Partial differential equations in action: from modelling to theory
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| Language: | English |
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[Cham], Switzerland
Springer
[2016]
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| Edition: | Third edition, reprinted with modifications |
| Series: | UNITEXT - La Matematica per il 3+2
volume 99 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xviii, 686 Seiten Diagramme |
| ISBN: | 9783319312378 |
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MARC
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| 100 | 1 | |a Salsa, Sandro |d 1950- |e Verfasser |0 (DE-588)1111815445 |4 aut | |
| 245 | 1 | 0 | |a Partial differential equations in action |b from modelling to theory |c Sandro Salsa |
| 250 | |a Third edition, reprinted with modifications | ||
| 264 | 1 | |a [Cham], Switzerland |b Springer |c [2016] | |
| 300 | |a xviii, 686 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a UNITEXT - La Matematica per il 3+2 |v volume 99 | |
| 650 | 4 | |a Mathematics | |
| 650 | 4 | |a Differential equations, partial | |
| 650 | 4 | |a Engineering mathematics | |
| 650 | 4 | |a Mathematical Modeling and Industrial Mathematics | |
| 650 | 4 | |a Partial Differential Equations | |
| 650 | 4 | |a Mathematical Applications in the Physical Sciences | |
| 650 | 4 | |a Appl.Mathematics/Computational Methods of Engineering | |
| 650 | 4 | |a Partial differential equations | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 4 | |a Mathematical models | |
| 650 | 4 | |a Applied mathematics | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Mathematisches Modell | |
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| 787 | 0 | 8 | |i Ergänzung |a Salsa, Sandro |t Partial differential equations in action |c complements and exercises |
| 830 | 0 | |a UNITEXT - La Matematica per il 3+2 |v volume 99 |w (DE-604)BV047304938 |9 99 | |
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Record in the Search Index
| _version_ | 1804179097013714944 |
|---|---|
| adam_text | Contents
1 Introduction.......................................................... 1
1.1 Mathematical Modelling........................................... 1
1.2 Partial Differential Equations................................... 2
1.3 Well Posed Problems.............................................. 5
1.4 Basic Notations and Facts........................................ 7
1.5 Smooth and Lipschitz Domains.................................... 12
1.6 Integration by Parts Formulas................................... 15
2 Diffusion........................................................... 17
2.1 The Diffusion Equation ......................................... 17
2.1.1 Introduction............................................. 17
2.1.2 The conduction of heat.................................. 18
2.1.3 Well posed problems (n = 1).............................. 20
2.1.4 A solution by separation of variables.................... 23
2.1.5 Problems in dimension n 1 ............................. 32
2.2 Uniqueness and Maximum Principles............................... 34
2.2.1 Integral method.......................................... 34
2.2.2 Maximum principles....................................... 36
2.3 The Fundamental Solution....................................... 39
2.3.1 Invariant transformations................................ 39
2.3.2 The fundamental solution (n — 1)......................... 41
2.3.3 The Dirac distribution................................... 43
2.3.4 The fundamental solution (n 1)......................... 47
2.4 Symmetric Random Walk (n = 1) .................................. 48
2.4.1 Preliminary computations................................. 49
2.4.2 The limit transition probability......................... 52
2.4.3 From random walk to Brownian motion...................... 54
2.5 Diffusion, Drift and Reaction .................................. 58
2.5.1 Random walk with drift................................... 58
2.5.2 Pollution in a channel................................... 60
2.5.3 Random walk with drift and reaction...................... 63
xii Contents
2.5.4 Critical dimension in a simple population dynamics....... 64
2.6 Multidimensional Random Walk.................................... 66
2.6.1 The symmetric case ...................................... 66
2.6.2 Walks with drift and reaction............................ TO
2.7 An Example of Reaction-Diffusion in Dimension n = 3............. 71
2.8 The Global Cauchy Problem (n = 1)............................... 76
2.8.1 The homogeneous case..................................... 76
2.8.2 Existence of a solution.................................. 78
2.8.3 The nonhomogeneous case. DuhameFs method................. 79
2.8.4 Global maximum principles and uniqueness................. 82
2.8.5 The proof of the existence theorem 2.12.................. 85
2.9 An Application to Finance. . ................................... 88
2.9.1 European options ........................................ 88
2.9.2 An evolution model for the price S....................... 89
2.9.3 The Black-Scholes equation............................... 91
2.9.4 The solutions............................................ 95
2.9.5 Hedging and self-financing strategy......................100
2.10 Some Nonlinear Aspects.........................................102
2.10.1 Nonlinear diffusion. The porous medium equation..........102
2.10.2 Nonlinear reaction. Fischer’s equation....................105
Problems..............................................................109
3 The Laplace Equation...................................................115
3.1 Introduction.....................................................115
3.2 Well Posed Problems. Uniqueness..................................116
3.3 Harmonic Functions...............................................118
3.3.1 Discrete harmonic functions ..............................118
