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Mattheij R.M.M. — Partial differential equations: modeling, analysis, computation
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Название: Partial differential equations: modeling, analysis, computationАвтор: Mattheij R.M.M. Аннотация: Partial differential equations (PDEs) are used to describe a large variety of physical phenomena, from fluid flow to electromagnetic fields, and are indispensable to such disparate fields as aircraft simulation and computer graphics. While most existing texts on PDEs deal with either analytical or numerical aspects of PDEs, this innovative and comprehensive textbook features a unique approach that integrates analysis and numerical solution methods and includes a third component—modeling—to address real-life problems. The authors believe that modeling can be learned only by doing; hence a separate chapter containing 16 user-friendly case studies of elliptic, parabolic, and hyperbolic equations is included and numerous exercises are included in all other chapters. Partial Differential Equations: Modeling, Analysis, Computation enables readers to deepen their understanding of a topic ubiquitous in mathematics and science and to tackle practical problems. The advent of fast computers and the development of numerical methods have enabled the modern engineer to use a large variety of packages to find numerical approximations to solutions of PDEs. Problems are usually standard and a thorough knowledge of a well-chosen subset of analytical and numerical tools and methodologies is necessary when dealing with real-life problems. When one is dealing with PDEs in practice, it becomes clear that both numerical and analytical treatments of the problem are needed. This comprehensive book is intended for graduate students in applied mathematics, engineering, and physics and may be of interest to advanced undergraduate students. Mathematicians, scientists, and engineers also will find the book useful. Contents List of Figures; List of Tables; Notation; Preface; Chapter 1: Differential and difference equations; Chapter 2: Characterization and classification; Chapter 3: Fourier theory; Chapter 4: Distributions and fundamental solutions; Chapter 5: Approximation by finite differences; Chapter 6: The Equations of continuum mechanics and electromagnetics; Chapter 7: The art of modeling; Chapter 8: The analysis of elliptic equations; Chapter 9: Numerical methods for elliptic equations; Chapter 10: Analysis of parabolic equations; Chapter 11: Numerical methods for parabolic equations; Chapter 12: Analysis of hyperbolic equations; Chapter 13: Numerical methods for scalar hyperbolic equations; Chapter 14: Numerical methods for hyperbolic systems; Chapter 15: Perturbation methods; Chapter 16: Modeling, analyzing, and simulating problems from practice; Appendices: Useful definitions and properties; Bibliography; Index.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2005Количество страниц: 665Добавлена в каталог: 19.10.2010Операции: Положить на полку |
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339 339 341 339 89 273 2-norm 98 Absence of free magnetic poles 123 Acoustic impedance 623 Acoustic liner 623 Adams - Bashforth 90 Adams - Moulton 90 Adams methods 90 ADI (Alternating Direction Implicit) 213 294 Adiabatic invariants 492 Adjoint operator 58 Advection-diffusion barrier 375 Advection-diffusion equation 295 373 Air-damped resonator 488 Aircraft engine noise 622 Alternating direction implicit (ADI) 213 294 Amp 122 Amplification factor 261 365 Amplitude error 371 Approximate Riemann solver 426 Artificial diffusion 378 Associated matrix norm 638 Asymptotic approximation 461 Asymptotic expansion 462 Asymptotic sequence 462 Asymptotically equal 463 Averaging 351 Backward difference 79 90 631 Backward difference formulae (BDF) 86 Backward Euler method 89 Band matrix 208 Barotropic flow 154 Base characteristic 15 BDF (Backward Difference Formulas) 90 632 Beam - Warming scheme 414 Block tridiagonal 208 Block-wave function 37 Blokhintzev invariant 567 Boundary basis vectors 616 Boundary condition 160 Boundary condition, absorbing 448 Boundary condition, nonreflecting 448 Boundary condition, top layer 616 Boundary layer 92 476 Box scheme 189 203 379 Buckley - Leverett equation 326 Catalytic reaction 555 Cauchy - Schwarz inequality 638 Cauchy problem 334 Cauchy's equation of motion 114 Causal 41 Causality condition 41 136 Cell-centered finite volume method 85 Central difference 79 80 CFL (Courant - Friedrichs - Lewy) condition 366 Characteristic equations 15 Characteristic polynomial 639 Characteristic strip equations 316 