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Partial Differential Equations

New Methods for Their Treatment and Solution

  • Richard Bellman
  • George Adomian

Part of the Mathematics and Its Applications book series (MAIA, volume 15)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Richard Bellman, George Adomian
    Pages 1-4
  3. Richard Bellman, George Adomian
    Pages 5-27
  4. Richard Bellman, George Adomian
    Pages 28-35
  5. Richard Bellman, George Adomian
    Pages 36-58
  6. Richard Bellman, George Adomian
    Pages 59-61
  7. Richard Bellman, George Adomian
    Pages 62-81
  8. Richard Bellman, George Adomian
    Pages 82-87
  9. Richard Bellman, George Adomian
    Pages 88-102
  10. Richard Bellman, George Adomian
    Pages 103-109
  11. Richard Bellman, George Adomian
    Pages 110-119
  12. Richard Bellman, George Adomian
    Pages 120-128
  13. Richard Bellman, George Adomian
    Pages 129-147
  14. Richard Bellman, George Adomian
    Pages 148-152
  15. Richard Bellman, George Adomian
    Pages 153-175
  16. Richard Bellman, George Adomian
    Pages 176-236
  17. Richard Bellman, George Adomian
    Pages 237-242
  18. Richard Bellman, George Adomian
    Pages 243-247
  19. Richard Bellman, George Adomian
    Pages 248-253
  20. Richard Bellman, George Adomian
    Pages 254-288
  21. Back Matter
    Pages 289-290

About this book

Introduction

The purpose of this book is to present some new methods in the treatment of partial differential equations. Some of these methods lead to effective numerical algorithms when combined with the digital computer. Also presented is a useful chapter on Green's functions which generalizes, after an introduction, to new methods of obtaining Green's functions for partial differential operators. Finally some very new material is presented on solving partial differential equations by Adomian's decomposition methodology. This method can yield realistic computable solutions for linear or non­ linear cases even for strong nonlinearities, and also for deterministic or stochastic cases - again even if strong stochasticity is involved. Some interesting examples are discussed here and are to be followed by a book dealing with frontier applications in physics and engineering. In Chapter I, it is shown that a use of positive operators can lead to monotone convergence for various classes of nonlinear partial differential equations. In Chapter II, the utility of conservation technique is shown. These techniques are suggested by physical principles. In Chapter III, it is shown that dyn~mic programming applied to variational problems leads to interesting classes of nonlinear partial differential equations. In Chapter IV, this is investigated in greater detail. In Chapter V, we show. that the use of a transformation suggested by dynamic programming leads to a new method of successive approximations.

Keywords

derivative difference equation differential equation functional equation minimum partial differential equation

Authors and affiliations

  • Richard Bellman
    • 1
    • 2
  • George Adomian
    • 2
  1. 1.Dept. of Electrical EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Center for Applied MathematicsThe University of GeorgiaAthensUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-009-5209-6
  • Copyright Information Springer Science+Business Media B.V. 1985
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-94-010-8804-6
  • Online ISBN 978-94-009-5209-6
  • Buy this book on publisher's site
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