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Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena. The present book introduces the reader to the general principles underlying these methods and then describes in detail their convergence properties when applied to ordinary differential equations, functional equations with (Volterra type) memory terms, delay equations, and differential-algebraic and integral-algebraic equations. Each chapter starts with a self-contained introduction to the relevant theory of the class of equations under consideration. Numerous exercises and examples are supplied, along with extensive historical and bibliographical notes utilising the vast annotated reference list of over 1300 items. In sum, Hermann Brunner has written a treatise that can serve as an introduction for students, a guide for users, and a comprehensive resource for experts.

Il Calcolo Frazionario ha recentemente guadagnato un crescente interesse nella letteratura economica e finanziaria. Per quanto riguarda i modelli economici, quelli di crescita sono stati modellati utilizzando una rappresentazione tramite derivate frazionarie. Questo tipo di equazioni non consente soluzioni in forma chiusa e quindi è necessario ricorrere a metodi numerici appropriati per ottenere approssimazioni accurate delle soluzioni. Per questo motivo, in questo contributo, proponiamo un approccio basato sulle cosiddette Physics Informed Neural Network per risolvere le equazioni integrali di Volterra di ordine frazionario. Alcuni esperimenti numerici mostrano l’accuratezza dell’algoritmo suggerito.

This book looks at the theories of Volterra integral and functional equations.

Seminar paper from the year 2015 in the subject Mathematics - Applied Mathematics, grade: A, , language: English, abstract: In scientific and engineering problems Volterra integral equations are always encountered. Applications of Volterra integral equations arise in areas such as population dynamics, spread of epidemics in the society, etc. The problem statement is to obtain a good numerical solution to such an integral equation. A brief theory of Volterra Integral equation, particularly, of weakly singular types, and a numerical method, the collocation method, for solving such equations, in particular Volterra integral equation of second kind, is handled in this paper. The principle of this method is to approximate the exact solution of the equation in a suitable finite dimensional space. The approximating space considered here is the polynomial spline space. In the treatment of the collocation method emphasis is laid, during discretization, on the mesh type. The approximating space applied here is the polynomial spline space. The discrete convergence properties of spline collocation solutions for certain Volterra integral equations with weakly singular kernels shall is analyzed. The order of convergence of spline collocation on equidistant mesh points is also compared with approximation on graded meshes. In particular, the attainable convergence orders at the collocation points are examined for certain choices of the collocation parameters.

The purpose of this paper is to show how Volterra integral equations may be studied within the framework of the theory of topological dynamics. Part I contains the basic theory, as local dynamical systems are discussed together with some of their elementary properties. The notation of compatible pairs of function spaces is introduced. Part II contains examples of compatible pairs, as these spaces are studied in some detail. Part III contains some applications of the first two parts.

In the history of mathematics there are many situations in which cal- lations were performed incorrectly for important practical applications. Let us look at some examples, the history of computing the number ? began in Egypt and Babylon about 2000 years BC, since then many mathematicians have calculated ? (e. g. , Archimedes, Ptolemy, Vi` ete, etc. ). The ?rst formula for computing decimal digits of ? was disc- ered by J. Machin (in 1706), who was the ?rst to correctly compute 100 digits of ?. Then many people used his method, e. g. , W. Shanks calculated ? with 707 digits (within 15 years), although due to mistakes only the ?rst 527 were correct. For the next examples, we can mention the history of computing the ?ne-structure constant ? (that was ?rst discovered by A. Sommerfeld), and the mathematical tables, exact - lutions, and formulas, published in many mathematical textbooks, were not veri?ed rigorously [25]. These errors could have a large e?ect on results obtained by engineers. But sometimes, the solution of such problems required such techn- ogy that was not available at that time. In modern mathematics there exist computers that can perform various mathematical operations for which humans are incapable. Therefore the computers can be used to verify the results obtained by humans, to discovery new results, to - provetheresultsthatahumancanobtainwithoutanytechnology. With respectto our example of computing?, we can mention that recently (in 2002) Y. Kanada, Y. Ushiro, H. Kuroda, and M.

Presents an aspect of activity in integral equations methods for the solution of Volterra equations for those who need to solve real-world problems. Since there are few known analytical methods leading to closed-form solutions, the emphasis is on numerical techniques. The major points of the analytical methods used to study the properties of the solution are presented in the first part of the book. These techniques are important for gaining insight into the qualitative behavior of the solutions and for designing effective numerical methods. The second part of the book is devoted entirely to numerical methods. The author has chosen the simplest possible setting for the discussion, the space of real functions of real variables. The text is supplemented by examples and exercises.

Unparalleled in scope compared to the literature currently available, the Handbook of Integral Equations, Second Edition contains over 2,500 integral equations with solutions as well as analytical and numerical methods for solving linear and nonlinear equations. It explores Volterra, Fredholm, WienerHopf, Hammerstein, Uryson, and other equa

I fell in love with integral equations about twenty years ago when I was working on my thesis, and I am still attracted by their mathematical beauty. This book will try to stimulate the reader to share this love with me. Having taught integral equations a number of times I felt a lack of a text which adequately combines theory, applications and numerical methods. Therefore, in this book I intend to cover each of these fields with the same weight. The first part provides the basic Riesz-Fredholm theory for equa tions of the second kind with compact opertors in dual systems including all functional analytic concepts necessary for developing this theory. The second part then illustrates the classical applications of integral equation methods to boundary value problems for the Laplace and the heat equation as one of the main historical sources for the development of integral equations, and also in troduces Cauchy type singular integral equations. The third part is devoted to describing the fundamental ideas for the numerical solution of integral equa tions. Finally, in a fourth part, ill-posed integral equations of the first kind and their regularization are studied in a Hilbert space setting. In order to make the book accessible not only to mathematicans but also to physicists and engineers I have planned it as self-contained as possible by requiring only a solid foundation in differential and integral calculus and, for parts of the book, in complex function theory.