Solutions Of Cauchy Research Articles

On the connection between the Runge-Kutta method and Picard's method

The Runge-Kutta method for the numerical solution of Cauchy's problem for a system of ordinary differential equations has an obvious iterative character. As demonstrated In this paper, this phenomenon arises from the connection between the Runge-Kutta method and Picard's iterative method. The estimation of error for the Runge-Kutta method is based on this connection.

On the approximate solution of Cauchy's problem for ordinary differential equations to a given number of correct figures. II

On the approximate solution of Cauchy's problem for ordinary differential equations to a given number of correct figures. II

The numerical solution of Cauchy's problem for linear second order equations of the parabolic type

Boundary value problems for linear elliptic and parabolic differential equations represent a familiar field of application for Monte Carlo methods. The means for realizing a method of this kind which are described in [1] utilize the random closeness to a fixed lattice covering the region at the boundary of which the conditions are specified. Another means of possible interest is based on the well-known close connection of the problems under discussion with random processes of a diffusion type. Since this latter approach appears not to have been employed as yet for computational purposes, a description of it follows below, as applied, for the sake of definiteness, to the Cauchy problem for linear parabolic equations.

On the number of operations performed in the solution of Cauchy' s problem for one hyperbolic equation

On the number of operations performed in the solution of Cauchy' s problem for one hyperbolic equation

On an algorithm for the solution of Cauchy's problem for partial differential equations of the first order

AN EFFICIENT and simple algorithm for the solution of Cauchy's problem for systems of partial differential equations with analytical coefficients is proposed in this paper.

On a method for the numerical solution of Cauchy's problem for divergent systems

This article is concerned with the question of the numerical solution of Cauchy's problem for divergent hyperbolic systems of equations with partial derivatives of the first order: ∂u ∂t + ∂ϑ (u, x, t) ∂x =ψ (u, x, t) , u ( x, 0) = u 0( x), where u, ϑ, ψ are n-dimensional vectors. Without any essential loss of generality we shall concentrate our attention on the homogeneous systems of the form ∂u ∂t + ∂ϑ (u) ∂x =0 , As usual, we look for the generalised solutions of the problem (3), (2), i.e. for piece-wise continuous functions u( x, t) satisfying the initial condition (2) and the system of integral conservation laws: ▪ which is a consequence of system (3) and is equivalent to it in the class of smooth coefficients ( C is an arbitrary piece-wise smooth contour in the ( x, t) plane). The method we suggest consists in reducing the problem (4), (2) to a problem with initial data for some system of equations which we shall call equations containing the trajectory density. The numerical calculation consists of the approximate construction of the trajectories and the calculation of the values of the function u( x, t) along them. The satisfactory results of numerical experiments indicate the possibility of applying the method to through calculation across the lines of discontinuity of the solution. It is not possible at present to consider this method as theoretically justified to a sufficient extent: there are no proofs of the existence and uniqueness of the solution of the basic problem, and questions connected with the transition from problem (3), (2) to the new problem have not been completely solved. The aim of this work is to throw light on the fundamentals of the method and to draw certain conclusions about its use.

A method for the numerical solution of the problem of the flow about conical bodies in a supersonic stream of gas

In this paper we examine a method for the approximate solution of the boundary value problem of the flow about smooth conical bodies in a supersonic stream of an ideal gas, and also discuss the results of some calculations. The inverse method is based on the multiple solution of Cauchy' s problem with successive correction of the analytically given initial data until the boundary conditions on the body are satisfied to a given accuracy. The results of the calculations are compared with those obtained by other methods [1], [2] and experimentally [3], [4].

