LEADER 05087nam 22006615 450 001 996466489703316 005 20200702030724.0 010 $a1-280-39157-X 010 $a9786613569493 010 $a3-642-04041-1 024 7 $a10.1007/978-3-642-04041-2 035 $a(CKB)2670000000003378 035 $a(SSID)ssj0000315625 035 $a(PQKBManifestationID)11238919 035 $a(PQKBTitleCode)TC0000315625 035 $a(PQKBWorkID)10255120 035 $a(PQKB)10921773 035 $a(DE-He213)978-3-642-04041-2 035 $a(MiAaPQ)EBC3064811 035 $a(PPN)149054173 035 $a(EXLCZ)992670000000003378 100 $a20100301d2010 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aSobolev Gradients and Differential Equations$b[electronic resource] /$fby john neuberger 205 $a2nd ed. 2010. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d2010. 215 $a1 online resource (XIII, 289 p.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v1670 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-642-04040-3 320 $aIncludes bibliographical references (p. [277]-286) and index. 327 $aSeveral 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 -- Ordinary Differential Equations and Sobolev Gradients -- Convexity and Gradient Inequalities -- Boundary and Supplementary Conditions -- Continuous Newton#x2019;s Method -- More About Finite Differences -- Sobolev Gradients for Variational Problems -- An Introduction to Sobolev Gradients in Non-Inner Product Spaces -- Singularities and a Simple Ginzburg-Landau Functional -- The Superconductivity Equations of Ginzburg-Landau -- Tricomi Equation: A Case Study -- Minimal Surfaces -- Flow Problems and Non-Inner Product Sobolev Spaces -- An Alternate Approach to Time-dependent PDEs -- Foliations and Supplementary Conditions I -- Foliations and Supplementary Conditions II -- Some Related Iterative Methods for Differential Equations -- An Analytic Iteration Method -- Steepest Descent for Conservation Equations -- Code for an Ordinary Differential Equation -- Geometric Curve Modeling with Sobolev Gradients -- Numerical Differentiation, Sobolev Gradients -- Steepest Descent and Newton#x2019;s Method and Elliptic PDE -- Ginzburg-Landau Separation Problems -- Numerical Preconditioning Methods for Elliptic PDEs -- More Results on Sobolev Gradient Problems -- Notes and Suggestions for Future Work. 330 $aA Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that functional taken relative to an underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. For discrete versions of partial differential equations, corresponding Sobolev gradients are seen to be vastly more efficient than ordinary gradients. In fact, descent methods with these gradients generally scale linearly with the number of grid points, in sharp contrast with the use of ordinary gradients. Aside from the first edition of this work, this is the only known account of Sobolev gradients in book form. Most of the applications in this book have emerged since the first edition was published some twelve years ago. What remains of the first edition has been extensively revised. There are a number of plots of results from calculations and a sample MatLab code is included for a simple problem. Those working through a fair portion of the material have in the past been able to use the theory on their own applications and also gain an appreciation of the possibility of a rather comprehensive point of view on the subject of partial differential equations. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v1670 606 $aMathematical analysis 606 $aAnalysis (Mathematics) 606 $aPartial differential equations 606 $aNumerical analysis 606 $aAnalysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12007 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 606 $aNumerical Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M14050 615 0$aMathematical analysis. 615 0$aAnalysis (Mathematics). 615 0$aPartial differential equations. 615 0$aNumerical analysis. 615 14$aAnalysis. 615 24$aPartial Differential Equations. 615 24$aNumerical Analysis. 676 $a515/.353 700 $aneuberger$b john$4aut$4http://id.loc.gov/vocabulary/relators/aut$0878610 906 $aBOOK 912 $a996466489703316 996 $aSobolev Gradients and Differential Equations$91961575 997 $aUNISA