3.3.2 Mean value properties ....................................122
3.3.3 Maximum principles........................................124
3.3.4 The Hopf principle........................................126
3.3.5 The Dirichlet problem in a disc. Poisson’s formula........127
3.3.6 Harnack’s inequality and Liouville’s theorem..............131
3.3.7 Analyticity of harmonic functions ........................133
3.4 A probabilistic solution of the Dirichlet problem................135
3.5 Sub/Superharmonic Functions. The Perron Method..................140
3.5.1 Sub/superlmrmonic functions...............................140
3.5.2 The method................................................142
3.5.3 Boundary behavior.........................................143
3.0 Fundamental Solution and Newtonian Potential.....................147
3.6.1 The fundamental solution .................................147
3.6.2 The Newtonian potential...................................148
3.6.3 A divergence-curl system. Helmholtz decomposition
formula...................................................151
3.7 The Green Function...............................................155
3.7.1 An integral identity......................................155
Contents
Xlll
3.7.2 Green’s function .........................................157
3.7.3 Green’s representation formula............................160
3.7.4 The Neumann function......................................161
3.8 Uniqueness in Unbounded Domains..................................163
3.8.1 Exterior problems.........................................163
3.9 Surface Potentials...............................................166
3.9.1 The double and single layer potentials ...................166
3.9.2 The integral equations of potential theory................171
Problems..............................................................174
Scalar Conservation Taws and First Order Equations...............179
4.1 Introduction.....................................................179
4.2 Linear Transport Equation .......................................180
4.2.1 Pollution in a channel....................................180
4.2.2 Distributed source........................................182
4.2.3 Extinction and localized source...........................183
4.2.4 Inflow and outflow characteristics. A stability estimate..185
4.3 Traffic Dynamics.................................................187
4.3.1 A macroscopic model.......................................187
4.3.2 The method of characteristics............................ 189
4.3.3 The green light problem...................................191
4.3.4 Traffic jam ahead....................................... 196
4.4 Weak (or Integral) Solutions.....................................199
4.4.1 The method of characteristics revisited...................199
4.4.2 Definition of weak solution ..............................202
4.4.3 Piecewise smooth functions and the Rankine-Hugoniot
condition.................................................205
4.5 An Entropy Condition.............................................209
4.6 The Riemann problem..............................................212
4.6.1 Convex/concave flux function..............................212
4.6.2 Vanishing viscosity method................................214
4.6.3 The viscous Burgers equation..............................218
4.6.4 Flux function with inflection points......................220
4.7 An Application to a Sedimentation Problem........................224
4.8 The Method of Characteristics for Quasilinear Equations..........230
4.8.1 Characteristics...........................................230
4.8.2 The Cauchy problem........................................232
4.8.3 Lagrange method of first integrals........................239
4.8.4 Underground flow..........................................241
4.9 General First Order Equations....................................244
4.9.1 Characteristic strips.....................................244
4.9.2 The Cauchy Problem........................................246
Problems............................................................. 251
XIV
Contents
5 Waves and Vibrations ...................................................259
5.1 General Concepts.................................................259
5.1.1 Types of waves.............................................259
5.1.2 Group velocity and dispersion relation....................261
5.2 Transversal Waves in a String....................................264
5.2.1 The model..................................................264
5.2.2 Energy.................................................. 266
5.3 The One-dimensional Wave Equation................................267
5.3.1 Initial and boundary conditions............................267
5.3.2 Separation of variables....................................269
5.4 The d’Alembert Formula...........................................275
5.4.1 The homogeneous equation...................................275
5.4.2 Generalized solutions and propagation of singularities...279
5.4.3 The fundamental solution...................................282
5.4.4 Nonhomogeneous equation. Duhamel’s method..................285
5.4.5 Dissipation and dispersion.................................286
5.5 Second Order Linear Equations....................................288
5.5.1 Classification.............................................288
5.5.2 Characteristics and canonical form ........................291
5.6 The Multi-dimensional Wave Equation (n 1)....................296
5.6.1 Special solutions..........................................296
5.6.2 Well posed problems. Uniqueness............................298
5.7 Two Classical Models.............................................302
5.7.1 Small vibrations of an elastic membrane....................302
5.7.2 Small amplitude sound waves................................306
5.8 The Global Cauchy Problem........................................310
5.8.1 Fundamental solution (n = 3) and strong Huygens5
