Formal and Analytic Solutions of Diff. Equations
Formal and Analytic Solutions of Diff. Equations

Author: Galina Filipuk

Publisher: Springer

Published: 2018-09-24

Total Pages: 273

ISBN-13: 3319991485

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These proceedings provide methods, techniques, different mathematical tools and recent results in the study of formal and analytic solutions to Diff. (differential, partial differential, difference, q-difference, q-difference-differential.... ) Equations. They consist of selected contributions from the conference "Formal and Analytic Solutions of Diff. Equations", held at Alcalá de Henares, Spain during September 4-8, 2017. Their topics include summability and asymptotic study of both ordinary and partial differential equations. The volume is divided into four parts. The first paper is a survey of the elements of nonlinear analysis. It describes the algorithms to obtain asymptotic expansion of solutions of nonlinear algebraic, ordinary differential, partial differential equations, and of systems of such equations. Five works on formal and analytic solutions of PDEs are followed by five papers on the study of solutions of ODEs. The proceedings conclude with five works on related topics, generalizations and applications. All contributions have been peer reviewed by anonymous referees chosen among the experts on the subject. The volume will be of interest to graduate students and researchers in theoretical and applied mathematics, physics and engineering seeking an overview of the recent trends in the theory of formal and analytic solutions of functional (differential, partial differential, difference, q-difference, q-difference-differential) equations in the complex domain.

Formal And Analytic Solutions Of Differential Equations
Formal And Analytic Solutions Of Differential Equations

Author: Galina Filipuk

Publisher: World Scientific

Published: 2022-03-03

Total Pages: 400

ISBN-13: 1800611374

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The book provides the reader with an overview of the actual state of research in ordinary and partial differential equations in the complex domain. Topics include summability and asymptotic study of both ordinary and partial differential equations, and also q-difference and differential-difference equations. This book will be of interest to researchers and students who wish to expand their knowledge of these fields.With the latest results and research developments and contributions from experts in their field, Formal and Analytic Solutions of Differential Equations provides a valuable contribution to methods, techniques, different mathematical tools, and study calculations.

Recent Trends in Formal and Analytic Solutions of Diff. Equations
Recent Trends in Formal and Analytic Solutions of Diff. Equations

Author: Galina Filipuk

Publisher: American Mathematical Society

Published: 2023-02-09

Total Pages: 240

ISBN-13: 147046604X

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This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28–July 2, 2021, and hosted by University of Alcalá, Alcalá de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.

Analytic Solutions of Functional Equations
Analytic Solutions of Functional Equations

Author: Sui Sun Cheng

Publisher: World Scientific

Published: 2008

Total Pages: 296

ISBN-13: 9812793348

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This book presents a self-contained and unified introduction to the properties of analytic functions. Based on recent research results, it provides many examples of functional equations to show how analytic solutions can be found.Unlike in other books, analytic functions are treated here as those generated by sequences with positive radii of convergence. By developing operational means for handling sequences, functional equations can then be transformed into recurrence relations or difference equations in a straightforward manner. Their solutions can also be found either by qualitative means or by computation. The subsequent formal power series function can then be asserted as a true solution once convergence is established by various convergence tests and majorization techniques. Functional equations in this book may also be functional differential equations or iterative equations, which are different from the differential equations studied in standard textbooks since composition of known or unknown functions are involved.

Functional Analysis, Sobolev Spaces and Partial Differential Equations
Functional Analysis, Sobolev Spaces and Partial Differential Equations

Author: Haim Brezis

Publisher: Springer Science & Business Media

Published: 2010-11-02

Total Pages: 600

ISBN-13: 0387709142

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This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.

Partial Differential Equations
Partial Differential Equations

Author: Walter A. Strauss

Publisher: John Wiley & Sons

Published: 2007-12-21

Total Pages: 467

ISBN-13: 0470054565

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Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.

Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations
Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations

Author: Werner Balser

Publisher: Springer Science & Business Media

Published: 2000

Total Pages: 314

ISBN-13: 0387986901

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Simple Ordinary Differential Equations may have solutions in terms of power series whose coefficients grow at such a rate that the series has a radius of convergence equal to zero. In fact, every linear meromorphic system has a formal solution of a certain form, which can be relatively easily computed, but which generally involves such power series diverging everywhere. In this book the author presents the classical theory of meromorphic systems of ODE in the new light shed upon it by the recent achievements in the theory of summability of formal power series.

Involution
Involution

Author: Werner M. Seiler

Publisher: Springer Science & Business Media

Published: 2009-10-26

Total Pages: 663

ISBN-13: 3642012876

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The book provides a self-contained account of the formal theory of general, i.e. also under- and overdetermined, systems of differential equations which in its central notion of involution combines geometric, algebraic, homological and combinatorial ideas.

Cohomological Analysis of Partial Differential Equations and Secondary Calculus
Cohomological Analysis of Partial Differential Equations and Secondary Calculus

Author: A. M. Vinogradov

Publisher: American Mathematical Soc.

Published: 2001-10-16

Total Pages: 268

ISBN-13: 9780821897997

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This book is dedicated to fundamentals of a new theory, which is an analog of affine algebraic geometry for (nonlinear) partial differential equations. This theory grew up from the classical geometry of PDE's originated by S. Lie and his followers by incorporating some nonclassical ideas from the theory of integrable systems, the formal theory of PDE's in its modern cohomological form given by D. Spencer and H. Goldschmidt and differential calculus over commutative algebras (Primary Calculus). The main result of this synthesis is Secondary Calculus on diffieties, new geometrical objects which are analogs of algebraic varieties in the context of (nonlinear) PDE's. Secondary Calculus surprisingly reveals a deep cohomological nature of the general theory of PDE's and indicates new directions of its further progress. Recent developments in quantum field theory showed Secondary Calculus to be its natural language, promising a nonperturbative formulation of the theory. In addition to PDE's themselves, the author describes existing and potential applications of Secondary Calculus ranging from algebraic geometry to field theory, classical and quantum, including areas such as characteristic classes, differential invariants, theory of geometric structures, variational calculus, control theory, etc. This book, focused mainly on theoretical aspects, forms a natural dipole with Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Volume 182 in this same series, Translations of Mathematical Monographs, and shows the theory "in action".