Cauchy problem

Now, the averaged hyperbolic model, Eq. (52), defines a characteristic initial-value problem (Cauchy problem). To complete the model, we need to specify Cm only along the characteristic curves x = 0 and f — 0. Thus, the initial and boundary conditions for the averaged model are obtained by taking the mixing-cup averages of Eqs. (31) and (32) ... 

Other possibilities exist to solve the frame invariant problem Cauchy-Maxwell equation uses the Cauchy tensor, C, which is also independent of the system of reference, the Lodge rubber-like liquid model uses the Finger tensor but contrarily to the Lodge model, it uses a generalized memory function ... 

Problem—Show that d2F/da2 > 0 so that Eq. (4-159) actually minimizes F. Hint the Cauchy inequality can be used to show that... 

The second-order difference equations. The Cauchy problem. Boundary-value problems. The second-order difference equation transforms into a more transparent form... 

It is necessary to specify two conditions for the complete posing of this or that problem. The assigned values of y and Ay suit us perfectly and lie in the background a widespread classification which will be used in the sequel. When equation (6) is put together with the values yi and A yi given at one point, they are referred to as the Cauchy problem. Combination of two conditions at different nonneighboring points with equation (6) leads to a boundary-value problem. 

In the case of the Cauchy problem with assigned values y and y, we have at our disposal the system of algebraic equations for constants Cj and... 

We claim that Ik is just the solution of the Cauchy problem for the second-order difference equation ... 

Example 1. The Cauchy problem for an ordinary differential equation ... 

The Cauchy problem for a system of differential equations of first order. Stability condition for Euler s scheme. We illustrate those ideas with concern of the Cauchy problem for the system of differential equations of first order... 

Of special interest is Euler s scheme for the Cauchy problem... 

We must show that Green s function specified in such a way exists and find its explicit representation similar to expression (4). With this aim, the functions a, and (3 will be declared to be solutions of the corresponding Cauchy problems... 

Explicit schemes for the Cauchy problem. The first-order equation... 

A Cauchy problem is said to be stable with respect to the initial data and right-hand side if... 

In conformity with the superposition principle ( is a linear operator), the stability of the Cauchy problem with respect to the right-hand side follows from the uniform stability with respect to the initial data... 

As a matter of fact, we will consider the set of solutions ykri )] of Cauchy problem (4) dependent on the input data 2/o/> -... 

Inequality (12) expresses the property of continuous dependence which is uniform in h and t of the Cauchy problem (4) upon the input data. Here and below the meaning of this property is stability. A difference scheme is said to be absolutely stable if it is stable for any r and h (not only for all sufficiently small ones). It is fairly common to distinguish the notion of stability with respect to the initial data and that with respect to the right-hand side. Scheme (4) is said to be stable with respect to the initial data if a solution to the homogeneous equation... 

In what follows the Cauchy problem (1) is supposed to be solvable,... 

Example 1 The Cauchy problem, being the most familiar one, comes first ... 

Here Go(a ,t) is a function of the heat source of the Cauchy problem associated with the one-dimensional heat conduction equation... 

The second reduction of the Cauchy problem. On the whole segment tj [Pg.625]

This means that the system of differential equations (55)-(56) generates an approximation of order 1 in a summarized sense to the Cauchy problem (51) under the extra restrictions on the existence and boundedness of the derivative A t)cPu/dt in some suitable norm. 

Is it possible or not to improve the accuracy in r without essential modifications of the composite Cauchy problem In an attempt to give a definite answer to this question, the composite Cauchy problem (55) is designated by the symbolism... 

The solution of Eq. (2.6) for infinite interval and delta-shaped initial distribution (2.8) is called the fundamental solution of Cauchy problem. If the initial value of the Markov process is not fixed, but distributed with the probability density Wo(x), then this probability density should be taken as the initial condition ... 

In terms of nonlinear dynamical systems, the second waveguide of the junction can be considered as a system that is initially more or less far from its stable point. The global dynamics of the system is directly related to the spatial transfomation of the total field behind the plane of junction. In structure A, the initial linear mode transforms into a nonlinear mode of the waveguide with the same width and refractive index. In the structure B, the initial filed distribution corresponds to a nonlinear mode of the first waveguide it differs from nonlinear mode of the second waveguide, however. The dynamics in both cases is complicated and involves nonlinear modes as well as radiation. Global dynamics of a non-integrable system usually requires numerical simulations. For the junctions, the Cauchy problem also cannot be solved analytically. 

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