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ON MIXED BOUNDARY VALUE PROBLEM FOR HYPERBOLIC SYSTEMS

ON MIXED BOUNDARY VALUE PROBLEM FOR HYPERBOLIC SYSTEMS DEMONSTRATIO MATHEMATICAVol. VNo 31973Jozef GrzegorczykON MIXED BOUNDARY VALUE PROBLEMFOR HYPERBOLIC SYSTEMS1. IntroductionIn the present paper the conditions of uniqueness andexistence of the solutions of mixed boundary value problemfor hyperbolic systems (of partial differential equations ofsecond order) are discussed. The obtained results generalizetheorems proved by O.A.Ladyzhenskaya in [1] to the case ofhyperbolic systems. Notation used throughout the paper andformulation of the problem are given in Section 2. Uniquenessis proved in Section 5, whereas the existence of the solutionand a method of its construction are considered in Section2. Formulation of the problemWe shall use the following notations: x = (xQ,x^,...,xm)is a point of (m+1) - dimensional space, where x Q is tiime-coordinate. The remaining space coordinates are brieflydenoted by x = (x1,... .x^) e h"1. Let a be an open boundedregion in ^ and let 2Q. be its boundary. We denote by Q == £i x (0,a) the set of points x = (xQ,x) and by S its lateralsurface.The system of differential equations considered in thispaper is of the formmmA u - u _ „ = F(x)x xAo oi(x)N+ao%+Ji.j-1i'1^ (x)is a vector function.where u(x) =u n (x)l 4 ^oV+r- 119 -2J.GrzegorczykThe matrix of scalar coefficients a ij( x ) is symmetrical,a.•L.(x)= http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON MIXED BOUNDARY VALUE PROBLEM FOR HYPERBOLIC SYSTEMS

Grzegorczyk, Jozef
Demonstratio Mathematica , Volume 5 (3): 24 – Jul 1, 1973
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Publisher
de Gruyter
Copyright
© by Jozef Grzegorczyk
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1973-0305
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. VNo 31973Jozef GrzegorczykON MIXED BOUNDARY VALUE PROBLEMFOR HYPERBOLIC SYSTEMS1. IntroductionIn the present paper the conditions of uniqueness andexistence of the solutions of mixed boundary value problemfor hyperbolic systems (of partial differential equations ofsecond order) are discussed. The obtained results generalizetheorems proved by O.A.Ladyzhenskaya in [1] to the case ofhyperbolic systems. Notation used throughout the paper andformulation of the problem are given in Section 2. Uniquenessis proved in Section 5, whereas the existence of the solutionand a method of its construction are considered in Section2. Formulation of the problemWe shall use the following notations: x = (xQ,x^,...,xm)is a point of (m+1) - dimensional space, where x Q is tiime-coordinate. The remaining space coordinates are brieflydenoted by x = (x1,... .x^) e h"1. Let a be an open boundedregion in ^ and let 2Q. be its boundary. We denote by Q == £i x (0,a) the set of points x = (xQ,x) and by S its lateralsurface.The system of differential equations considered in thispaper is of the formmmA u - u _ „ = F(x)x xAo oi(x)N+ao%+Ji.j-1i'1^ (x)is a vector function.where u(x) =u n (x)l 4 ^oV+r- 119 -2J.GrzegorczykThe matrix of scalar coefficients a ij( x ) is symmetrical,a.•L.(x)=

Journal

Demonstratio Mathematica – de Gruyter

Published: Jul 1, 1973

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