The Wave Equation with Memory
In this chapter, we study wave front asymptotics of solutions of wave equations with memory. Sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, and 3.9 are devoted to the one-dimensional case. In Sects. 3.10 and 3.11, we deal with the case of two and three
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Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity
Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity
Alexander A. Lokshin
Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity
Alexander A. Lokshin Department of Mathematics and Informatics in Primary School Moscow Pedagogical University Moscow, Russia
ISBN 978-981-15-8577-7 ISBN 978-981-15-8578-4 https://doi.org/10.1007/978-981-15-8578-4
(eBook)
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Preface
This book is devoted to convolution type equations, which occur in linear wave problems of hereditary elasticity. The main mathematical tool used below is the Fourier–Laplace transform. The possibility of making use of the Fourier–Laplace transform, when solving convolution type equations, is evident. After applying the mentioned transform, we get a simple formula for the transformed solution of the equation considered. However the essence of matter is to derive from this formula the description of behavior of the solution itself: to find its support, asymptotics, etc. We must note that, as a rule, the Fourier–Laplace transform is used only formally in papers on wave problems of hereditary elasticity, whereas profound mathematical theorems (such as the Paley–Wiener theorem and Tauberian theorems) are neglected. The purpose of this book is to construct a rigorous mathematical approach to linear hereditary problems of wave propagation theory and to demonstrate usefulness of profound mathematical theorems in hereditary mechanics. Chapter 1
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