FIO-product representation of solutions to symmetrizable hyperbolic systems

The solution operator of a symmetrizable hyperbolic system is approximated by a multi-product of Fourier integral operators. Exact convergence is obtained as we let the number of operators in the product grow to infinity, which yields an alternative representation of the solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems

An associative commutative algebra of distributions that contains homogeneous and associated homogeneous distributions is constructed. This algebra is used to analyze generalized solutions to strictly hyperbolic partial differential equations. Possible types of singularities are studied and the necessary (analogues of Hugoniót conditions for shock waves) and sufficient conditions for the existence of such solutions are obtained. This paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matermaticheskaya Fizika, Vol. 114, No. 1, pp. 3–55, January, 1998.     

Propagation of the singularities of solutions of initial-boundary-value problems for first-order hyperbolic systems

One proves a certain variant of the lemma on the finite velocity of the propagation of singularities for the solutions of first order hyperbolic systems. This result has been already used in a series of papers on the spectral asymptotics of elliptic operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 138–149, 1985.     

Generalized solutions to nonlinear first-order systems

In the present work we reinterpret a result of approximate solutions to a nonlinear first order system in the framework of Colombeau's theory, defining an algebra of generalized germs where the problem $$u_t = f(x,t,u,u_x ), u|_{t = 0} = u_o (x)$$ has a unique solution, wheref andu ₀ are vector-valued functions.     

Global analytic solutions to hyperbolic systems

The aim of this paper is to extend to some classes of systems the global existence of analytic solutions to scalar equations of Kirchhoff type. Received March 17, 1997.     

Generalized solutions to semilinear hyperbolic systems

In this article semilinear hyperbolic first order systems in two variables are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that these systems admit unique global solutions in the Colombeau algebraG(²). In particular, this provides unique generalized solutions for arbitrary distributions as initial data. The solution inG(²) is shown to be consistent with the locally integrable or the distributional solutions, when they exist.     

BREAKDOWN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C1 solution to Cauchy problem.     

Discontinuous solutions to hyperbolic systems under operator splitting

Two-dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time-steps exactly solves the component one-dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to “pseudo acoustic waves” and a nonphysical stationary wave. The implications of these findings for the design of high-resolution computing schemes are discussed.     

Exact boundary observability for nonautonomous first-order quasilinear hyperbolic systems

By means of the theory on the semiglobal C¹ solution to the mixed initial-boundary value problem for first-order quasilinear hyperbolic systems, we establish the local exact boundary observability for general nonautonomous first-order quasilinear hyperbolic systems without zero eigenvalues and reveal the essential difference between nonautonomous hyperbolic systems and autonomous hyperbolic systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Representations of the solutions of the first-order elliptic and hyperbolic systems via harmonic and wave functions respectively

ABSTRACT

In this paper representations of the solutions of all first-order elliptic and hyperbolic systems in three-dimensional and four-dimensional spaces are obtained through the derivatives of harmonic and wave functions, respectively.     

Blow-up of periodic solutions to reducible quasilinear hyperbolic systems

Under general assumption, we prove that the smooth solution to reducible quasilinear hyperbolic system with periodic initial data in general develop singularities in finite time, provided that the total variation on one period of the initial data is small.     

Simple digital quantum algorithm for symmetric first-order linear hyperbolic systems

Numerical Algorithms - This paper is devoted to the derivation of a digital quantum algorithm for the Cauchy problem for symmetric first-order linear hyperbolic systems, thanks to the reservoir...