principle.................................................310
5.8.2 The Kirchhoff formula......................................313
5.8.3 The Cauchy problem in dimension 2.......................316
5.9 The Cauchy Problem with Distributed Sources......................318
5.9.1 Retarded potentials (n = 3)................................318
5.9.2 Radiation from a moving point source.......................320
5.10 An Application to Thermoacoustic Tomography......................324
5.11 Linear Water Waves...............................................328
5.11.1 A model for surface waves.................................328
5.11.2 Dimensionless formulation and linearization...............332
5.11.3 Deep water waves..........................................334
5.11.4 Interpretation of the solution............................336
5.11.5 Asymptotic behavior.......................................338
5.11.6 The method of stationary phase............................340
Problems..............................................................342
( ont ont s
xv
G Elements of Functional Analysis......................................, M7
0.1 Motivations......................................................547
0.2 Norms and Hanacli Space1«.........................................’5ad
0.5 Hilbert Spares...................................................558
0.1 Project ions and Hases...........................................505
0.4.1 1 Vojert ions.............................................505
0.4.2 Hases.....................................................507
0.5 Linear Operators and Duality.....................................ATA
0.5.1 Linear operators .........................................AT A
0.5.2 Funot ionals and dual space...............................577
0.5.5 44k* adjoint of a bounded operator........................579
0.0 Abstract Variational Problems....................................582
0.0.1 Bilinear forms and t lx* Lax-Milgram Theorem..............582
0.0.2 Minimization of quadratic functionals.....................587
0.0.5 Approximation and Galerkin method ..................... .588
0.7 Compactness and Weak Convergence.................................591
0.7.1 Compactness ..............................................591
0.7.2 Compactness in CJ(f2) and in IJ}(£2) .....................592
0.7.5 Weak convergence and compactness..........................595
0.7.4 Compact, operators........................................597
0.8 The Fredholm Alternative.........................................599
0.8.1 Hilbert, triplets.........................................599
0.8.2 Solvability for abstract variational problems ............402
0.8.5 Fredholm’s alternative....................................405
0.9 Spectral Theory for Symmetric Bilinear Forms....................407
0.9.1 Spectrum of a, mat rix....................................407
0.9.2 Separation of variables revisited....................... 407
0.9.5 Spectrum of a compact self-adjoint operator...............408
0.9.4 Application to abstract variational problems..............411
0.10 Fixed Points Theorems...........................................410
0.10.1 The Cont raction Mapping Theorem.........................417
0.10.2 The Sehauder Theorem ....................................418
0.10.5 The Leray-Schauder Theorem...............................420
Problems ............................................................421
7 Distributions and Sobolev Spaces......................................427
7.1 Distribut ions. Preliminary Ideas................................427
7.2 Test Functions and Modifiers ....................................429
7.5 Distributions....................................................455
7.4 Calculus.........................................................458
7.4.1 The derivative in the* sense of distributions ............458
7.4.2 Gradient, divergence, Laplaeian...........................440
7.5 Operations wit h Dit ribut ions .................................445
7.5.1 Multiplication. Leibniz rule..............................445
7.5.2 Composition...............................................444
448
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Contents
7.5.3 Division...........................................
7.5.4 Convolution........................................
7.5.5 Tensor or direct product ..........................
7.6 Tempered Distributions and Fourier Transform..............
7.6.1 Tempered distributions.............................
7.6.2 Fourier transform in S ............................
7.6.3 Fourier transform in L2............................
7.7 Sobolev Spaces............................................
7.7.1 An abstract construction...........................
7.7.2 The space H1 (f2)..................................
7.7.3 The space Hq (12)..................................
7.7.4 The dual of H£(f2).................................
7.7.5 The spaces Lfm (12), m l...........................
7.7.6 Calculus rules.....................................
7.7.7 Fourier transform and Sobolev spaces...............
7.8 Approximations by Smooth Functions and Extensions.........
7.8.1 Local approximations...............................
7.8.2 Extensions and global approximations...............
7.9 Traces....................................................
7.9.1 Traces of functions in ......................
7.9.2 Traces of functions in Hrn(i2).....................
7.9.3 Trace spaces.......................................
7.10 Compactness and Embeddings................................
7.10.1 Rellich’s theorem..................................
7.10.2 Poincare’s inequalities............................
7.10.3 Sobolev inequality in 1R™ .........................
7.10.4 Bounded domains....................................
7.11 Spaces Involving Time.....................................
7.11.1 Functions with values into Hilbert spaces..........
7.11.2 Sobolev spaces involving time......................
Problems.......................................................