On the approximate solution of Cauchy's problem for ordinary differential equations to a given number of correct figures

On the approximate solution of Cauchy's problem for ordinary differential equations to a given number of correct figures

The asymptote to the numerical solution of Cauchy's problem for one quasi-linear equation of the first order

The asymptote to the numerical solution of Cauchy's problem for one quasi-linear equation of the first order

Green's Distributions and the Cauchy Problem for the Iterated Klein-Gordon Operator

An explicit form of the homogeneous Green's function for the multi-dimensional iterated Klein-Gordon operator is obtained. By a direct calculation from its Fourier representation, the Green's function is expressed as a one-dimensional, infinite integral of the Sonine type. Although this integral is classically divergent when the order of the operator is less than the number of space dimensions, it can be treated rigorously under these conditions using the concepts of distribution analysis. A generalized Sonine integral is developed and the result applied to obtaining an explicit expression for the Green's function, which is now to be regarded as a distribution in the sense of Schwartz. Using a distribution introduced for this purpose, the Green's function is written in a form which explicitly displays its singularities on the light cone. The well-known difference between even- and odd-dimensional spaces is reflected in the nature of these singularities. The singularities appearing for an odd number of space dimensions consist of a finite linear combination of derivatives of the Dirac delta function δ(s2) where s is the space-time distance. The highest derivative appearing is of order ½(n − 2l − 1) with n giving the number of space dimensions and 2l giving the order of the operator. The singular part for even-dimensional spaces consists of a polynomial in 1/s of degree n − 2l + 1. No singularities appear when the order of the operator is greater than the number of dimensions. The general solution of Cauchy's problem for the iterated Klein-Gordon operator is obtained in convolution form. An explicit solution for the ordinary Klein-Gordon equation is presented in a form which exhibits separately the contributions due to the singular part and the regular part of the Green's function.

The solution of Cauchy's problem for a system of quasi-linear equations in many independent variables

The solution of Cauchy's problem for a system of quasi-linear equations in many independent variables

Solution of cauchy’s problem for the wave equation $$\left( {\frac{{\partial ^2 }}{{\partial t^2 }} - \nabla ^2 + k^2 } \right)\psi = 0$$

A method given by Copson has been generalised here to obtain a solution of Cauchy’s problem for the wave equation ∂2 u/∂t 2−∇2 u+k 2 u=0 in any odd number of spatial dimensions. The method does not involve the use of any device for evaluating the divergent integrals.

Cauchy's problem and the problem of a piston for one-dimensional, non-steady motions of gas (automodel motion)

Considered here is the automodel problem for the equations of one-dimensional, non-steady motions of ideal non-heat-conducting perfect gas (Cauchy's problem) and the problem of automodel motions of gas with properties mentioned above, created by symmetrical expanding piston contained in the gas. Automodel presentation of the first problem was first given by Sedov [1]. Special cases of this problem were considered by Sedov [2] (problem concerning the focusing of gas in one point and about its expansion from a point) and Kompaneetz [4]. The second problem, for the case where the velocity of the piston is constant, was discussed by Sedov [2]. The case when the piston velocity is a power function of time was dealt with by Krasheninnikova [3]. In this article we show that: (1) For certain given initial functions, the solution of Cauchy's problem cannot be continued for all instants of time t. (2) Under certain values of the exponent in the formula for piston velocity, the solution for the second problem does not exist.

The solution of Cauchy’s problem for a third-order linear hyperoblic differential equation by means of Riesz integral

The solution of Cauchy’s problem for a third-order linear hyperoblic differential equation by means of Riesz integral

Errata: The Solution of Cauchy's Problem for Two Totally Hyperbolic Linear Differential Equations by Means of Riesz Integrals

Errata: The Solution of Cauchy's Problem for Two Totally Hyperbolic Linear Differential Equations by Means of Riesz Integrals

The Solution of Cauchy's Problem for Two Totally Hyperbolic Linear Differential Equations by Means of Riesz Integrals

The Solution of Cauchy's Problem for Two Totally Hyperbolic Linear Differential Equations by Means of Riesz Integrals

On the Existence and Uniqueness of the Solution of Cauchy's Problem for a System of Two First Order Partial Differential Equations

On the Existence and Uniqueness of the Solution of Cauchy's Problem for a System of Two First Order Partial Differential Equations

The solution of Cauchy's Problem for linear partial differential equations, with constant coefficients, by means of integrals involving complex variables

The object of this paper is to show how the theory of integrals involving complex variables may be applied to the integration of linear partial differential equations, possessing real, distinct characteristics and constant coefficients. The problem considered is a Cauchy problem (with analytic data)—typical of the equation of real characteristics and the method taken is that of Riemann. For simplicity of exposition, the second order hyperbolic equation is considered, but the results are given in such a form as to indicate an obvious generalisation to equations of higher order.