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Techniques of Functional Analysis for Differential and Integral Equations

Techniques of Functional Analysis for Differential and Integral Equations
A Book

by Paul Sacks

  • Publisher : Academic Press
  • Release : 2017-05-16
  • Pages : 320
  • ISBN : 0128114576
  • Language : En, Es, Fr & De
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Techniques of Functional Analysis for Differential and Integral Equations describes a variety of powerful and modern tools from mathematical analysis, for graduate study and further research in ordinary differential equations, integral equations and partial differential equations. Knowledge of these techniques is particularly useful as preparation for graduate courses and PhD research in differential equations and numerical analysis, and more specialized topics such as fluid dynamics and control theory. Striking a balance between mathematical depth and accessibility, proofs involving more technical aspects of measure and integration theory are avoided, but clear statements and precise alternative references are given . The work provides many examples and exercises drawn from the literature. Provides an introduction to mathematical techniques widely used in applied mathematics and needed for advanced research in ordinary and partial differential equations, integral equations, numerical analysis, fluid dynamics and other areas Establishes the advanced background needed for sophisticated literature review and research in differential equations and integral equations Suitable for use as a textbook for a two semester graduate level course for M.S. and Ph.D. students in Mathematics and Applied Mathematics

Integral Equations

Integral Equations
Theory and Numerical Treatment

by Wolfgang Hackbusch

  • Publisher : Birkhäuser
  • Release : 2012-12-06
  • Pages : 362
  • ISBN : 3034892152
  • Language : En, Es, Fr & De
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The theory of integral equations has been an active research field for many years and is based on analysis, function theory, and functional analysis. On the other hand, integral equations are of practical interest because of the «boundary integral equation method», which transforms partial differential equations on a domain into integral equations over its boundary. This book grew out of a series of lectures given by the author at the Ruhr-Universitat Bochum and the Christian-Albrecht-Universitat zu Kiel to students of mathematics. The contents of the first six chapters correspond to an intensive lecture course of four hours per week for a semester. Readers of the book require background from analysis and the foundations of numeri cal mathematics. Knowledge of functional analysis is helpful, but to begin with some basic facts about Banach and Hilbert spaces are sufficient. The theoretical part of this book is reduced to a minimum; in Chapters 2, 4, and 5 more importance is attached to the numerical treatment of the integral equations than to their theory. Important parts of functional analysis (e. g. , the Riesz-Schauder theory) are presented without proof. We expect the reader either to be already familiar with functional analysis or to become motivated by the practical examples given here to read a book about this topic. We recall that also from a historical point of view, functional analysis was initially stimulated by the investigation of integral equations.

Approximation Methods for Solutions of Differential and Integral Equations

Approximation Methods for Solutions of Differential and Integral Equations
A Book

by V. K. Dzyadyk

  • Publisher : Walter de Gruyter GmbH & Co KG
  • Release : 2018-11-05
  • Pages : 329
  • ISBN : 3110944693
  • Language : En, Es, Fr & De
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This book is the result of 20 years of investigations carried out by the author and his colleagues in order to bring closer and, to a certain extent, synthesize a number of well-known results, ideas and methods from the theory of function approximation, theory of differential and integral equations and numerical analysis. The book opens with an introduction on the theory of function approximation and is followed by a new approach to the Fredholm integral equations to the second kind. Several chapters are devoted to the construction of new methods for the effective approximation of solutions of several important integral, and ordinary and partial differential equations. In addition, new general results on the theory of linear differential equations with one regular singular point, as well as applications of the various new methods are discussed.

