Cauchy Initial Value Research Articles

On the Existence of a Solution to the Cauchy Initial Boundary Value Problem

On the Existence of a Solution to the Cauchy Initial Boundary Value Problem

A nonlinear boundary problem by the inventory’s optimization of a productive system with not wholly quadratic costs

In this paper a costs functional is assumed asymmetrical for the production and the inventory charges: As a state variable, the deviation u(t) = Iid(t)− Ire(t) between the inventories has been chosen. Obviously the manufacturer wishes to keep the cost-functional C(u) as a minimum. The relevant Euler-Lagrange time-equation ü(t) =α2u(t) + 2u(t)3, is faced with the initial condition u(0) = u0 ≠ 0 ; and with the final condition u(T) = 0 meaning that after a time T > 0 the deviation u(t) is re-absorbed at all. In the article the above nonlinear boundary value problem is solved through a much more tractable nonlinear (Cauchy) initial value problem, plus a “shooting”. The integration is carried out by means of the Jacobi elliptic functions, which do here their appearance in a micro-economic context for the first time. After having so detected the “optimal” deviation time law u(t) , a new production plan is then designed for driving the deviation till to its extinction within T .

Space-time means and solutions to a class of nonlinear parabolic equations

Cauchy problem and initial boundary value problem for nonlinear parabolic equation inCB([0,T):L p ) orL q (0,T; L p ) type space are considered. Similar to wave equation and dispersive wave equation, the space-time means for linear parabolic equation are shown and a series of nonlinear estimates for some nonlinear functions are obtained by space-time means. By Banach fixed point principle and usual iterative technique a local mild solution of Cauchy problem or IBV problem is constructed for a class of nonlinear parabolic equations inCB([0,T);L p orL q (0,T; L p ) with ϕ(x)∈L r . In critical nonlinear case it is also proved thatT can be taken as infinity provided that ||ϕ(x)||r is sufficiently small, where (p,q,r) is an admissible triple.

Generalized harmonic gauge conditions in General Relativity as field equations for lapse and shift

AbstractThe prescription of the harmonicity quantities to a spacetime metric may be read, in the ADM decomposition, as a system of evolution equations. Cauchy's initial value problem for this system is investigated here. Additionally, the differential‐geometric substance behind the ADM scheme is analysed.

Extrapolation applied to certain discretization methods solving the initial value problem for hyperbolic differential equations

The procedurediwiex presented in this paper provides an approximate solution to Cauchy's initial value problem for general hyperbolic systems of first order. The procedurecharex can be applied to the initial value problem for a hyperbolic system of quasi-linear differential equations. This second method is a kind of method of characteristics. It produces a solution for the whole domain of determinancy. Both procedures use extrapolation to the limit.

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.

Stability of flat thermal flux in a slab reactor

Using the qualitative theory of differential equations the equivalence of the boundary value and the Cauchy's initial value problem is demonstrated for the slab flat flux reactor. It is found that there exists an interval of the initial (central) fuel concentrations for which the initial value problem (with two algebraic conditions) describes a critical flat flux reactor while solutions outside this interval describe undercritical systems of a nonmultiplying medium.

Riemann representation method in viscoelasticity II. Cauchy's initial value problem

Riemann representation method in viscoelasticity II. Cauchy's initial value problem