Generalized Cauchy Problem Research Articles

The Fourier method for nonsmooth initial data

Application of the Fourier method to very general linear hyperbolic Cauchy problems having nonsmooth initial data is considered, both theoretically and computationally. In the absence of smoothing, the Fourier method will, in general, be globally inaccurate, and perhaps unstable. Two main results are proven: the first shows that appropriate smoothing techniques applied to the equation gives stability; and the second states that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a very small easily characterized exceptional set. A particular implementation of the smoothing method is discussed; and the results of its application to several test problems are presented, and compared with solutions obtained without smoothing.

Average boundary conditions in Cauchy problems

We consider the general Cauchy problem with initial data in a Hilbert space and with a formal dissipative linear generator. A complete parametrization is known of the (abstract) boundary conditions which make this problem well set. We exhibit a distinguished subset BE of the set B of boundary conditions and demonstrate explicitly that the evolution associated with each B in B can be represented as a (time independent) average over the evolutions associated with B′ in BE. Applications are discussed to Schrödinger equations in bounded regions or with singular potentials.

Approximate solution of a generalized Cauchy problem by the method of averaging functional corrections

Approximate solution of a generalized Cauchy problem by the method of averaging functional corrections

The construction of some solutions of the generalized Cauchy problem for nonlinear ordinary differential equations

The construction of some solutions of the generalized Cauchy problem for nonlinear ordinary differential equations

Existence of a null Chaplygin approximation for a solution of the generalized Cauchy problem

In this paper two theorems are proved on the existence of a null Chaplygin approximation for a solution of the generalized Cauchy problem, from which the solvability of this problem in the class of continuous functions follows.

THE STABILIZATION OF SOLUTIONS OF THE GENERALIZED CAUCHY PROBLEM FOR ULTRAPARABOLIC EQUATIONS

By using an integral representation of the solution of the generalized Cauchy problem for ultraparabolic equations, a necessary and sufficient condition for the stabilization of the solution has been obtained for the class of positive initial functionals. In the class of bounded with respect to translation functionals, it has been proved that a necessary and sufficient condition for the stabilization of the solution in a weak sense is the existence of a generalized spherical limiting mean.