Variational Formulation of Elliptic Problems...................
8.1 Elliptic Equations........................................
8.2 Notions of Solutions......................................
8.3 Problems for the Poisson Equation ........................
8.3.1 Dirichlet problem..................................
8.3.2 Neumann, Robin and mixed problems..................
8.3.3 Eigenvalues and eigenfunctions of the Laplace operator
8.3.4 An asymptotic stability result.....................
8.4 General Equations in Divergence Form......................
8.4.1 Basic assumptions..................................
8.4.2 Dirichlet problem..................................
8.4.3 Neumann problem....................................
8.4.4 Robin and mixed problems...........................
Contents xvii
8.5 Weak Maximum Principles .........................................531
8.6 Regularity..................................................... 536
Problems.............................................................544
9 Further Applications .................................................551
9.1 A Monotone Iteration Scheme for Semilinear Equations ............551
9.2 Equilibrium of a Plate......................................... 554
9.3 The Linear Elastostatic System ..................................556
9.4 The Stokes System ..............................................561
9.5 The Stationary Navier Stokes Equations...........................566
9.5.1 Weak formulation and existence of a solution.............566
9.5.2 Uniqueness.............................................. 569
9.6 A Control Problem................................................571
9.6.1 Structure of the problem..................................571
9.6.2 Existence and uniqueness of an optimal pair...............572
9.6.3 Lagrange multipliers and optimality conditions............574
9.6.4 An iterative algorithm.................................. 575
Problems.............................................................576
10 Weak Formulation of Evolution Problems................................581
10.1 Parabolic Equations.............................................581
10.2 The Cauchy-Dirichlet Problem for the Heat Equation.............583
10.3 Abstract Parabolic Problems.....................................586
10.3.1 Formulation..............................................586
10.3.2 Energy estimates. Uniqueness and stability...............589
10.3.3 The Faedo-Galerkin approximations........................591
10.3.4 Existence................................................592
10.4 Parabolic PDEs..................................................593
10.4.1 Problems for the heat equation...........................593
10.4.2 General Equations........................................596
10.4.3 Regularity...............................................598
10.5 Weak Maximum Principles ........................................GOO
10.6 The Wave Equation...............................................602
10.6.1 Hyperbolic Equations.....................................602
10.6.2 The Cauchy-Dirichlet problem.............................603
10.6.3 The method of Faedo-Galerkin.............................605
10.6.4 Solution of the approximate problem......................606
10.6.5 Energy estimates.........................................607
10.6.6 Existence, uniqueness and stability......................609
Problems..........................................................611 11
11 Systems of Conservation Laws..........................................615
11.1 Introduction ...................................................615
11.2 Linear Hyperbolic Systems ......................................620
11.2.1 Characteristics..........................................620
xviii Contents
11.2.2 Classical solutions of the Cauchy problem................621
11.2.3 Homogeneous systems with constant coefficients. The
Riemann problem..........................................623
11.3 Quasilinear Conservation Laws..................................627
11.3.1 Characteristics and Riemann invariants...................627
11.3.2 Weak (or integral) solutions and the Rankine-Hugoniot
condition................................................630
11.4 The Riemann Problem............................................631
11.4.1 Rarefaction curves and waves. Genuinely nonlinear systems .633
11.4.2 Solution of the Riemann problem by a single rarefaction
wave.....................................................636
11.4.3 Lax entropy condition. Shock waves and contact
discontinuities .........................................638
11.4.4 Solution of the Riemann problem by a single ¿-shock......640
11.4.5 The linearly degenerate case.............................642
11.4.6 Local solution of the Riemann problem....................643
11.5 The Riemann Problem for the p-system...........................644
11.5.1 Shockwaves...............................................644
11.5.2 Rarefaction waves........................................646
11.5.3 The solution in the general case.........................649
Problems............................................................653
Appendix A. Fourier Series...............................................657
A.l Fourier Coefficients...........................................657
A. 2 Expansion in Fourier Series....................................660
Appendix B. Measures and Integrals ......................................663
B. l Lebesgue Measure and Integral..................................663
B.1.1 A counting problem .......................................663
B.l.2 Measures and measurable functions.........................665