Special Functions and Analysis of Differential Equations

Special Functions and Analysis of Differential Equations
A Book

by Praveen Agarwal,Ravi P Agarwal,Michael Ruzhansky

  • Publisher : CRC Press
  • Release : 2020-09-08
  • Pages : 354
  • ISBN : 1000078566
  • Language : En, Es, Fr & De
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Differential Equations are very important tools in Mathematical Analysis. They are widely found in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamical systems, economics, biology, and so on. Recently there has been an increasing interest in and widely-extended use of differential equations and systems of fractional order (that is, of arbitrary order) as better models of phenomena in various physics, engineering, automatization, biology and biomedicine, chemistry, earth science, economics, nature, and so on. Now, new unified presentation and extensive development of special functions associated with fractional calculus are necessary tools, being related to the theory of differentiation and integration of arbitrary order (i.e., fractional calculus) and to the fractional order (or multi-order) differential and integral equations. This book provides learners with the opportunity to develop an understanding of advancements of special functions and the skills needed to apply advanced mathematical techniques to solve complex differential equations and Partial Differential Equations (PDEs). Subject matters should be strongly related to special functions involving mathematical analysis and its numerous applications. The main objective of this book is to highlight the importance of fundamental results and techniques of the theory of complex analysis for differential equations and PDEs and emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Specific topics include but are not limited to Partial differential equations Least squares on first-order system Sequence and series in functional analysis Special functions related to fractional (non-integer) order control systems and equations Various special functions related to generalized fractional calculus Operational method in fractional calculus Functional analysis and operator theory Mathematical physics Applications of numerical analysis and applied mathematics Computational mathematics Mathematical modeling This book provides the recent developments in special functions and differential equations and publishes high-quality, peer-reviewed book chapters in the area of nonlinear analysis, ordinary differential equations, partial differential equations, and related applications.

Functional Analysis for the Applied Sciences

Functional Analysis for the Applied Sciences
A Book

by Gheorghe Moroşanu

  • Publisher : Springer Nature
  • Release : 2019-12-27
  • Pages : 432
  • ISBN : 3030271536
  • Language : En, Es, Fr & De
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This advanced graduate textbook presents main results and techniques in Functional Analysis and uses them to explore other areas of mathematics and applications. Special attention is paid to creating appropriate frameworks towards solving significant problems involving differential and integral equations. Exercises at the end of each chapter help the reader to understand the richness of ideas and methods offered by Functional Analysis. Some of the exercises supplement theoretical material, while others relate to the real world. This textbook, with its friendly exposition, focuses on different problems in physics and other applied sciences and uniquely provides solutions to most of the exercises. The text is aimed toward graduate students and researchers in applied mathematics, physics, and neighboring fields of science.

Methods in Nonlinear Integral Equations

Methods in Nonlinear Integral Equations
A Book

by R Precup

  • Publisher : Springer Science & Business Media
  • Release : 2013-03-09
  • Pages : 218
  • ISBN : 9401599866
  • Language : En, Es, Fr & De
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Methods in Nonlinear Integral Equations presents several extremely fruitful methods for the analysis of systems and nonlinear integral equations. They include: fixed point methods (the Schauder and Leray-Schauder principles), variational methods (direct variational methods and mountain pass theorems), and iterative methods (the discrete continuation principle, upper and lower solutions techniques, Newton's method and the generalized quasilinearization method). Many important applications for several classes of integral equations and, in particular, for initial and boundary value problems, are presented to complement the theory. Special attention is paid to the existence and localization of solutions in bounded domains such as balls and order intervals. The presentation is essentially self-contained and leads the reader from classical concepts to current ideas and methods of nonlinear analysis.

Integral Equations, Boundary Value Problems and Related Problems

Integral Equations, Boundary Value Problems and Related Problems
Dedicated to Professor Chien-Ke Lu on the Occasion of His 90th Birthday : Yinchuan, Ningxia, China, 19-23 August 2012

by Xing Li

  • Publisher : World Scientific
  • Release : 2013
  • Pages : 285
  • ISBN : 9814452882
  • Language : En, Es, Fr & De
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In this volume, we report new results about various theories and methods of integral equation, boundary value problems for partial differential equations and functional equations, and integral operators including singular integral equations, applications of boundary value problems and integral equations to mechanics and physics, numerical methods of integral equations and boundary value problems, theories and methods for inverse problems of mathematical physics, Clifford analysis and related problems.

