Partial differential equations in action : from modelling to theory / Sandro Salsa.

Format:

Book

Author/Creator:

Salsa, S.

Contributor:

Classes of 1883 and 1884 Fund.

Series:

Universitext

Language:

English

Subjects (All):

Differential equations, Partial.

Physical Description:

xv, 556 pages : illustrations ; 24 cm.

Place of Publication:

Milan : Springer, 2008.

Summary:

This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines like Applied Mathematics, Physics and Engineering.

The main purpose is on the one hand to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences; on the other hand to give them a solid theoretical background for numerical methods, such as finite elements.

At the end of each chapter, a number of exercises at different level of complexity is included. The most demanding problems are supplied with answers or hints.

The exposition if flexible enough to allow substantial changes in the presentation of the arguments without compromising the comprehension and to facilitate a selection of topics for a one or two semester course.

Contents:

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 10

1.6 Integration by Parts Formulas 11

2 Diffusion 13

2.1 The Diffusion Equation 13

2.1.2 The conduction of heat 14

2.1.3 Well posed problems (n = 1) 16

2.1.4 A solution by separation of variables 19

2.1.5 Problems in dimension n > 1 27

2.2 Uniqueness 30

2.2.1 Integral method 30

2.2.2 Maximum principles 31

2.3 The Fundamental Solution 34

2.3.1 Invariant transformations 34

2.3.2 Fundamental solution (n = 1) 36

2.3.3 The Dirac distribution 39

2.3.4 Fundamental solution (n > 1) 42

2.4 Symmetric Random Walk (n = 1) 43

2.4.1 Preliminary computations 44

2.4.2 The limit transition probability 47

2.4.3 From random walk to Brownian motion 49

2.5 Diffusion, Drift and Reaction 52

2.5.1 Random walk with drift 52

2.5.2 Pollution in a channel 54

2.5.3 Random walk with drift and reaction 57

2.6 Multidimensional Random Walk 58

2.6.1 The symmetric case 58

2.6.2 Walks with drift and reaction 62

2.7 An Example of Reaction-Diffusion (n = 3) 62

2.8 The Global Cauchy Problem (n = 1) 68

2.8.1 The homogeneous case 68

2.8.2 Existence of a solution 69

2.8.3 The non homogeneous case. Duhamel's method 71

2.8.4 Maximum principles and uniqueness 74

2.9 An Application to Finance 77

2.9.1 European options 77

2.9.2 An evolution model for the price S 77

2.9.3 The Black-Scholes equation 80

2.9.4 The solutions 83

2.9.5 Hedging and self-financing strategy 88

2.10 Some Nonlinear Aspects 90

2.10.1 Nonlinear diffusion. The porous medium equation 90

2.10.2 Nonlinear reaction. Fischer's equation 93

3 The Laplace Equation 102

3.2 Well Posed Problems. Uniqueness 103

3.3 Harmonic Functions 105

3.3.1 Discrete harmonic functions 105

3.3.2 Mean value properties 109

3.3.3 Maximum principles 110

3.3.4 The Dirichlet problem in a circle. Poisson's formula 113

3.3.5 Harnack's inequality and Liouville's theorem 117

3.3.6 A probabilistic solution of the Dirichlet problem 118

3.3.7 Recurrence and Brownian motion 122

3.4 Fundamental Solution and Newtonian Potential 124

3.4.1 The fundamental solution 124

3.4.2 The Newtonian potential 126

3.4.3 A divergence-curl system. Helmholtz decomposition formula 128

3.5 The Green Function 132

3.5.1 An integral identity 132

3.5.2 The Green function 133

3.5.3 Green's representation formula 135

3.5.4 The Neumann function 137

3.6 Uniqueness in Unbounded Domains 139

3.6.1 Exterior problems 139

3.7 Surface Potentials 141

3.7.1 The double and single layer potentials 142

3.7.2 The integral equations of potential theory 146

4 Scalar Conservation Laws and First Order Equations 156

4.2 Linear Transport Equation 157

