## Partial Differential Equations – Mikhailov

In this post we will see the book Partial Differential Equations by V. P. Mikhailov.

This book has developed from courses of lectures given by the
author over a period of years to the students of the Moscow PhysicoTechnical
Institute. It is intended for the students having basic
knowledge of mathematical analysis, algebra and the theory of
ordinary differential equations to the extent of a university course.

Except Chapter I, where some general questions regarding partial
differential equations have been examined, the material has been
arranged so as to correspond to the basic types of equations. The
central role in the book is played by Chapter IV, the largest of all,
which discusses elliptic equations. Chapters V and VI are devoted
to the hyperbolic and parabolic equations.

The method used in this book for investigating the boundary value
problems and, partly, the Cauchy problem is based on the notion
of generalized solution which enables us to examine equations with
variable coefficients with the same ease as the simplest equations:
Poisson’s equation, wave equation and heat equation. Apart from
discussing the questions of existence and uniqueness of solutions of
the basic boundary value problems, considerable space has been
devoted to the approximate methods of solving these equations:
Ritz’s method in the case of elliptic equations and Galerkin’s
method for hyperbolic and parabolic equations.

The book was translated from the Russian by P. C. Sinha and was first published by Mir Publishers in 1978.

PDF | OCR | Cover | 600 dpi | Bookmarked | Paginated | 22.4 MB (15.6 MB Zipped) | 408 pages

(Note: IA file parameters maybe different.)

You can get the book here (IA) and here (filecloud).

Password, if needed: mirtitles

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CONTENTS
\Preface 7
CHAPTER I
INTRODUCTION. CLASSIFICATION OF EQUATIONS. FORMULATION OF SOME PROBLEMS

§1. The Cauchy Problem. Kovalevskaya’s Theorem 12
§2. Classification of Linear Differential Equations of the Second Order 31
§3. Formulation of Some Problems 34

Problems on Chapter I 41
Suggested Reading on Chapter I 41

CHAPTER II
THE LEBESGUE INTEGRAL AND SOME QUESTIONS OF FUNCTIONAL ANALYSIS

§1. The Lebesgue Integral 42
§2. Normed Linear Spaces. Hilbert Space 64
§3. Linear Operators. Compact Sets. Completely Continuous Operators 72
§4. Linear Equations in a Hilbert Space 85
§5. Selfadjoint Completely Continuous Operators 94

CHAPTER III
FUNCTION SPACES

§1. Spaces of Continuous and Continuously Differentiable Functions 101
§2. Spaces of Integrable Functions 104
§3. Generalized Derivatives 111
§4. Spaces Hk(Q) 121
§5. Properties of Functions Belonging to H1(Q) and H1(Q) 135
§6. Properties of Functions Belonging to Hk(Q) 149
§7. Spaces cr,0andC2s,8• SpacesHT,0andH28,8 155
§8. Examples of Operators in Function Spaces 161

Problems on Chapter III 166
Suggested Reading on Chapter III 168

CHAPTER IV
ELLIPTIC EQUATIONS

§1. Generalized Solutions of Boundary-Value Problems. Eigenvalue Problems 169
§2. Smoothness of Generalized Solutions. Classical Solutions 208
§3. Classical Solutions of Laplace’s and Poisson’s Equations 232

Problems on Chapter IV 261
Suggested Reading on Chapter IV 264

CHAPTER V
HYPERBOLIC EQUATIONS

§1. Properties of Solutions of Wave Equation. The Cauchy Problem for Wave Equation 266
§2. Mixed Problems 284
§3. Generalized Solution of the Cauchy Problem 328

Problems on Chapter V 339
Suggested Reading on Chapter V 341

CHAPTER VI
PARABOLIC EQUATIONS

§1. Properties of Solutions of Heat Equation. The Cauchy Problem for Heat Equation 342
§2. Mixed Problems 362

Problems on Chapter VI 388
Suggested Reading on Chapter VI 391

Index 392

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### 4 Responses to Partial Differential Equations – Mikhailov

1. reporteroz says:

Hello,

Many thanks for posting this book but your post link appears dead, you might already be on it. Also the download links point to the *Problems in Higher Mathematics *by Minorsky book.

Cheers,

Neva

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• The Mitr says:

Ah, it got published by a mistake. Will rectify it soon with proper links.

D

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2. Ariel says:

Thank you for posting these references, they are as wonderful today as they have always been. Someday someone will explain it better, but for now I think that there is something about Mir that is very special, a rare combination of simplicity, generosity, and the highest levels of intellectual achievement. When so many cultures throughout the world were told to learn English, or German, or French, if they wanted to access this knowledge–there was Mir providing a lovely counterexample.

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• shrikant says:

Dear Ariel,
you are absolutely correct.
shrikant

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