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|a 9783540695943
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|a neuberger, john
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|a Sobolev Gradients and Differential Equations
|h Elektronische Ressource
|c by john neuberger
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| 250 |
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|a 1st ed. 1997
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| 260 |
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|a Berlin, Heidelberg
|b Springer Berlin Heidelberg
|c 1997, 1997
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| 300 |
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|a VIII, 152 p
|b online resource
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|a Several gradients -- Comparison of two gradients -- Continuous steepest descent in Hilbert space: Linear case -- Continuous steepest descent in Hilbert space: Nonlinear case -- Orthogonal projections, Adjoints and Laplacians -- Introducing boundary conditions -- Newton's method in the context of Sobolev gradients -- Finite difference setting: the inner product case -- Sobolev gradients for weak solutions: Function space case -- Sobolev gradients in non-inner product spaces: Introduction -- The superconductivity equations of Ginzburg-Landau -- Minimal surfaces -- Flow problems and non-inner product Sobolev spaces -- Foliations as a guide to boundary conditions -- Some related iterative methods for differential equations -- A related analytic iteration method -- Steepest descent for conservation equations -- A sample computer code with notes
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| 653 |
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|a Numerical Analysis
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| 653 |
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|a Numerical analysis
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| 653 |
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|a Differential Equations
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| 653 |
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|a Differential equations
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| 041 |
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|a eng
|2 ISO 639-2
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| 989 |
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|b SBA
|a Springer Book Archives -2004
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|a Lecture Notes in Mathematics
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| 028 |
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|a 10.1007/BFb0092831
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| 856 |
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|u https://doi.org/10.1007/BFb0092831?nosfx=y
|x Verlag
|3 Volltext
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| 082 |
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|a 515.35
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| 520 |
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|a A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. Equal emphasis is placed on numerical and theoretical matters. Several concrete applications are made to illustrate the method. These applications include (1) Ginzburg-Landau functionals of superconductivity, (2) problems of transonic flow in which type depends locally on nonlinearities, and (3) minimal surface problems. Sobolev gradient constructions rely on a study of orthogonal projections onto graphs of closed densely defined linear transformations from one Hilbert space to another. These developments use work of Weyl, von Neumann and Beurling
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