Solutions Of Cauchy Research Articles

A method of stable solution of Cauchy's problem for elliptic equations

A method of stable solution of Cauchy's problem for elliptic equations

Random Evolutions with Underlying Semi-Markov Processes

of the random evolution. Extensions of the initial definition and its uses have been completed by various authors and are reported in the surveys of Hersh [13] and Pinsky [27]. In this paper we extend the definition of random evolution to the case in which the underlying process is a semi-Markov process. We prove new representation theorems for solutions of various abstract integral equations in terms of this generalized random evolution. In particular, we use a special type of random evolution with underlying semi-Markov process to give a new representation for the solution of abstract Cauchy problems of the type treated by Griego-Hersh and, motivated through this representation, we generate new limit theorems of 'generalized- central-limit-theorem type' for the abstract Cauchy solutions. In Section one we present background results on Markov renewal processes and semi-Markov processes which we need in this development. In Section two we define the random evolution with underlying semi-Markov process and related notions; we then prove conditioning results for the random evolutions and representation theorems for solutions of abstract integral equations in terms of

A uniqueness theorem for a generalized solution of a system of two quasilinear equations in the class of bounded, measurable functions

A theorem is formulated and proved regarding the uniqueness of a generalized solution of Cauchy's for a hyperbolic system consisting of two first-order quasilinear equations with one spatial variable. Admissible solutions are bounded, measurable functions satisfying an additional condition of entropy type.

Bounds of solutions of Cauchy's problem for second-order parabolic equations independent of dimension

Bounds of solutions of Cauchy's problem for second-order parabolic equations independent of dimension

Extrapolation applied to the method of characteristics for a first order system of two partial differential equations

The proceduresivp presented in this paper calculates an approximate solution of Cauchy's initial value problem for hyperbolic systems of the form (1) (see below). The discretization which proceeds along the characteristics is performed using the midpoint rule started by Euler's method. To provide an algorithm of high accuracy the numerical solution is improved by step size extrapolation. This paper contains anAlgol program completed by examples of the use and test results.

Solution of Cauchy's functional equation on a restricted domain

Solution of Cauchy's functional equation on a restricted domain

On a method of solving the initial value problem for the wave equation

The wave equation is formally reduced to Laplace's equation by the change of variable x0 = it, where i = -1. In this paper we shall derive the well-known formula for the solution of Cauchy's problem of the wave equation from the integral representations of the solutions of Dirichlet and Neumann problems of Laplace's equation in the half-plane. Our method can be viewed as a hyperfunction-theoretic approach.

Structure of solution of Cauchy's equation for a linear differential system on a manifold

Structure of solution of Cauchy's equation for a linear differential system on a manifold

UNIQUENESS CLASSES FOR THE SOLUTION OF GOURSAT'S PROBLEM

Uniqueness classes for the solution of Goursat's problem, which consists in giving Cauchy initial conditions for each of the variables , , are studied for linear partial differential equations with constant coefficients with two groups of variables: time and space . The results obtained generalize a well-known theorem of Gel'fand and ?ilov on uniqueness classes for the solution of Cauchy's problem.Bibliography: 10 items.

Approximate solution of the differential equation 𝑦^{”}=𝑓(𝑥,𝑦) with spline functions

An approximate spline is constructed for the solution of Cauchy’s problem regarding a second-order differential equation. The existence, uniqueness and convergence of the approximate spline solution are investigated.

Absolutely continuous solutions of Cauchy’s problem foru xy =f(x, y, u, u x ,u y )

Absolutely continuous solutions of Cauchy’s problem foru xy =f(x, y, u, u x ,u y )

New Integral Representations for Solutions of Cauchy's Problem for Abstract Parabolic Equations

A study has been made of Cauchy's problem for a class of abstract parabolic differential equations. Our study is based upon a transformation that maps solutions of second-order abstract Cauchy problems into solutions of first-order. This is a preliminary report on some of the results obtained. These results include (1) new representations for solutions of abstract parabolic equations, (2) applications to the heat equation, (3) a discussion of solution representations for second-order abstract Cauchy problems, and (4) applications to nonwell posed (in the sense of Hadamard) Cauchy problems.

Approximate solution of Cauchy's problem for laplace's equation applicable to the problem of shaping of dense spatially inhomogeneous beams of charged particles: PMM vol. 35, no. 4, pp. 656–668

