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Algebraic Approaches to Partial Differential Equations

  • Book
  • © 2013

Overview

Authors:
  1. Xiaoping Xu

    1. , Institute of Mathematics, Academy of Mathematics and System Scienc, Beijing, China, People's Republic

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  • Fundamental algebraic techniques of solving PDEs
  • Exact solutions to physical equations
  • Accessibility to general audience
  • Includes supplementary material: sn.pub/extras

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Table of contents (10 chapters)

  1. Front Matter

    Pages I-XXIV
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  2. Ordinary Differential Equations

    1. Front Matter

      Pages 1-1
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    2. First-Order Ordinary Differential Equations

      • Xiaoping Xu
      Pages 3-16

    3. Higher Order Ordinary Differential Equations

      • Xiaoping Xu
      Pages 17-35

    4. Special Functions

      • Xiaoping Xu
      Pages 37-63

  3. Partial Differential Equations

    1. Front Matter

      Pages 65-65
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    2. First-Order or Linear Equations

      • Xiaoping Xu
      Pages 67-140

    3. Nonlinear Scalar Equations

      • Xiaoping Xu
      Pages 141-178

    4. Nonlinear Schrödinger and Davey–Stewartson Equations

      • Xiaoping Xu
      Pages 179-211

    5. Dynamic Convection in a Sea

      • Xiaoping Xu
      Pages 213-230

    6. Boussinesq Equations in Geophysics

      • Xiaoping Xu
      Pages 231-267

    7. Navier–Stokes Equations

      • Xiaoping Xu
      Pages 269-316

    8. Classical Boundary Layer Problems

      • Xiaoping Xu
      Pages 317-383

  4. Back Matter

    Pages 385-394
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Keywords

About this book

This book presents the various algebraic techniques for solving partial differential equations to yield exact solutions, techniques developed by the author in recent years and with emphasis on physical equations such as: the Maxwell equations, the Dirac equations, the KdV equation,  the KP equation,  the nonlinear Schrodinger equation,  the Davey and Stewartson equations, the Boussinesq equations in geophysics,  the Navier-Stokes equations and the boundary layer problems.  In order to solve them, I have employed the grading technique, matrix differential operators, stable-range of nonlinear terms, moving frames, asymmetric assumptions,  symmetry transformations,  linearization techniques  and  special functions. The book is self-contained and requires only a minimal understanding of calculus and linear algebra, making it accessible to a broad audience in the fields of mathematics, the sciences and engineering. Readers may find the exact solutions and mathematical skills needed in their own research.

Reviews

From the book reviews:

“Professor Xu’s treatise is an interesting addition to the literature on exact/explicit solutions to nonlinear partial differential equations. … the present book will be useful for a large audience of applied mathematicians, specialists in numerical analysis, physicists and engineers.” (Enrique G. Reyes, Mathematical Reviews, August, 2014)

Authors and Affiliations

  • , Institute of Mathematics, Academy of Mathematics and System Scienc, Beijing, China, People's Republic

    Xiaoping Xu

About the author

The author received his Ph.D. from Rutgers University, USA in 1992. He is currently a research professor at the Chinese Academy of Sciences’ Institute of Mathematics, and  has been working on representation theory and applied partial differential equations for twenty years, during which he has published over fifty substantial research papers and two monographs on mathematics.

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