B.1.3 The Lebesgue integral.....................................667
B.l.4 Some fundamental theorems.................................668
B. 2 Probability spaces, random variables and their
integrals.......................................................670
Appendix C. Identities and Formulas......................................673
C. l Gradient, Divergence, Curl, Laplacian..........................673
C.2 Formulas.......................................................675
References...............................................................677
Index
681
|
| any_adam_object | 1 |
| author | Salsa, Sandro 1950- |
| author_GND | (DE-588)1111815445 |
| author_facet | Salsa, Sandro 1950- |
| author_role | aut |
| author_sort | Salsa, Sandro 1950- |
| author_variant | s s ss |
| building | Verbundindex |
| bvnumber | BV045306739 |
| classification_rvk | SK 500 SK 540 |
| ctrlnum | (OCoLC)958032245 (DE-599)BVBBV045306739 |
| dewey-full | 003.3 510 |
| dewey-hundreds | 000 - Computer science, information, general works 500 - Natural sciences and mathematics |
| dewey-ones | 003 - Systems 510 - Mathematics |
| dewey-raw | 003.3 510 |
| dewey-search | 003.3 510 |
| dewey-sort | 13.3 |
| dewey-tens | 000 - Computer science, information, general works 510 - Mathematics |
| discipline | Informatik Mathematik |
| edition | Third edition, reprinted with modifications |
| format | Book |
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| genre | 1\p (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV045306739 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:14:26Z |
| institution | BVB |
| isbn | 9783319312378 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030693776 |
| oclc_num | 958032245 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-20 DE-11 DE-83 |
| owner_facet | DE-355 DE-BY-UBR DE-20 DE-11 DE-83 |
| physical | xviii, 686 Seiten Diagramme |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | Springer |
| record_format | marc |
| series | UNITEXT - La Matematica per il 3+2 |
| series2 | UNITEXT - La Matematica per il 3+2 |
| spelling | Salsa, Sandro 1950- Verfasser (DE-588)1111815445 aut Partial differential equations in action from modelling to theory Sandro Salsa Third edition, reprinted with modifications [Cham], Switzerland Springer [2016] xviii, 686 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier UNITEXT - La Matematica per il 3+2 volume 99 Mathematics Differential equations, partial Engineering mathematics Mathematical Modeling and Industrial Mathematics Partial Differential Equations Mathematical Applications in the Physical Sciences Appl.Mathematics/Computational Methods of Engineering Partial differential equations Mathematical physics Mathematical models Applied mathematics Mathematik Mathematische Physik Mathematisches Modell Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Erscheint auch als Online-Ausgabe 978-3-319-31238-5 Ergänzung Salsa, Sandro Partial differential equations in action complements and exercises UNITEXT - La Matematica per il 3+2 volume 99 (DE-604)BV047304938 99 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030693776&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Salsa, Sandro 1950- Partial differential equations in action from modelling to theory UNITEXT - La Matematica per il 3+2 Mathematics Differential equations, partial Engineering mathematics Mathematical Modeling and Industrial Mathematics Partial Differential Equations Mathematical Applications in the Physical Sciences Appl.Mathematics/Computational Methods of Engineering Partial differential equations Mathematical physics Mathematical models Applied mathematics Mathematik Mathematische Physik Mathematisches Modell Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4123623-3 |
| title | Partial differential equations in action from modelling to theory |
| title_auth | Partial differential equations in action from modelling to theory |
| title_exact_search | Partial differential equations in action from modelling to theory |
| title_full | Partial differential equations in action from modelling to theory Sandro Salsa |
| title_fullStr | Partial differential equations in action from modelling to theory Sandro Salsa |
| title_full_unstemmed | Partial differential equations in action from modelling to theory Sandro Salsa |
| title_short | Partial differential equations in action |
| title_sort | partial differential equations in action from modelling to theory |
| title_sub | from modelling to theory |
| topic | Mathematics Differential equations, partial Engineering mathematics Mathematical Modeling and Industrial Mathematics Partial Differential Equations Mathematical Applications in the Physical Sciences Appl.Mathematics/Computational Methods of Engineering Partial differential equations Mathematical physics Mathematical models Applied mathematics Mathematik Mathematische Physik Mathematisches Modell Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Mathematics Differential equations, partial Engineering mathematics Mathematical Modeling and Industrial Mathematics Partial Differential Equations Mathematical Applications in the Physical Sciences Appl.Mathematics/Computational Methods of Engineering Partial differential equations Mathematical physics Mathematical models Applied mathematics Mathematik Mathematische Physik Mathematisches Modell Partielle Differentialgleichung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030693776&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV047304938 |
| work_keys_str_mv | AT salsasandro partialdifferentialequationsinactionfrommodellingtotheory |