Download or read book Collocation Methods for Volterra Integral and Related Functional Differential Equations written by Hermann Brunner and published by Cambridge University Press. This book was released on 2004-11-15 with total page 620 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Volterra Integral Equations written by Hermann Brunner and published by Cambridge University Press. This book was released on 2017-01-20 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations (VIEs), ranging from Volterra's fundamental contributions and the resulting classical theory to more recent developments that include Volterra functional integral equations with various kinds of delays, VIEs with highly oscillatory kernels, and VIEs with non-compact operators. It will act as a 'stepping stone' to the literature on the advanced theory of VIEs, bringing the reader to the current state of the art in the theory. Each chapter contains a large number of exercises, extending from routine problems illustrating or complementing the theory to challenging open research problems. The increasingly important role of VIEs in the mathematical modelling of phenomena where memory effects play a key role is illustrated with some 30 concrete examples, and the notes at the end of each chapter feature complementary references as a guide to further reading.
Download or read book Spectral Methods written by Jie Shen and published by Springer Science & Business Media. This book was released on 2011-08-25 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.
Download or read book Multidimensional Weakly Singular Integral Equations written by Gennadi Vainikko and published by Springer. This book was released on 2006-11-15 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: The final aim of the book is to construct effective discretization methods to solve multidimensional weakly singular integral equations of the second kind on a region of Rn e.g. equations arising in the radiation transfer theory. To this end, the smoothness of the solution is examined proposing sharp estimates of the growth of the derivatives of the solution near the boundary G. The superconvergence effect of collocation methods at the collocation points is established. This is a book for graduate students and researchers in the fields of analysis, integral equations, mathematical physics and numerical methods. No special knowledge beyond standard undergraduate courses is assumed.
Download or read book Solving ODEs with MATLAB written by Lawrence F. Shampine and published by Cambridge University Press. This book was released on 2003-04-28 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This concise text, first published in 2003, is for a one-semester course for upper-level undergraduates and beginning graduate students in engineering, science, and mathematics, and can also serve as a quick reference for professionals. The major topics in ordinary differential equations, initial value problems, boundary value problems, and delay differential equations, are usually taught in three separate semester-long courses. This single book provides a sound treatment of all three in fewer than 300 pages. Each chapter begins with a discussion of the 'facts of life' for the problem, mainly by means of examples. Numerical methods for the problem are then developed, but only those methods most widely used. The treatment of each method is brief and technical issues are minimized, but all the issues important in practice and for understanding the codes are discussed. The last part of each chapter is a tutorial that shows how to solve problems by means of small, but realistic, examples.
Download or read book Numerical Methods for Delay Differential Equations written by Alfredo Bellen and published by OUP Oxford. This book was released on 2003-03-20 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main purpose of the book is to introduce the readers to the numerical integration of the Cauchy problem for delay differential equations (DDEs). Peculiarities and differences that DDEs exhibit with respect to ordinary differential equations are preliminarily outlined by numerous examples illustrating some unexpected, and often surprising, behaviours of the analytical and numerical solutions. The effect of various kinds of delays on the regularity of the solution is described and some essential existence and uniqueness results are reported. The book is centered on the use of Runge-Kutta methods continuously extended by polynomial interpolation, includes a brief review of the various approaches existing in the literature, and develops an exhaustive error and well-posedness analysis for the general classes of one-step and multistep methods. The book presents a comprehensive development of continuous extensions of Runge-Kutta methods which are of interest also in the numerical treatment of more general problems such as dense output, discontinuous equations, etc. Some deeper insight into convergence and superconvergence of continuous Runge-Kutta methods is carried out for DDEs with various kinds of delays. The stepsize control mechanism is also developed on a firm mathematical basis relying on the discrete and continuous local error estimates. Classical results and a unconventional analysis of "stability with respect to forcing term" is reviewed for ordinary differential equations in view of the subsequent numerical stability analysis. Moreover, an exhaustive description of stability domains for some test DDEs is carried out and the corresponding stability requirements for the numerical methods are assessed and investigated. Alternative approaches, based on suitable formulation of DDEs as partial differential equations and subsequent semidiscretization are briefly described and compared with the classical approach. A list of available codes is provided, and illustrative examples, pseudo-codes and numerical experiments are included throughout the book.
Download or read book Spectral and High order Methods with Applications written by Jie Shen and published by . This book was released on 2006 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: 中国科学院科学出版基金资助出版。
Download or read book Abel Integral Equations written by Rudolf Gorenflo and published by Springer. This book was released on 2006-11-14 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: In many fields of application of mathematics, progress is crucially dependent on the good flow of information between (i) theoretical mathematicians looking for applications, (ii) mathematicians working in applications in need of theory, and (iii) scientists and engineers applying mathematical models and methods. The intention of this book is to stimulate this flow of information. In the first three chapters (accessible to third year students of mathematics and physics and to mathematically interested engineers) applications of Abel integral equations are surveyed broadly including determination of potentials, stereology, seismic travel times, spectroscopy, optical fibres. In subsequent chapters (requiring some background in functional analysis) mapping properties of Abel integral operators and their relation to other integral transforms in various function spaces are investi- gated, questions of existence and uniqueness of solutions of linear and nonlinear Abel integral equations are treated, and for equations of the first kind problems of ill-posedness are discussed. Finally, some numerical methods are described. In the theoretical parts, emphasis is put on the aspects relevant to applications.
