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AuthorDautray, Robert. author
TitleMathematical Analysis and Numerical Methods for Science and Technology [electronic resource] : Volume 2 Functional and Variational Methods / by Robert Dautray, Jacques-Louis Lions
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 2000
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Descript XVI, 590 p. online resource


These 6 volumes - the result of a 10 year collaboration between the authors, two of France's leading scientists and both distinguished international figures - compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the "Methoden der mathematischen Physik" by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form. The advent of large computers has in the meantime revolutionised methods of computation and made this gap in the literature intolerable: the objective of the present work is to fill just this gap. Many phenomena in physical mathematics may be modeled by a system of partial differential equations in distributed systems: a model here means a set of equations, which together with given boundary data and, if the phenomenon is evolving in time, initial data, defines the system. The advent of high-speed computers has made it possible for the first time to calculate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every facet of technical and industrial activity has been affected by these developments. Modeling by distributed systems now also supports work in many areas of physics (plasmas, new materials, astrophysics, geophysics), chemistry and mechanics and is finding increasing use in the life sciences


III. Functional Transformations -- A. Some Transformations Useful in Applications -- ยง1. Fourier Series and Dirichletโ{128}{153}s Problem -- ยง2. The Mellin Transform -- ยง3. The Hankel Transform -- Review of Chapter III A -- B. Discrete Fourier Transforms and Fast Fourier Transforms -- ยง1. Introduction -- ยง2. Acceleration of the Product of a Matrix by a Vector -- ยง3. The Fast Fourier Transform of Cooley and Tukey -- ยง4. The Fast Fourier Transform of Good-Winograd -- ยง5. Reduction of the Number of Multiplications -- ยง6. Fast Fourier Transform in Two Dimensions -- ยง7. Some Applications of the Fast Fourier Transform -- Review of Chapter III B -- IV. Sobolev Spaces -- ยง1. Spaces H1(?), Hm(?) -- ยง2. The Space Hs(?n) -- ยง3. Sobolevโ{128}{153}s Embedding Theorem -- ยง4. Density and Trace Theorems for the Spaces Hm(?), (m ? ? * = ?\z0{) -- ยง5. The Spaces H-m(?) for all m ? ? -- ยง6. Compactness -- ยง7. Some Inequalities in Sobolev Spaces -- ยง8. Supplementary Remarks -- Review of Chapter IV -- Appendix: The Spaces Hs(?) with ? the โ{128}{156}Regularโ{128}{157} Boundary of an Open Set ? in ?n -- V. Linear Differential Operators -- ยง1. Generalities on Linear Differential Operators -- ยง2. Linear Differential Operators with Constant Coefficients -- ยง3. Cauchy Problem for Differential Operators with Constant Coefficients -- ยง4. Local Regularity of Solutions* -- ยง5. The Maximum Principle * -- Review of Chapter V -- VI. Operators in Banach Spaces and in Hilbert Spaces -- ยง1. Review of Functional Analysis: Banach and Hilbert Spaces -- ยง2. Linear Operators in Banach Spaces -- ยง3. Linear Operators in Hilbert Spaces -- Review of Chapter VI -- VII. Linear Variational Problems. Regularity -- ยง1. Elliptic Variational Theory -- ยง2. Examples of Second Order Elliptic Problems -- ยง3. Regularity of the Solutions of Variational Problems -- Review of Chapter VII -- Appendix. โ{128}{156}Distributionsโ{128}{157} -- ยง1. Definition and Basic Properties of Distributions -- ยง2. Convolution of Distributions -- ยง3. Fourier Transforms -- Table of Notations -- of Volumes 1, 3โ{128}{147}6

Mathematics Partial differential equations System theory Numerical analysis Calculus of variations Physics Mathematics Partial Differential Equations Numerical Analysis Theoretical Mathematical and Computational Physics Systems Theory Control Calculus of Variations and Optimal Control; Optimization


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