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AuthorTutschke, Wolfgang. author
TitleSolution of Initial Value Problems in Classes of Generalized Analytic Functions [electronic resource] / by Wolfgang Tutschke
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1989
Connect tohttp://dx.doi.org/10.1007/978-3-662-09943-8
Descript 188 p. online resource

SUMMARY

The purpose of the present book is to solve initial value problems in classes of generalized analytic functions as well as to explain the functional-analytic background material in detail. From the point of view of the theory of partial differential equations the book is intendยญ ed to generalize the classicalCauchy-Kovalevskayatheorem, whereas the functional-analytic background connected with the method of successive approximations and the contraction-mapping principle leads to the conยญ cept of so-called scales of Banach spaces: 1. The method of successive approximations allows to solve the initial value problem du CTf = f(t,u), (0. 1) u(O) = u , (0. 2) 0 where u = u(t) ist real o. r vector-valued. It is well-known that this method is also applicable if the function u belongs to a Banach space. A completely new situation arises if the right-hand side f(t,u) of the differential equation (0. 1) depends on a certain derivative Du of the sought function, i. e. , the differential equation (0,1) is replaced by the more general differential equation du dt = f(t,u,Du), (0. 3) There are diff. erential equations of type (0. 3) with smooth right-hand sides not possessing any solution to say nothing about the solvability of the initial value problem (0,3), (0,2), Assume, for instance, that the unknown function denoted by w is complex-valued and depends not only on the real variable t that can be interpreted as time but also on spacelike variables x and y, Then the differential equation (0


CONTENT

0. Introduction -- 1. Initial Value Problems in Banach Spaces -- 2. Scales of Banach Spaces -- 3. Solution of Initial Value Problems in Scales of Banach Spaces -- 4. The Classical Cauchy-Kovalevskaya Theorem -- 5. The Holmgren Theorem -- 6. Basic Properties of Generalized Analytic Functions -- 7. Initial Value Problems with Generalized Analytic Initial Functions -- 8. Contraction-Mapping Principles in Scales of Banach Spaces -- 9. Further Existence Theorems for Initial Value Problems in Scales of Banach Spaces -- 10. Further Uniqueness Theorems -- References


Mathematics Mathematical analysis Analysis (Mathematics) Physics Mathematics Analysis Theoretical Mathematical and Computational Physics



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