Download or read book New Trends in Differential and Difference Equations and Applications written by Feliz Manuel Minhós and published by MDPI. This book was released on 2019-10-14 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Special Issue aims to be a compilation of new results in the areas of differential and difference Equations, covering boundary value problems, systems of differential and difference equations, as well as analytical and numerical methods. The objective is to provide an overview of techniques used in these different areas and to emphasize their applicability to real-life phenomena, by the inclusion of examples. These examples not only clarify the theoretical results presented, but also provide insight on how to apply, for future works, the techniques used.
Download or read book Introduction to Integral Equations with Applications written by Abdul J. Jerri and published by . This book was released on 1985 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abdul Jerri has revised his highly applied book to make it even more useful for scientists and engineers, as well as mathematicians. Covering the fundamental ideas and techniques at a level accessible to anyone with a solid undergraduate background in calculus and differential equations, Dr. Jerri clearly demonstrates how to use integral equations to solve real-world engineering and physics problems. This edition provides precise guidelines to the basic methods of solutions, details more varied numerical methods, and substantially boosts the total of practical examples and exercises. Plus, it features added emphasis on the basic theorems for the existence and uniqueness of solutions of integral equations and points out the interrelation between differentiation and integration.
Download or read book Weighted Inequalities of Hardy Type written by Alois Kufner and published by World Scientific. This book was released on 2003 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inequalities play an important role in almost all branches of mathematics as well as in other areas of science and engineering. This book surveys the present state of the theory of weighted integral inequalities of Hardy type, including modifications concerning Hardy-Steklov operators, and some basic results about Hardy type inequalities and their limit (Carleman-Knopp type) inequalities. It also describes some rather new fields such as higher order and fractional order Hardy type inequalities and integral inequalities on the cone of monotone functions together with some applications and open problems. The book can serve as a reference and a source of inspiration for researchers working in these and related areas, but could also be used for advanced graduate courses.
Download or read book Inequalities Involving Functions and Their Integrals and Derivatives written by Dragoslav S. Mitrinovic and published by Springer Science & Business Media. This book was released on 1991-07-31 with total page 606 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides a comprehensive, up-to-date survey of inequalities that involve a relationship between a function and its derivatives or integrals. The book is divided into 18 chapters, some of which are devoted to specific inequalities such as those of Kolmogorov-Landau, Wirtinger, Hardy, Carlson, Hilbert, Caplygin, Lyapunov, Gronwell and others. Over 800 references to the literature are cited; proofs are given when these provide insight into the general methods involved; and applications, especially to the theory of differential equations, are mentioned when appropriate. This volume will interest all those whose work involves differential and integral equations. It can also be recommended as a supplementary text.
Download or read book Elements of Applied Bifurcation Theory written by Yuri Kuznetsov and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 648 pages. Available in PDF, EPUB and Kindle. Book excerpt: Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.
Download or read book Acta Numerica 2004 Volume 13 written by Arieh Iserles and published by Cambridge University Press. This book was released on 2004-06-03 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: An annual volume presenting substantive survey articles in numerical mathematics and scientific computing.
Download or read book Spectral Methods written by Claudio Canuto and published by Springer Science & Business Media. This book was released on 2007-09-23 with total page 585 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the publication of "Spectral Methods in Fluid Dynamics" 1988, spectral methods have become firmly established as a mainstream tool for scientific and engineering computation. The authors of that book have incorporated into this new edition the many improvements in the algorithms and the theory of spectral methods that have been made since then. This latest book retains the tight integration between the theoretical and practical aspects of spectral methods, and the chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is also greatly expanded.
Download or read book Analytical and Numerical Methods for Volterra Equations written by Peter Linz and published by SIAM. This book was released on 1985-01-01 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Parabolic Equations in Biology written by Benoît Perthame and published by Springer. This book was released on 2015-09-09 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents several fundamental questions in mathematical biology such as Turing instability, pattern formation, reaction-diffusion systems, invasion waves and Fokker-Planck equations. These are classical modeling tools for mathematical biology with applications to ecology and population dynamics, the neurosciences, enzymatic reactions, chemotaxis, invasion waves etc. The book presents these aspects from a mathematical perspective, with the aim of identifying those qualitative properties of the models that are relevant for biological applications. To do so, it uncovers the mechanisms at work behind Turing instability, pattern formation and invasion waves. This involves several mathematical tools, such as stability and instability analysis, blow-up in finite time, asymptotic methods and relative entropy properties. Given the content presented, the book is well suited as a textbook for master-level coursework.