A solution of the problem of shaping (Sect. 1) of a number of three-dimensional beams is given in the form of asymptotic expansions. The results are compared with exact expressions which determine the shaping electrodes for a plane flow along circular trajectories (Sect. 2). From the paraxial approximation for the electrostatic beams, cases which satisfy the conditions of a full spatial charge on the emitter, without disturbing the regularity of this approximation (Sect. 3) are selected. Quasi-axially symmetric beams (Sect. 4) and a quasi-cylindrical domain of arbitrary section (Sect. 5) are considered. The hydrodynamic theory of intense beams of charged particles represents one of the branches of the mechanics of continuous medium. However, the asymptotic methods, although used widely and for a long time in other branches of mechanics, began to find application in this field only recently. The inverse problem or the problem of synthesis which appears when a system with desired characteristics is constructed, consists of two parts; the internal problem, which deals with solutions of the equations of the beam, and the external problem, connected with determining the shaping electrodes which provide the realization of the computed flow. The Cauchy problem for the Laplace equation represents the mathematical expression for the latter. For the solution of the internal problem for narrow beams the asymptotic method of the extension or the narrow strip type [4,5] have been successfully used in [1–3], although the study of the problem of shaping was complicated by the existence of singularities at the flow boundary. An approximate solution of this problem with singularities present in the Cauchy conditions based on the multiscale and factorization method, is given below.

Errors in approximate solutions of Cauchy's problem for a first-order quasilinear equation

The proximity is investigated of the solution of Cauchy's problem for the equation u t ɛ +(ϕ(uɛ))x= ɛu xx ɛ (ϕ″(uɛ) > 0) to the solution of Cauchy's problem for the equation ut+ (ϕ(u))x= 0, when the solution of the latter problem has a finite number of lines of discontinuity in the strip 0 ≤ t ≤ T. It is proved that, everywhere outside a fixed neighborhood of the lines of discontinuity, we have |uɛ−u| ≤ Ce, where the constant C is independent ofe. Similar inequalities are derived for the first derivatives of uɛ−u.

DIRECT SOLUTION OF GENERAL ONE-DIMENSIONAL LINEAR PARABOLIC EQUATION VIA AN ABSTRACT PLANCHEREL FORMULA

We use the formalism of Fourier transformation to derive a functional calculus for the generator of an abstract group. As an application we obtain a new direct solution of Cauchy's problem for one-dimensional stationary parabolic equations of arbitrary order with bounded coefficients.

Asymptotic form of the solution of cauchy's problem for an equation with slowly varying coefficients

Asymptotic form of the solution of cauchy's problem for an equation with slowly varying coefficients

Uniqueness in cauchy's problem for elliptic equations with double characteristics

This paper is concerned with the question of local uniqueness of solutions of Cauchy's problem for elliptic partial differential equations with characteristics of multiplicity not greater than two. Let P(x,~) be a homogencous,mth degree elliptic polynomial in ~=(~1,~:2 ..... ~n) with complex coefficients of class C 2 in a closed neighborhood D of the origin in Rn. For a given vector O#NERn, we define S(N) to be the class of all C m surfaces having normal N at the origin. We are interested in determining conditions on P which insure tha t solutions of Cauchy's problem for the equation P(x, D)u +Q(x, D)u = 0

On suspension flow in a porous medium

The phenomenon of suspension flow in porous medium is accompanied by the time-space change of the concentration N of solid particles suspended in dispersing fluid, as well as by the change of the concentration P of particles deposited (trapped) in porous medium. These changes are due to the trapping of suspended particles by porous medium i.e. to the, so-called, colmatage phenomenon, as well as due to the breaking away of particles of this medium i.e. to the, so-called, scouring phenomenon. The system of the equations, which describes the transport of the volume of suspended particles and the kinetics of the colmatage-scouring process, determines the functions N and P. In the equation describing the transport phenomenon there is taken into account the phenomenon of diffusion of particles suspended in dispersing fluid. The kinetics equation of the colmatage-scouring has been obtained by averaging the stochastic process (of a ‘birth-death’ type) assumed as the mathematical model of the phenomenon concerned. If the diffusion effect is neglected the mathematical model comes to a system of hyperbolic equations. There has been obtained a particular solution of this system important from the point of view of experimental investigations. In particular, the case of colmatage phenomenon due to the impulse inject of suspension is interesting. The characteristics of the system of equations mentioned above form the lines along which the discontinuities of boundary—initial conditions are propagating. The experiments show the effect of a wash-out of these discontinuities. This suggests the consideration of a system of equations in which the diffusion effect is taken into account. The consideration of this diffusion effect leads to a non-linear parabolic equation. The particular solution of this equation has been obtained for certain boundary-initial conditions. The solution of Cauchy's problem related to the colmatage-scouring phenomenon is interesting from the point of view of experimental investigations. The basic solution of this problem describes the colmatage-scouring phenomenon due to the initial condition in the form of an impulse inject.

On the theory of emersion and impulse waves. A rigorous solution, with finite energy, of the cauchy-poisson problem

In this note a rigorous solution of Cauchy and Poisson problem is reached for the case of irrotational flow of a perfect fluid with initial elevation having a finite energy. The problem is solved both in the plane surface case, in which the solution can be given in the form of known trascendentes, and in the case of axial simmetry, in which the solution is obtained by means of rapidly converging series infolds.

One-sided bounds in investigations of solutions of Cauchy's limit problem for differential equations with unbounded operators

One-sided bounds in investigations of solutions of Cauchy's limit problem for differential equations with unbounded operators