati equations. Assuming uniform bounds for the norm of the matrix coefficients of the continued fraction, the minimal and maximal solutions of the corresponding algebraic Riccati equation can be accurately enclosed.     

A new efficient DQ algorithm for the solution of elliptic problems in higher dimensions

Two new efficient algorithms are developed to approximate the derivatives of sufficiently smooth functions. The new techniques are based on differential quadrature method with quartic B-spline bases as test functions. To obtain the weighting coefficients of differential quadrature method (DQM), we use the midpoints of a uniform partition mixed with near-boundary grid points. This enables us to obtain the weighting coefficients without adding the new extra relations. By obtaining the error bounds, it is proved that the method in its classic form is non-optimal. Then, some new weighting coefficients are constructed to obtain higher accuracy. By obtaining the error bounds, it is proved that the new algorithm is superconvergent. Afterwards, by defining some new symbols, we find a way to approximate the partial derivatives of multivariate functions. Also, some approximations are constructed to the mixed derivatives of multivariate functions. Finally, the applicability of the methods is examined by solving some well-known problems of partial differential equations. Some examples of 2D and 3D biharmonic, Poisson, and convection-diffusion equations are solved and compared to the existing methods to show the efficiency of the proposed algorithms.     

On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions

We prove the existence of positive solutions of second-order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions. Obtained results yield positive periodic solutions of the equation on the whole real axis, provided that the coefficients are periodic.     

EXPONENTIAL GROWTH AND OSCILLATION OF NONLINEAR HOMOGENEOUS DIFFERENCEEQUATIONS

1IntroductionConsiderthenonlinearhomogeneousdifferenceequationfwhereashi>0,I--1,''5m,qandrarequotientsofpositiveoddintegers,p20,r=p q,a13'',amarepositiveintegerswithal"<''相似文献   

Nonlocal boundary-value problems for systems on linear partial differential equations

We study the classical well-posedness of problems with nonlocal two-point conditions for typeless systems of linear partial differential equations with variable coefficients in a cylindrical domain. We prove metric theorems on lower bounds for small denominators that appear in the construction of solutions of such problems.