Linear Integral Equations

Linear Integral Equations
A Book

by Rainer Kress

  • Publisher : Springer Science & Business Media
  • Release : 2013-12-04
  • Pages : 412
  • ISBN : 1461495938
  • Language : En, Es, Fr & De
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This book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter. For this third edition in order to make the introduction to the basic functional analytic tools more complete the Hahn–Banach extension theorem and the Banach open mapping theorem are now included in the text. The treatment of boundary value problems in potential theory has been extended by a more complete discussion of integral equations of the first kind in the classical Holder space setting and of both integral equations of the first and second kind in the contemporary Sobolev space setting. In the numerical solution part of the book, the author included a new collocation method for two-dimensional hypersingular boundary integral equations and a collocation method for the three-dimensional Lippmann-Schwinger equation. The final chapter of the book on inverse boundary value problems for the Laplace equation has been largely rewritten with special attention to the trilogy of decomposition, iterative and sampling methods Reviews of earlier editions: "This book is an excellent introductory text for students, scientists, and engineers who want to learn the basic theory of linear integral equations and their numerical solution." (Math. Reviews, 2000) "This is a good introductory text book on linear integral equations. It contains almost all the topics necessary for a student. The presentation of the subject matter is lucid, clear and in the proper modern framework without being too abstract." (ZbMath, 1999)

Recent Advances in Integral Equations

Recent Advances in Integral Equations
A Book

by Francisco Bulnes

  • Publisher : BoD – Books on Demand
  • Release : 2019-07-24
  • Pages : 100
  • ISBN : 183880658X
  • Language : En, Es, Fr & De
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Integral equations are functional equations in which an unknown function appears under an integral sign. This can involve aspects of function theory and their integral transforms when the unknown function appears with a functional non-degenerated kernel under the integral sign. The close relation between differential and integral equations does that in some functional analysis, and function theory problems may be formulated either way. This book establishes the fundamentals of integral equations and considers some deep research aspects on integral equations of first and second kind, operator theory applied to integral equations, methods to solve some nonlinear integral equations, and singular integral equations, among other things. This is the first volume on this theme, hoping that other volumes of this important functional analysis theme and operator theory to formal functional equations will be realized in the future.

Functional Analysis and Numerical Mathematics

Functional Analysis and Numerical Mathematics
A Book

by Lothar Collatz

  • Publisher : Academic Press
  • Release : 2014-05-12
  • Pages : 494
  • ISBN : 1483264009
  • Language : En, Es, Fr & De
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Functional Analysis and Numerical Mathematics focuses on the structural changes which numerical analysis has undergone, including iterative methods, vectors, integral equations, matrices, and boundary value problems. The publication first examines the foundations of functional analysis and applications, including various types of spaces, convergence and completeness, operators in Hilbert spaces, vector and matrix norms, eigenvalue problems, and operators in pseudometric and other special spaces. The text then elaborates on iterative methods. Topics include the fixed-point theorem for a general iterative method in pseudometric spaces; special cases of the fixed-point theorem and change of operator; iterative methods for differential and integral equations; and systems of equations and difference methods. The manuscript takes a look at monotonicity, inequalities, and other topics, including monotone operators, applications of Schauder's theorem, matrices and boundary value problems of monotone kind, discrete Chebyshev approximation and exchange methods, and approximation of functions. The publication is a valuable source of data for mathematicians and researchers interested in functional analysis and numerical mathematics.

Introduction to Nonlinear Differential and Integral Equations

Introduction to Nonlinear Differential and Integral Equations
A Book

by Harold Thayer Davis

  • Publisher : Courier Corporation
  • Release : 1962-01-01
  • Pages : 566
  • ISBN : 9780486609713
  • Language : En, Es, Fr & De
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Topics covered include differential equations of the 1st order, the Riccati equation and existence theorems, 2nd order equations, elliptic integrals and functions, nonlinear mechanics, nonlinear integral equations, more. Includes 137 problems.

