Partial Differential Equations in Action

by Salsa, Sandro

• ISBN13: 9788847007512
• ISBN10: 8847007518
• Edition: 1st
• Format: Paperback
• Copyright: 2008-02-13
• Publisher: Springer Verlag

Summary

This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines. The main purpose is on the one hand to train students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other hand to give them a solid theoretical background for numerical methods. At the end of each chapter, a number of exercises at different level of complexity is included

Table of Contents

Preface	p. v
Introduction	p. 1
Mathematical Modelling	p. 1
Partial Differential Equations	p. 2
Well Posed Problems	p. 5
Basic Notations and Facts	p. 7
Smooth and Lipschitz Domains	p. 10
Integration by Parts Formulas	p. 11
Diffusion	p. 13
The Diffusion Equation	p. 13
Introduction	p. 13
The conduction of heat	p. 14
Well posed problems (n = 1)	p. 16
A solution by separation of variables	p. 19
Problems in dimension n > 1	p. 27
Uniqueness	p. 30
Integral method	p. 30
Maximum principles	p. 31
The Fundamental Solution	p. 34
Invariant transformations	p. 34
Fundamental solution (n = 1)	p. 36
The Dirac distribution	p. 39
Fundamental solution (n > 1)	p. 42
Symmetric Random Walk (n = 1)	p. 43
Preliminary computations	p. 44
The limit transition probability	p. 47
From random walk to Brownian motion	p. 49
Diffusion, Drift and Reaction	p. 52
Random walk with drift	p. 52
Pollution in a channel	p. 54
Random walk with drift and reaction	p. 57
Multidimensional Random Walk	p. 58
The symmetric case	p. 58
Walks with drift and reaction	p. 62
An Example of Reaction-Diffusion (n = 3)	p. 62
The Global Cauchy Problem (n = 1)	p. 68
The homogeneous case	p. 68
Existence of a solution	p. 69
The non homogeneous case. Duhamel's method	p. 71
Maximum principles and uniqueness	p. 74
An Application to Finance	p. 77
European options	p. 77
An evolution model for the price S	p. 77
The Black-Scholes equation	p. 80
The solutions	p. 83
Hedging and self-financing strategy	p. 88
Some Nonlinear Aspects	p. 90
Nonlinear diffusion. The porous medium equation	p. 90
Nonlinear reaction. Fischer's equation	p. 93
Problems	p. 97
The Laplace Equation	p. 102
Introduction	p. 102
Well Posed Problems. Uniqueness	p. 103
Harmonic Functions	p. 105
Discrete harmonic functions	p. 105
Mean value properties	p. 109
Maximum principles	p. 110
The Dirichlet problem in a circle. Poisson's formula	p. 113
Harnack's inequality and Liouville's theorem	p. 117
A probabilistic solution of the Dirichlet problem	p. 118
Recurrence and Brownian motion	p. 122
Fundamental Solution and Newtonian Potential	p. 124
The fundamental solution	p. 124
The Newtonian potential	p. 126
A divergence-curl system. Helmholtz decomposition formula	p. 128
The Green Function	p. 132
An integral identity	p. 132
The Green function	p. 133
Green's representation formula	p. 135
The Neumann function	p. 137
Uniqueness in Unbounded Domains	p. 139
Exterior problems	p. 139
Surface Potentials	p. 141
The double and single layer potentials	p. 142
The integral equations of potential theory	p. 146
Problems	p. 150
Scalar Conservation Laws and First Order Equations	p. 156
Introduction	p. 156
Linear Transport Equation	p. 157
Pollution in a channel	p. 157
Distributed source	p. 159
Decay and localized source	p. 160
Inflow and outflow characteristics. A stability estimate	p. 162
Traffic Dynamics	p. 164
A macroscopic model	p. 164
The method of characteristics	p. 165
The green light problem	p. 168
Traffic jam ahead	p. 172
Integral (or Weak) Solutions	p. 174
The method of characteristics revisited	p. 174
Definition of integral solution	p. 177
The Rankine-Hugoniot condition	p. 179
The entropy condition	p. 183
The Riemann problem	p. 185
Vanishing viscosity method	p. 186
The viscous Burger equation	p. 189
The Method of Characteristics for Quasilinear Equations	p. 192
Characteristics	p. 192
The Cauchy problem	p. 194
Lagrange method of first integrals	p. 202
Underground flow	p. 205
General First Order Equations	p. 207
Characteristic strips	p. 207
The Cauchy Problem	p. 210
Problems	p. 214
Waves and Vibrations	p. 221
General Concepts	p. 221
Types of waves	p. 221
Group velocity and dispersion relation	p. 223
Transversal Waves in a String	p. 226
The model	p. 226
Energy	p. 228
The One-dimensional Wave Equation	p. 229
Initial and boundary conditions	p. 229
Separation of variables	p. 231
The d'Alembert Formula	p. 236
The homogeneous equation	p. 236
Generalized solutions and propagation of singularities	p. 240
The fundamental solution	p. 244
Non homogeneous equation. Duhamel's method	p. 246
Dissipation and dispersion	p. 247
Second Order Linear Equations	p. 249
Classification	p. 249
Characteristics and canonical form	p. 252
Hyperbolic Systems with Constant Coefficients	p. 257
The Multi-dimensional Wave Equation (n > 1)	p. 261
Special solutions	p. 261
Well posed problems. Uniqueness	p. 263
Two Classical Models	p. 266
Small vibrations of an elastic membrane	p. 266
Small amplitude sound waves	p. 270
The Cauchy Problem	p. 274
