Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients

Summary I give a sufficient condition in order that a Dirichlet problem is solvable in H ² (Ω) for a class of linear second order elliptic partial differential equations. Such a class includes some particular cases for which the result is known.

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Sunto Si prova una condizione sufficiente affinchè un problema di Dirichlet sia risolubile in H ² (Ω) per una classe di equazioni differenziali alle derivate parziali lineari ellittiche del secondo ordine. Tale classe comprende alcuni casi particolari per i quali il risultato è noto.

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The present work was written while the author was a member of the ? Centro di Matematica e Fisica Teorica del C.N.R. ? at the University of Genova, directed by professorJ. Cecconi.

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Entrata in Redazione il 25 febbraio 1971.     

Finding numerical solution of linear partial differential equations of first order with real coefficients

Applying Bittner's operational calculus we present a method to give approximate solutions of linear partial differential equations of first order

PERTURBATION RESULTS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

Two perturbation results on the linear differential function f" ∏(z)A(z)f = 0 areobtained, where ∏(z) and A(z) are periodic entire functions with period 2πi and σe(∏) ＜σe(A).     

Stability criterion for second order linear impulsive differential equations with periodic coefficients

In this paper we obtain instability and stability criteria for second order linear impulsive differential equations with periodic coefficients. Further, a Lyapunov type inequality is also established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.     

Reflected backward doubly stochastic differential equations with discontinuous coefficients

In this paper, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous (left or right continuous) generator. We obtain an existence theorem and a comparison theorem for solutions of the class of RBDSDEs.     

Difference methods for stochastic differential equations with discontinuous coefficients

In this paper it is proved that the solution of the stochastic (vector)-differential equation dx(t) = a(t,x(t))dt + b(t,x(t)) dw(t) can be approximated in a weak sense (convergence of distributions) by the Euler method x$sub:n + l$esub:=x$sub:n$esub: + a(t$sub:n$esub:,x$sub:n$esub:)ßt + b(t$sub:n$esub:,x$sub:n$esub:)w$sub:n$esub:, where the w$sub:n$esub:are independent, normally distributed random variables with mean zero and variance ßt, if the coefficients are Lipschitz-continuous outside of a finite set of switching curves and b is a uniformly positive definite matrix.     

Stability properties of solutions of linear second order differential equations with random coefficients

The equation

     

高阶亚纯系数非齐次线性微分方程的复振荡

对一类高阶亚纯系数非齐次线性微分方程的复振荡进行了研究,应用值分布的理论和方法,得到了它的亚纯解的超级,二级不同零点收剑指数的精确估计.     

Pathology of infinite systems of linear first order differential equations with constant coefficients

Summary The properties of solutions of finite systems are analyzed and it is shown by examples how these properties may be distorted and ultimately lost in passing from finite to infinite systems. To Enrico Bompiani on his Scientific Jubilee This research was supported in part by United States Army through its Office of Ordnance Research under Grant DA-ORD-12. — Most of the results have been presented to various organisations such as Section A of the American Association for the Advancement of Science and mathematics colloquia at the universities of Leningrad, Moscow, and Toronto.     

On the supports of solutions of linear partial differential equations with analytic coefficients

Sunto In questo lavoro vengono dimostrati, con un metodo basato sui funzionali analitici, il teorema di Holmgren e varie sue estensioni. Il punto essenziale è che nelle dimostrazioni sono usate soltanto transformazioni lineari. Così è messo in luce che la variabile ? tempo ? ha un ruolo completamente diverso dal ruolo delle variabili ? spaziali ? per l'unicità del problema lineare di Cauchy. Viene anche studiato il problema globale di Cauchy in un semispazio.

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Entrato in Redazione il 1° aprile 1971.     

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.     

Second order abstract differential equations with singular coefficients

Summary We study a large class of second order linear abstract differential equations, whose coefficients can be singular. In the framework of suitable « weighted » spaces, we prove some existence and uniqueness results for generalized and ordinary solutions of initial value problems for such equations.This work was partially supported by the G.N.A.F.A. and the Istituto di Analisi Numerica of the C.N.R. (Italy).     

Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients

Conservative linear equations arise in many areas of application, including continuum mechanics or high-frequency geometrical optics approximations. This kind of equation admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finite-difference numerical schemes and introduce an appropriate concept of consistency with the continuous problem. Some basic examples including computational results are also supplied.

     

Asymptotic behavior of solutions of second order parabolic partial differential equations with unbounded coefficients

Classical solutions of a second order parabolic partial differential equation are considered in unbounded domains. The coefficients are allowed to have unbounded growth as the space variables tend to infinity. A Phragmèn-Lindelöf principle is proved for such equations. That principle is used together with comparison functions to derive sufficient conditions for the asymptotic decay in time of solutions.     

Convergence of finite element method for linear second‐order wave equations with discontinuous coefficients

A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L^∞(L²) and L^∞(H¹) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L^∞(H¹) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

First boundary-value problem for second-order elliptic-parabolic equations with discontinuous coefficients

In this paper, the first boundary value problem for the second-order degenerated ellipticparabolic equations in nondivergent form is considered. Unique strong (almost everywhere) solvability of this problem in the corresponding weight Sobolev space is proved.     

On stochastic partial differential equations with Unbounded coefficients

Second-order stochastic partial differential equations of parabolic type are considered. Generalizations of known theorems on existence, uniqueness and on approximations are presented. Thus, in particular the case of unbounded coefficients in investigated. Some examples illustrating the usefulness of the results are also given.