Applied Mechanics Reviews

Applied Mechanics Reviews
A Book

by Anonim

  • Publisher : Unknown Publisher
  • Release : 1979
  • Pages : 329
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Analysis of Approximation Methods for Differential and Integral Equations

Analysis of Approximation Methods for Differential and Integral Equations
A Book

by Hans-Jürgen Reinhardt

  • Publisher : Springer
  • Release : 1985-10-07
  • Pages : 398
  • ISBN : 9780387962146
  • Language : En, Es, Fr & De
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This book is primarily based on the research done by the Numerical Analysis Group at the Goethe-Universitat in Frankfurt/Main, and on material presented in several graduate courses by the author between 1977 and 1981. It is hoped that the text will be useful for graduate students and for scientists interested in studying a fundamental theoretical analysis of numerical methods along with its application to the most diverse classes of differential and integral equations. The text treats numerous methods for approximating solutions of three classes of problems: (elliptic) boundary-value problems, (hyperbolic and parabolic) initial value problems in partial differential equations, and integral equations of the second kind. The aim is to develop a unifying convergence theory, and thereby prove the convergence of, as well as provide error estimates for, the approximations generated by specific numerical methods. The schemes for numerically solving boundary-value problems are additionally divided into the two categories of finite difference methods and of projection methods for approximating their variational formulations.

Integral Equations: A Practical Treatment, from Spectral Theory to Applications

Integral Equations: A Practical Treatment, from Spectral Theory to Applications
A Book

by David Porter,David S. G. Stirling,Porter David

  • Publisher : Cambridge University Press
  • Release : 1990-09-28
  • Pages : 372
  • ISBN : 9780521337427
  • Language : En, Es, Fr & De
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The book aims to tackle the solution of integral equations using a blend of abstract 'structural' results and more direct, down-to-earth mathematics.

Applied Functional Analysis

Applied Functional Analysis
Numerical Methods, Wavelet Methods, and Image Processing

by Abul Hasan Siddiqi

  • Publisher : CRC Press
  • Release : 2003-09-19
  • Pages : 660
  • ISBN : 9780203913017
  • Language : En, Es, Fr & De
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The methods of functional analysis have helped solve diverse real-world problems in optimization, modeling, analysis, numerical approximation, and computer simulation. Applied Functional Analysis presents functional analysis results surfacing repeatedly in scientific and technological applications and presides over the most current analytical and n

Survey of Applicable Mathematics

Survey of Applicable Mathematics
A Book

by K. Rektorys

  • Publisher : Springer
  • Release : 2013-12-19
  • Pages : 1723
  • ISBN : 9401583080
  • Language : En, Es, Fr & De
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This major two-volume handbook is an extensively revised, updated second edition of the highly praised Survey of Applicable Mathematics, first published in English in 1969. The thirty-seven chapters cover all the important mathematical fields of use in applications: algebra, geometry, differential and integral calculus, infinite series, orthogonal systems of functions, Fourier series, special functions, ordinary differential equations, partial differential equations, integral equations, functions of one and several complex variables, conformal mapping, integral transforms, functional analysis, numerical methods in algebra and in algebra and in differential boundary value problems, probability, statistics, stochastic processes, calculus of variations, and linear programming. All proofs have been omitted. However, theorems are carefully formulated, and where considered useful, are commented with explanatory remarks. Many practical examples are given by way of illustration. Each of the two volumes contains an extensive bibliography and a comprehensive index. Together these two volumes represent a survey library of mathematics which is applicable in many fields of science, engineering, economics, etc. For researchers, students and teachers of mathematics and its applications.

