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Partial differential equations and operators

Fundamental solutions and semigroups Part I

Partial differential equations and operators
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ISBN: 978-87-403-0346-9
1 edition
Pages : 169
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About the book

  1. Reviews
  2. Description
  3. Content
  4. About the Author


victor okai ★★★★★

It is very good and useful to scientists and engineers.


In this book, which is basically self-contained, we concentrate on partial differential equations in mathematical physics and on operator semigroups with their generators. A central theme is a thorough treatment of distribution theory. This is done via convolution products, Fourier transforms, and fundamental solutions of partial differential operators with constant coefficients. Linear initial value problems are treated via operator semigroups. A relationship between so-called Feller-Dynkin semigroups and Markov processes is described. Finally, Feynman-Kac semigroups are introduced.


The Author


  1. Distributions, differential operators and examples
    1. Introduction
    2. Topics to be treated in this book
    3. Partition of unity
    4. Test functions and distributions
    5. Tempered distributions and Fourier transforms
    6. Examples of Fourier transforms
  2. Fundamental solutions
    1. Introduction and examples
  3. Fundamental solutions of the wave operator
    1. Fundamental solutions of the wave operator in one space dimension
    2. Fundamental solutions of the wave equationin several space dimensions
  4. Proofs of some main results
    1. Convolution products: formulation of some results
    2. Fourier transform and its inverse
    3. Theorem of Malgrange and Ehrenpreis
    4. Sobolev theory
    5. Elliptic operators
    6. 1 Sobolev spaces
    7. Paley-Wiener theorems
    8. Multiplicative distributions
  5. To see Section 5–9 download Partial differential equations and operators: Part II

  6. Operators in Hilbert space Part
    1. Some results in Banach algebras
    2. Closed linear operators
  7. Operator semigroups and Markov processes
    1. Generalities on semigroups
    2. Examples
    3. Markov processes
    4. Feynman-Kac semigroups Part II
    5. Harmonic functions on a strip
  8. Elements of functional analysis
    1. Theorem of Hahn-Banach
    2. Banach-Steinhaus theorems: barreled spaces
  9. Subjects for further research and presentations
  10. Bibliography

About the Author

Since 2009 the author is retired from the University of Antwerp. Until the present day his teaching duties include a course on ``Partial Differential Equations and Operators’’ and one on ``Advanced Stochastic Processes’’. In the sixties the author was a student at the Catholic University of Nijmegen, Netherlands (nowadays Radboud University), and he earned his Ph.D. from the University of Hawaii, USA, (1971). Since 1972 he has been a member of the academic staff of the University of Antwerp, Department of Mathematics and Computer Science, Belgium. Most of his professional life he has been teaching courses in analysis and stochastic processes. His research lies in the area of stochastic analysis. A recent book authored by him is Markov Processes, Feller Semigroups and Evolution Equations, published by WSPC, Singapore, 2011, of about 800 pages.

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