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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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Partial Differential Equations and Boundary Value Problems (Hardcover, 1998 ed.)
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Partial Differential Equations and Boundary Value Problems (Hardcover, 1998 ed.)
Series: Mathematics and Its Applications, 441
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The material of the present book has been used for graduate-level
courses at the University of Ia i during the past ten years. It is
a revised version of a book which appeared in Romanian in 1993 with
the Publishing House of the Romanian Academy. The book focuses on
classical boundary value problems for the principal equations of
mathematical physics: second order elliptic equations (the Poisson
equations), heat equations and wave equations. The existence theory
of second order elliptic boundary value problems was a great
challenge for nineteenth century mathematics and its development
was marked by two decisive steps. Undoubtedly, the first one was
the Fredholm proof in 1900 of the existence of solutions to
Dirichlet and Neumann problems, which represented a triumph of the
classical theory of partial differential equations. The second step
is due to S. 1. Sobolev (1937) who introduced the concept of weak
solution in partial differential equations and inaugurated the
modern theory of boundary value problems. The classical theory
which is a product ofthe nineteenth century, is concerned with
smooth (continuously differentiable) sollutions and its methods
rely on classical analysis and in particular on potential theory.
The modern theory concerns distributional (weak) solutions and
relies on analysis of Sob ole v spaces and functional methods. The
same distinction is valid for the boundary value problems
associated with heat and wave equations. Both aspects of the theory
are present in this book though it is not exhaustive in any sense.
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