Library of Congress Subject Headings

Library of Congress Subject Headings
A Book

by Library of Congress

  • Publisher : Unknown Publisher
  • Release : 1994
  • Pages : 329
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Nonlinear Functional Analysis and Applications

Nonlinear Functional Analysis and Applications
Proceedings of an Advanced Seminar Conducted by the Mathematics Research Center, the University of Wisconsin, Madison, October 12-14, 1970

by Louis B. Rall

  • Publisher : Elsevier
  • Release : 2014-05-10
  • Pages : 594
  • ISBN : 1483272443
  • Language : En, Es, Fr & De
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Nonlinear Functional Analysis and Applications provides information pertinent to the fundamental aspects of nonlinear functional analysis and its application. This book provides an introduction to the basic concepts and techniques of this field. Organized into nine chapters, this book begins with an overview of the possibilities for applying ideas from functional analysis to problems in analysis. This text then provides a systematic exposition of several aspects of differential calculus in norms and topological linear spaces. Other chapters consider the various settings in nonlinear functional analysis in which differentials play a significant role. This book discusses as well the generalized inverse for a bounded linear operator, whose range is not necessarily closed. The final chapter deals with the equations of hydrodynamics, which are usually highly nonlinear and difficult to solve. This book is a valuable resource for mathematicians. Readers who are interested in nonlinear functional analysis will also find this book useful.

Linear Integral Equations

Linear Integral Equations
Theory & Technique

by Ram P. Kanwal

  • Publisher : Springer Science & Business Media
  • Release : 2012-11-07
  • Pages : 318
  • ISBN : 1461460123
  • Language : En, Es, Fr & De
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Many physical problems that are usually solved by differential equation methods can be solved more effectively by integral equation methods. Such problems abound in applied mathematics, theoretical mechanics, and mathematical physics. This uncorrected soft cover reprint of the second edition places the emphasis on applications and presents a variety of techniques with extensive examples.Originally published in 1971, Linear Integral Equations is ideal as a text for a beginning graduate level course. Its treatment of boundary value problems also makes the book useful to researchers in many applied fields.

Green's Functions and Boundary Value Problems

Green's Functions and Boundary Value Problems
A Book

by Ivar Stakgold,Michael J. Holst

  • Publisher : John Wiley & Sons
  • Release : 2011-03-01
  • Pages : 736
  • ISBN : 0470906529
  • Language : En, Es, Fr & De
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Praise for the Second Edition "This book is an excellent introduction to the wide field of boundary value problems."—Journal of Engineering Mathematics "No doubt this textbook will be useful for both students and research workers."—Mathematical Reviews A new edition of the highly-acclaimed guide to boundary value problems, now featuring modern computational methods and approximation theory Green's Functions and Boundary Value Problems, Third Edition continues the tradition of the two prior editions by providing mathematical techniques for the use of differential and integral equations to tackle important problems in applied mathematics, the physical sciences, and engineering. This new edition presents mathematical concepts and quantitative tools that are essential for effective use of modern computational methods that play a key role in the practical solution of boundary value problems. With a careful blend of theory and applications, the authors successfully bridge the gap between real analysis, functional analysis, nonlinear analysis, nonlinear partial differential equations, integral equations, approximation theory, and numerical analysis to provide a comprehensive foundation for understanding and analyzing core mathematical and computational modeling problems. Thoroughly updated and revised to reflect recent developments, the book includes an extensive new chapter on the modern tools of computational mathematics for boundary value problems. The Third Edition features numerous new topics, including: Nonlinear analysis tools for Banach spaces Finite element and related discretizations Best and near-best approximation in Banach spaces Iterative methods for discretized equations Overview of Sobolev and Besov space linear Methods for nonlinear equations Applications to nonlinear elliptic equations In addition, various topics have been substantially expanded, and new material on weak derivatives and Sobolev spaces, the Hahn-Banach theorem, reflexive Banach spaces, the Banach Schauder and Banach-Steinhaus theorems, and the Lax-Milgram theorem has been incorporated into the book. New and revised exercises found throughout allow readers to develop their own problem-solving skills, and the updated bibliographies in each chapter provide an extensive resource for new and emerging research and applications. With its careful balance of mathematics and meaningful applications, Green's Functions and Boundary Value Problems, Third Edition is an excellent book for courses on applied analysis and boundary value problems in partial differential equations at the graduate level. It is also a valuable reference for mathematicians, physicists, engineers, and scientists who use applied mathematics in their everyday work.