4.2.1 Pollution in a channel 157

4.2.2 Distributed source 159

4.2.3 Decay and localized source 160

4.2.4 Inflow and outflow characteristics. A stability estimate 162

4.3 Traffic Dynamics 164

4.3.1 A macroscopic model 164

4.3.2 The method of characteristics 165

4.3.3 The green light problem 168

4.3.4 Traffic jam ahead 172

4.4 Integral (or Weak) Solutions 174

4.4.1 The method of characteristics revisited 174

4.4.2 Definition of integral solution 177

4.4.3 The Rankine-Hugoniot condition 179

4.4.4 The entropy condition 183

4.4.5 The Riemann problem 185

4.4.6 Vanishing viscosity method 186

4.4.7 The viscous Burger equation 189

4.5 The Method of Characteristics for Quasilinear Equations 192

4.5.1 Characteristics 192

4.5.2 The Cauchy problem 194

4.5.3 Lagrange method of first integrals 202

4.5.4 Underground flow 205

4.6 General First Order Equations 207

4.6.1 Characteristic strips 207

4.6.2 The Cauchy Problem 210

5 Waves and Vibrations 221

5.1 General Concepts 221

5.1.1 Types of waves 221

5.1.2 Group velocity and dispersion relation 223

5.2 Transversal Waves in a String 226

5.2.1 The model 226

5.2.2 Energy 228

5.3 The One-dimensional Wave Equation 229

5.3.1 Initial and boundary conditions 229

5.3.2 Separation of variables 231

5.4 The d'Alembert Formula 236

5.4.1 The homogeneous equation 236

5.4.2 Generalized solutions and propagation of singularities 240

5.4.3 The fundamental solution 244

5.4.4 Non homogeneous equation. Duhamel's method 246

5.4.5 Dissipation and dispersion 247

5.5 Second Order Linear Equations 249

5.5.1 Classification 249

5.5.2 Characteristics and canonical form 252

5.6 Hyperbolic Systems with Constant Coefficients 257

5.7 The Multi-dimensional Wave Equation (n > 1) 261

5.7.1 Special solutions 261

5.7.2 Well posed problems. Uniqueness 263

5.8 Two Classical Models 266

5.8.1 Small vibrations of an elastic membrane 266

5.8.2 Small amplitude sound waves 270

5.9 The Cauchy Problem 274

5.9.1 Fundamental solution (n = 3) and strong Huygens' principle 274

5.9.2 The Kirchhoff formula 277

5.9.3 Cauchy problem in dimension 2 279

5.9.4 Non homogeneous equation. Retarded potentials 281

5.10 Linear Water Waves 282

5.10.1 A model for surface waves 282

5.10.2 Dimensionless formulation and linearization 286

5.10.3 Deep water waves 288

5.10.4 Interpretation of the solution 290

5.10.5 Asymptotic behavior 292

5.10.6 The method of stationary phase 293

6 Elements of Functional Analysis 302

6.1 Motivations 302

6.2 Norms and Banach Spaces 307

6.3 Hilbert Spaces 311

6.4 Projections and Bases 316

6.4.1 Projections 316

6.4.2 Bases 320

6.5 Linear Operators and Duality 326

6.5.1 Linear operators 326

6.5.2 Functionals and dual space 328

6.5.3 The adjoint of a bounded operator 331

6.6 Abstract Variational Problems 334

6.6.1 Bilinear forms and the Lax-Milgram Theorem 334

6.6.2 Minimization of quadratic functionals 339

6.6.3 Approximation and Galerkin method 340

6.7 Compactness and Weak Convergence 343

6.7.1 Compactness 343

6.7.2 Weak convergence and compactness 344

6.7.3 Compact operators 348

6.8 The Fredholm Alternative 350

6.8.1 Solvability for abstract variational problems 350

6.8.2 Fredholm's Alternative 354

6.9 Spectral Theory for Symmetric Bilinear Forms 356

6.9.1 Spectrum of a matrix 356

6.9.2 Separation of variables revisited 357

6.9.3 Spectrum of a compact self-adjoint operator 358

6.9.4 Application to abstract variational problems 360

7 Distributions and Sobolev Spaces 367

7.1 Distributions. Preliminary Ideas 367

7.2 Test Functions and Mollifiers 369

7.3 Distributions 373

7.4 Calculus 377

7.4.1 The derivative in the sense of distributions 377

7.4.2 Gradient, divergence, lapacian 379