Fundamental solution (n = 3) and strong Huygens' principle	p. 274
The Kirchhoff formula	p. 277
Cauchy problem in dimension 2	p. 279
Non homogeneous equation. Retarded potentials	p. 281
Linear Water Waves	p. 282
A model for surface waves	p. 282
Dimensionless formulation and linearization	p. 286
Deep water waves	p. 288
Interpretation of the solution	p. 290
Asymptotic behavior	p. 292
The method of stationary phase	p. 293
Problems	p. 296
Elements of Functional Analysis	p. 302
Motivations	p. 302
Norms and Banach Spaces	p. 307
Hilbert Spaces	p. 311
Projections and Bases	p. 316
Projections	p. 316
Bases	p. 320
Linear Operators and Duality	p. 326
Linear operators	p. 326
Functionals and dual space	p. 328
The adjoint of a bounded operator	p. 331
Abstract Variational Problems	p. 334
Bilinear forms and the Lax-Milgram Theorem	p. 334
Minimization of quadratic functionals	p. 339
Approximation and Galerkin method	p. 340
Compactness and Weak Convergence	p. 343
Compactness	p. 343
Weak convergence and compactness	p. 344
Compact operators	p. 348
The Fredholm Alternative	p. 350
Solvability for abstract variational problems	p. 350
Fredholm's Alternative	p. 354
Spectral Theory for Symmetric Bilinear Forms	p. 356
Spectrum of a matrix	p. 356
Separation of variables revisited	p. 357
Spectrum of a compact self-adjoint operator	p. 358
Application to abstract variational problems	p. 360
Problems	p. 362
Distributions and Sobolev Spaces	p. 367
Distributions. Preliminary Ideas	p. 367
Test Functions and Mollifiers	p. 369
Distributions	p. 373
Calculus	p. 377
The derivative in the sense of distributions	p. 377
Gradient, divergence, lapacian	p. 379
Multiplication, Composition, Division, Convolution	p. 382
Multiplication. Leibniz rule	p. 382
Composition	p. 384
Division	p. 385
Convolution	p. 386
Fourier Transform	p. 388
Tempered distributions	p. 388
Fourier transform in S'	p. 391
Fourier transform in L[superscript 2]	p. 393
Sobolev Spaces	p. 394
An abstract construction	p. 394
The space H[superscript 1] ([Omega])	p. 396
The space H[superscript 1 subscript 0] ([Omega])	p. 399
The dual of H[superscript 1 subscript 0]([Omega])	p. 401
The spaces H[superscript m] ([Omega]), m > 1	p. 403
Calculus rules	p. 404
Fourier Transform and Sobolev Spaces	p. 405
Approximations by Smooth Functions and Extensions	p. 406
Local approximations	p. 406
Estensions and global approximations	p. 407
Traces	p. 411
Traces of functions in H[superscript 1] ([Omega])	p. 411
Traces of functions in H[superscript m] ([Omega])	p. 414
Trace spaces	p. 415
Compactness and Embeddings	p. 418
Rellich's theorem	p. 418
Poincare's inequalities	p. 419
Sobolev inequality in R[superscript n]	p. 420
Bounded domains	p. 422
Spaces Involving Time	p. 424
Functions with values in Hilbert spaces	p. 424
Sobolev spaces involving time	p. 425
Problems	p. 428
Variational Formulation of Elliptic Problems	p. 431
Elliptic Equations	p. 431
The Poisson Problem	p. 433
Diffusion, Drift and Reaction (n = 1)	p. 435
The problem	p. 435
Dirichlet conditions	p. 435
Neumann, Robin and mixed conditions	p. 439
Variational Formulation of Poisson's Problem	p. 444
Dirichlet problem	p. 444
Neumann, Robin and mixed problems	p. 447
Eigenvalues of the Laplace operator	p. 451
An asymptotic stability result	p. 453
General Equations in Divergence Form	p. 454
Basic assumptions	p. 454
Dirichlet problem	p. 455
Neumann problem	p. 461
Robin and mixed problems	p. 463
Weak Maximum Principles	p. 465
Regularity	p. 467
Equilibrium of a plate	p. 473
A Monotone Iteration Scheme for Semilinear Equations	p. 475
A Control Problem	p. 478
Structure of the problem	p. 478
Existence and uniqueness of an optimal pair	p. 480
Lagrange multipliers and optimality conditions	p. 481
An iterative algorithm	p. 483
Problems	p. 485
Weak Formulation of Evolution Problems	p. 492
Parabolic Equations	p. 492
Diffusion Equation	p. 493
The Cauchy-Dirichlet problem	p. 493
Faedo-Galerkin method (I)	p. 496
Solution of the approximate problem	p. 497
Energy estimates	p. 498
Existence, uniqueness and stability	p. 500
Regularity	p. 503
The Cauchy-Neuman problem	p. 505
Cauchy-Robin and mixed problems	p. 507
A control problem	p. 509
General Equations	p. 512
Weak formulation of initial value problems	p. 512
Faedo-Galerkin method (II)	p. 514
The Wave Equation	p. 517
Hyperbolic Equations	p. 517
The Cauchy-Dirichlet problem	p. 518
Faedo-Galerkin method (III)	p. 520
Solution of the approximate problem	p. 521
Energy estimates	p. 522
Existence, uniqueness and stability	p. 525
Problems	p. 528
Fourier Series	p. 531
Fourier coefficients	p. 531
Expansion in Fourier series	p. 534
Measures and Integrals	p. 537
Lebesgue Measure and Integral	p. 537
A counting problem	p. 537
Measures and measurable functions	p. 539
The Lebesgue integral	p. 541
Some fundamental theorems	p. 542
Probability spaces, random variables and their integrals	p. 543
Identities and Formulas	p. 545
Gradient, Divergence, Curl, Laplacian	p. 545
Formulas	p. 547
References	p. 549
Index	p. 553
Table of Contents provided by Ingram. All Rights Reserved.