偏微分方程式のための方法(テキスト)<br>Methods for Partial Differential Equations : Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models (2018)

個数:
電子版価格
¥12,152
  • 電子版あり

偏微分方程式のための方法(テキスト)
Methods for Partial Differential Equations : Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models (2018)

  • 提携先の海外書籍取次会社に在庫がございます。通常3週間で発送いたします。
    重要ご説明事項
    1. 納期遅延や、ご入手不能となる場合が若干ございます。
    2. 複数冊ご注文の場合は、ご注文数量が揃ってからまとめて発送いたします。
    3. 美品のご指定は承りかねます。

    ●3Dセキュア導入とクレジットカードによるお支払いについて
  • 【入荷遅延について】
    世界情勢の影響により、海外からお取り寄せとなる洋書・洋古書の入荷が、表示している標準的な納期よりも遅延する場合がございます。
    おそれいりますが、あらかじめご了承くださいますようお願い申し上げます。
  • ◆画像の表紙や帯等は実物とは異なる場合があります。
  • ◆ウェブストアでの洋書販売価格は、弊社店舗等での販売価格とは異なります。
    また、洋書販売価格は、ご注文確定時点での日本円価格となります。
    ご注文確定後に、同じ洋書の販売価格が変動しても、それは反映されません。
  • 製本 Hardcover:ハードカバー版/ページ数 456 p.
  • 商品コード 9783319664552

Full Description

This book provides an overview of different topics related to the theory of partial differential equations. Selected exercises are included at the end of each chapter to prepare readers for the "research project for beginners" proposed at the end of the book. It is a valuable resource for advanced graduates and undergraduate students who are interested in specializing in this area.

The book is organized in five parts:

In Part 1 the authors review the basics and the mathematical prerequisites, presenting two of the most fundamental results in the theory of partial differential equations: the Cauchy-Kovalevskaja theorem and Holmgren's uniqueness theorem in its classical and abstract form. It also introduces the method of characteristics in detail and applies this method to the study of Burger's equation.

Part 2 focuses on qualitative properties of solutions to basic partial differential equations, explaining the usual properties of solutions to elliptic, parabolic and hyperbolic equations for the archetypes Laplace equation, heat equation and wave equation as well as the different features of each theory. It also discusses the notion of energy of solutions, a highly effective tool for the treatment of non-stationary or evolution models and shows how to define energies for different models.

Part 3 demonstrates how phase space analysis and interpolation techniques are used to prove decay estimates for solutions on and away from the conjugate line. It also examines how terms of lower order (mass or dissipation) or additional regularity of the data may influence expected results.

Part 4 addresses semilinear models with power type non-linearity of source and absorbing type in order to determine critical exponents: two well-known critical exponents, the Fujita exponent and the Strauss exponent come into play. Depending on concrete models these critical exponents divide the range of admissible powers in classes which make it possible toprove quite different qualitative properties of solutions, for example, the stability of the zero solution or blow-up behavior of local (in time) solutions.
The last part features selected research projects and general background material.

Contents

Part 1.- Introduction.- Part 2.- Partial differential equations in models.- Basics for partial differential equations.- The Cauchy-Kovalevskaja theorem.- Holmgren's uniqueness theorem.- Method of characteristics.- Burger's equation.- Laplace equation - properties of solutions - starting point of elliptic theory.- Heat equation - properties of solutions - starting point of parabolic theory.- Wave equation - properties of solutions - starting point of hyperbolic theory.- Energies of solutions - one of the most important quantities.- Part 3.- Phase space analysis for heat equation.- Phase space analysis and smoothing for Schrödinger equations.- Phase space analysis for wave models.- Phase space analysis for plate models.- The method of stationary phase and applications.- Part 4.- Semilinear heat models.- Semilinear classical damped wave models.- Semilinear wave models with a special structural dissipation.- Semilinear classical wave models.- Semilinear Schrödinger models.- Linear hyperbolic systems.- Part 5.- Research projects for beginners.- Background material.

最近チェックした商品