7.5 Multiplication, Composition, Division, Convolution 382

7.5.1 Multiplication.

Leibniz rule 382

7.5.2 Composition 384

7.5.3 Division 385

7.5.4 Convolution 386

7.6 Fourier Transform 388

7.6.1 Tempered distributions 388

7.6.2 Fourier transform in S' 391

7.6.3 Fourier transform in L[superscript 2] 393

7.7 Sobolev Spaces 394

7.7.1 An abstract construction 394

7.7.2 The space H[superscript 1] ([Omega]) 396

7.7.3 The space H[superscript 1 subscript 0] ([Omega]) 399

7.7.4 The dual of H[superscript 1 subscript 0]([Omega]) 401

7.7.5 The spaces H[superscript m] ([Omega]), m > 1 403

7.7.6 Calculus rules 404

7.7.7 Fourier Transform and Sobolev Spaces 405

7.8 Approximations by Smooth Functions and Extensions 406

7.8.1 Local approximations 406

7.8.2 Estensions and global approximations 407

7.9 Traces 411

7.9.1 Traces of functions in H[superscript 1] ([Omega]) 411

7.9.2 Traces of functions in H[superscript m] ([Omega]) 414

7.9.3 Trace spaces 415

7.10 Compactness and Embeddings 418

7.10.1 Rellich's theorem 418

7.10.2 Poincare's inequalities 419

7.10.3 Sobolev inequality in R[superscript n] 420

7.10.4 Bounded domains 422

7.11 Spaces Involving Time 424

7.11.1 Functions with values in Hilbert spaces 424

7.11.2 Sobolev spaces involving time 425

8 Variational Formulation of Elliptic Problems 431

8.1 Elliptic Equations 431

8.2 The Poisson Problem 433

8.3 Diffusion, Drift and Reaction (n = 1) 435

8.3.1 The problem 435

8.3.2 Dirichlet conditions 435

8.3.3 Neumann, Robin and mixed conditions 439

8.4 Variational Formulation of Poisson's Problem 444

8.4.1 Dirichlet problem 444

8.4.2 Neumann, Robin and mixed problems 447

8.4.3 Eigenvalues of the Laplace operator 451

8.4.4 An asymptotic stability result 453

8.5 General Equations in Divergence Form 454

8.5.2 Dirichlet problem 455

8.5.3 Neumann problem 461

8.5.4 Robin and mixed problems 463

8.5.5 Weak Maximum Principles 465

8.6 Regularity 467

8.7 Equilibrium of a plate 473

8.8 A Monotone Iteration Scheme for Semilinear Equations 475

8.9 A Control Problem 478

8.9.1 Structure of the problem 478

8.9.2 Existence and uniqueness of an optimal pair 480

8.9.3 Lagrange multipliers and optimality conditions 481

8.9.4 An iterative algorithm 483

9 Weak Formulation of Evolution Problems 492

9.1 Parabolic Equations 492

9.2 Diffusion Equation 493

9.2.1 The Cauchy-Dirichlet problem 493

9.2.2 Faedo-Galerkin method (I) 496

9.2.3 Solution of the approximate problem 497

9.2.4 Energy estimates 498

9.2.5 Existence, uniqueness and stability 500

9.2.6 Regularity 503

9.2.7 The Cauchy-Neuman problem 505

9.2.8 Cauchy-Robin and mixed problems 507

9.2.9 A control problem 509

9.3 General Equations 512

9.3.1 Weak formulation of initial value problems 512

9.3.2 Faedo-Galerkin method (II) 514

9.4 The Wave Equation 517

9.4.1 Hyperbolic Equations 517

9.4.2 The Cauchy-Dirichlet problem 518

9.4.3 Faedo-Galerkin method (III) 520

9.4.4 Solution of the approximate problem 521

9.4.5 Energy estimates 522

9.4.6 Existence, uniqueness and stability 525

Appendix A Fourier Series 531

A.1 Fourier coefficients 531

A.2 Expansion in Fourier series 534

Appendix B Measures and Integrals 537

B.1 Lebesgue Measure and Integral 537

B.1.1 A counting problem 537

B.1.2 Measures and measurable functions 539

B.1.3 The Lebesgue integral 541

B.1.4 Some fundamental theorems 542

B.1.5 Probability spaces, random variables and their integrals 543

Appendix C Identities and Formulas 545

C.1 Gradient, Divergence, Curl, Laplacian 545

C.2 Formulas 547.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Classes of 1883 and 1884 Fund.

ISBN:

9788847007512

8847007518

8847007526

9788847007529

OCLC:

191891483