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Partial Differential Equations

Partial Differential Equations


  • Date Published: May 1987
  • availability: Available
  • format: Paperback
  • isbn: 9780521277594

£ 74.99

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About the Authors
  • This book is a rigorous introduction to the abstract theory of partial differential equations. The main prerequisite is familiarity with basic functional analysis: more advanced topics such as Fredholm operators, the Schauder fixed point theorem and Bochner integrals are introduced when needed, and the book begins by introducing the necessary material from the theory of distributions and Sobolev spaces. Using such techniques, the author presents different methods available for solving elliptic, parabolic and hyperbolic equations. He also considers the difference process for the practical solution of a partial differential equation, emphasising that it is possible to solve them numerically by simple methods. Many examples and exercises are provided throughout, and care is taken to explain difficult points. Advanced undergraduates and graduate students will appreciate this self-contained and practical introduction.

    Reviews & endorsements

    'The work under discussion is a nice presentation of PDEs and proves to be a valuable source for undergraduate and graduate students at all levels. It highlights the importance of studying differential equations both in the setting of classical solutions as well as weak solutions.' Marius Ghergu, European Mathematical Society

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    Product details

    • Date Published: May 1987
    • format: Paperback
    • isbn: 9780521277594
    • length: 532 pages
    • dimensions: 227 x 151 x 34 mm
    • weight: 0.774kg
    • availability: Available
  • Table of Contents

    Part I. Sobolev Spaces:
    1. Notation, basic properties, distributions
    2. Geometric assumptions for the domain
    3. Definitions and density properties for the Sobolev-Slobodeckii spaces
    4. The transformation theorem and Sobolev spaces on differentiable manifolds
    5. Definition of Sobolev spaces by the Fourier transformation and extension theorems
    6. Continuous embeddings and Sobolev's lemma
    7. Compact embeddings
    8. The trace operator
    9. Weak sequential compactness and approximation of derivatives by difference quotients
    Part II. Elliptic Differential Operators:
    10. Linear differential operators
    11. The Lopatinskil-Sapiro condition and examples
    12. Fredholm operators
    13. The main theorem and some theorems on the index of elliptic boundary value problems
    14. Green's formulae
    15. The adjoint boundary value problem and the connection with the image space of the original operator
    16. Examples
    Part III. Strongly Elliptic Differential Operators and the Method of Variations:
    17. Gelfand triples, the Law-Milgram, V-elliptic and V-coercive operators
    18. Agmon's condition
    19. Agmon's theorem: conditions for the V-coercion of strongly elliptic differential operators
    20. Regularity of the solutions of strongly elliptic equations
    21. The solution theorem for strongly elliptic equations and examples
    22. The Schauder fixed point theorem and a non-linear problem
    23. Elliptic boundary value problemss for unbounded regions
    Part IV. Parabolic Differential Operators:
    24. The Bochner integral
    25. Distributions with values in a Hilbert space H and the space W
    26. The existence and uniqueness of the solution of a parabolic differential equation
    27. The regularity of solutions of the parabolic differential equation
    28. Examples
    Part V. Hyperbolic Differential Operators:
    29. Existence and uniqueness of the solution
    30. Regularity of the solutions of the hyperbolic differential equation
    Part VI. Difference Processes for the Calculation of the Solution of the Partial Differential Equation:
    32. Functional analytic concepts for difference processes
    33. Difference processes for elliptic differential equations and for the wave equation
    34. Evolution equations
    Function and distribution spaces

  • Author

    J. Wloka

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