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The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

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The "Atlantis Studies in Dynamical Systems" publishes monographs in the area of dynamical systems, written by leading experts in the field and useful for both students and researchers. Books with a theoretical nature will be published alongside books emphasizing applications. This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher.Printed on acid-free paper PrefaceMany problems arising in applied mathematics or mathematical physics, can be formulated in two ways namely as differential equations and as integral equations. In the differential equation approach, the boundary conditions have to be imposed externally, whereas in the case of integral equations, the boundary conditions are incorporated within the formulation, and this confers a valuable advantage to the latter method. Moreover, the integral equation approach leads quite naturally to the solution of the problem as an infinite series, known as the Neumann expansion, the Adomian decomposition method, and the series solution method in which the successive terms arise from the application of an iterative procedure. The proof of the convergence of this series under appropriate conditions presents an interesting exercise in an elementary analysis.This book encompasses recent developments of integral equations on time scales. For many population models biological reasons suggest using their difference analogues. For instance, North American big game populations have discrete birth pulses, not continuous births as is assumed by differential equations. Mathematical reasons also suggest using difference equations-they are easier to construct and solve in a computer spreadsheet. North American large mammal populations do not have continuous population growth, but rather discrete birth pulses, so the differential equation form of the logistic equation will not be convenient. Age-structured models add complexity to a population model, but make the model more realistic, in that essential features of the population growth process are captured by the model. They are used difference equations to define the population model because discrete age classes require difference equations for simple solutions. The discrete models can be investigated using integral equations in the case when the time scale is the set of the natural numbers. A powerful method introduced by Poincaré for examining the motion of dynamical systems is that of a Poincaré section. This method can be investigated using integral equations on the set of the natural numbers. The total charge on the capacitor can be investigated with an integral equation on the set of the harmonic numbers.This book contains elegant analytical and numerical methods. This book is intended for the use in the field of integral equations and dynamic calculus on time v

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