## Operational Methods

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Author: V. P. Maslov

Added by: mirtitles

Added Date: 2018-10-13

Language: English

Subjects: mathematics, operational method, lagranian, hamilton-jacobi equations, cauchy problem, dissipation, theorem, t-products, asymptotic methods, calculus, Noncommutative, Operators, operational calculus

Collections: mir-titles, additional collections

Pages Count: 300

PPI Count: 300

PDF Count: 1

Total Size: 455.48 MB

PDF Size: 34.23 MB

Extensions: epub, pdf, gz, html, zip, torrent

Year: 1976

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### Description

In this post, we will see the book Operational Methods by V. P. Maslov.

...This book is devoted to one, but sufficiently general operational method, which absorbs many operational methods known to date and allows for the uniform solution of both classical problems, involving differential equations with partial derivatives, and the absolutely new problems of mathematical physics, including those connected with non-linear equations in partial derivatives.

...This book on operational methods should be accessible to senior course students of mathematics and physics faculties at universities and departments of applied mathematics. This means that only a knowledge of classical analysis is required of the reader. The book provides explanations in sufficient volume of such concepts as the theory of Banach algebras of distributions (Chapter I), the theory of linear differential and difference equations (Sees. 1, 2, and 3 of Introduction), the theory of non-linear equations of the first order with partial derivatives (Chapter IV). This material may be also of use to the reader who is already familiar with these questions, because rather often it is not presented in traditional style, and adapted for further reference. The reader who studies the book thoroughly will be equipped to carry on independent research in the modern theory of linear, non-linear differential and differential-difference equations with partial derivatives.

...This book has been written in such a way as to serve the widest possible circle of readers. It is suitable for two methods of study. The reader, who seeks to avoid fine assessments and passing to the limit and only wishes to master the practical techniques for obtaining asymptotic solutions, may omit that part of the book which is devoted to functional analysis.

...The most effective way of mastering the subject, however, consists rather in first reading Introduction and then reading all the book in succession. The reader should nevertheless be warned that all these methods are not at all easy, because the book provides a new operational calculus-the calculus of ordered operators.

The book was translated from the Russian by V. Golo, N. Kulman and G. Voropaeva and was published by Mir in 1976.

Credits to the original uploader for the scan, in this link we have converted to pdf from djvu, added bookmarks and cover.

The Internet Archive Link

CONTENTS

Preface 7

Introduction to Operational Calculus 13

Sec. 1. Solution of Ordinary Differential Equations by the Heaviside Operational Method 13

Sec. 2. Difference Equations 20

Sec. 3. Solution of Systems of Differential Equations by the Heaviside Operational Method 22

Sec. 4. Algebra of Convergent Power Series of Noncommutative Operators 24

Sec. 5. Spectrum of a Pair of Ordered Operators 35

Sec. 6. Algebras with \mu-Structures 40

Sec. 7. An Example of a Solution of a Differential Equation 56

Sec. 8. Passage of the Equation of Oscillations of a Crystal Lattice into a Wave Equation 58

Sec. 9. The Concept of a Quasi-Inverse Operator and Formulation of

the Main Theorem 100

Chapter I Functions of a Regular Operator 147

Sec. 1. Certain Spaces of Continuous Functions and Related Spaces 149

Sec. 2. Embedding Theorems 154

Sec. 3. The Algebra of Functions of a Generator 158

Sec. 4. The Extension of the Class of Possible Symbols 173

Sec. 5. Homomorphism of Asymptotic Formulas. The Method of Stationary Phase 181

Sec. 6. The Spectrum of a Generator 188

Sec. 7. Regular Operators 194

Sec. 8. The Generalized Eigenfunctions and Associated Functions 198

Sec. 9. Self-Adjoint Operators as Transformers in the Schmidt Space 205

Chapter II Calculus of Noncommutative Operators 210

Sec. 1. Preliminary Definitions 210

Sec. 2. The Functions of Two Noncommutative Self-Adjoint Operators 224

Sec. 3. The Functions of Noncommutative Operators 228

Sec. 4. The Spectrum of a Vector-Operator 231

Sec. 5. Theorem on Homomorphism 239

Sec. 6. Problems 242

Sec. 7. Differentiation of the Functions of an Operator Depending on a Parameter 251

Sec. 8. Formulas of Commutation 256

Sec. 9. Growing Symbols 261

Sec. 10. The Factor-Spectrum 265

Sec. 11. The Functions of Components of a Lie Nilpotent Algebra and Their Representations 266

Chapter III Asymptotic Methods 273

Sec. 1. Canonical Transformations of Pseudodifferential Operators 273

Sec. 2. The Homomorphism of Asymptotic Formulas 294

Sec. 3. The Geometrical Interpretation of the Method of Stationary

Phase 301

Sec. 4. The Canonical Operator on an Unclosed Curve 303

Sec. 5. The Method of Stationary Phase 312

Sec. 6. The Canonical Operator on the Unclosed Curve Depending on Parameters Defined Correct to 0 ( 1/\omega ) 315

Sec. 7. V-Objects on the Curve 321

Sec. 8. The Canonical Operator on the Family of Unclosed Curves 327

Sec. 9. The Canonical Operator on the Family of Closed Curves 333

Sec. 10. An Example of Commutation of a Canonical Operator with a Hamiltonian 339

Sec. 11. Commutation of a Hamiltonian with a Canonical Operator 346

Sec. 12. The General Canonical Transformation of the Pseudodifferential Operator 348

Chapter IV Generalized Hamilton-Jacobi Equations 355

Sec. 1. Hamilton-Jacobi Equations with Dissipation 356

Sec. 2. The Lagrangean Manifold with a Complex Germ 360

Sec. 3. y-Atlases and the Dissipativity Inequality 372

Sec. 4. Solution of the Hamilton-Jacobi Equation with Dissipation 378

Sec. 5. Preservation of the Dissipativity Inequality. Bypassing Focuses Operation 386

Sec. 6. Solution of Transfer Equation with Dissipation 401

Chapter V Canonical Operator on a Lagrangean Manifold with a Complex Germ and Proof of the Main Theorem 419

Sec. 1. Quantum Bypassing Focuses Operation 419

Sec. 2. Commutation Formulas for a Complex Exponential and a Hamiltonian 440

Sec. 3. C-Lagrangean Manifolds and the Index of a Complex Germ 452

Sec. 4. Canonical Operator 469

Sec. 5. Proof of the Main Theorem 482

Appendix to Sec. 5 493

Sec. 6. Cauchy Problem for Systems with Complex Characteristics 503

Sec. 7. Quasi-Inverse of Operators with Matrix Symbols 519

Appendix. Spectral Expansion of T-products 545

Index 557