Characteristic variables 18 330 Characteristics 15 308 Charpit's equations 316 Classical Runge - Kutta 89 Closure 212 Collocation 78 Complementary error function 144 236 Conductive flux 85 Cone of normals 315 Conservation equation 109 311 Conservation law 86 Conservation of angular momentum 113 Conservation of electric charge 123 Conservation of energy 115 Conservation of linear momentum 113 Conservative 46 Conservative scheme 390 Consistency 98 Consistency, explicit Euler scheme 263 Constitutive equations 116 Constitutive relations 109 Contact angle 120 Contact discontinuity 340 Continuum physics 109 Control volume 85 Convective flux 85 Convergence, explicit Euler scheme 264 Convergence, pointwise 631 Convergence, uniform 631 Convolution theorem 39 Correspondence principle 477 Coulomb's law 122 Courant - Friedrichs - Lewy (CEL) condition 366 Courant number 364 Crank - Nicolson scheme 272 Critical rarefaction wave 438 Critical state 439 Cylindrical symmetry 83 D'Alembert solution 350 Dam break problem 435 Damped Newton 218 Damping factor 218 Davidenko equation 222 Decibel 147 Delta function 54 Descent direction 218 Difference equation 76 Difference scheme 80 Diffusion number 261 Dimensionally homogeneous 138 Dimensionless parameter 138 646 Dirichlet boundary condition 160 Discrete dispersion relation 364 Discrete Green's function 214 Discrete pressure gradient 619 Discretisation 77 Discretisation, backward differences 619 Discretisation, forward differences 619 dispersion 45 Displacement vector 110 Distinguished limit 464 477 distribution 54 Distribution, regular 54 Divergence theorem 642 DuFort - Frankel scheme 305 Duhamel integral 63 64 Eckert number 150 Edge condition 136 Eigenvalue 639 Eigenvector 639 Eikonal equation 315 Elasticity 120 Electromagnetic continuity conditions 127 Electrophoresis 599 Energy 484 Energy velocity 45 Enthalpy 118 248 entropy 118 Entropy condition 325 Entropy solution 325 Equispaced 72 Equivalent norms 638 Error function 236 Essential root 91 383 Essential solution 91 383 Euclidean vector space 644 Euler number 150 Expansion shock 323 Explicit Euler method 76 89 Explicit Euler scheme 261 Explicit method 89 Explicit midpoint rule 91 Faraday - Henry's law 122 Fick's law 118 Finite difference 73 Fisher's equation 252 Flux 85 Flux function 310 Flux limiter 397 Flux vector 331 Forward difference 79 Forward Euler Method 89 Fourier cosine series 37 Fourier discrete transform 43 Fourier mode 34 Fourier number 150 Fourier series 34 35 Fourier sine series 37 Fourier transform 38 Fourier's law 117 Fredholm's alternative 62 frequency 34 261 Galloping transmission lines 531 Gauss - Jacobi iteration 209 Gauss - Seidel iteration 209 Gauss's theorem 642 Generalised eigenvalue problem 327 Generalized Riemann problem 404 Genuinely nonlinear 337 Geometric multiplicity 640 Geopotential 342 Glass oven 613 Glass production 581 Global discretisation error 193 263 Global discretisation error, implicit Euler method 271 Global discretisation error, vector 99 Godunov scheme 388 423 Godunov scheme, wave propagation form 425 Green's first identity 642 Green's function 61 174 Groundwater flow 540 Group velocity 45 565 Growth factor 103 Harmonic function 160 Harten - Hyman entropy fix 439 heat capacity 119 Heat equation 24 233 260 Heat flow in a bar 469 Helmholtz equation 8 159 Helmholtz number 150 High-resolution flux 398 High-resolution scheme 391 398 Homogeneous 214 Hooke's law 120 Hyperbolic equation 308 Hyperbolic equation, conservation form 310 Hyperbolic equation, linear 308 Hyperbolic equation, quasi-linear 310 Hyperbolic equation, semilinear 309 Hyperbolic system 329 Hyperbolic system, characteristic form 331 Hyperbolic system, conservation form 331 Hyperbolic system, linear 328 Hyperbolic system, normal form 331 Hyperbolic system, quasi-linear 330 Implicit Euler method 76 89 Implicit Euler scheme 270 Implicit method 89 90 Incremental form 392 Infinity norm 98 Inherent scaling 138 Initial curve 15 Initial value 96 Inner expansion 476 Integral surface 13 Interior layers 92 Internal layer 476 Interpolation points 72 Isotachophoresis 599 Iteration matrix 208 Iterative methods 208 Jordan normal form 640 Joukowski transformation 153 Joule heating 124 Kelvin - Voigt model 121 Kelvin's theorem 126 2005 9 1 Kutta condition 154 Lagrangian basis polynomials 72 Lagrangian interpolation polynomial 72 Laplace equation 24 159 Laser drilling 589 Latent heat 119 Lax - Friedrichs scheme 414 Lax - Wendroff scheme 368 Layers 92 Leapfrog scheme 382 Left eigenvectors 327 Limiter, function 397 Limiter, superbee 402 Limiter, van Leer 402 Lindstedt - Poincar 473 Linearly degenerate 338 Local discretisation error 78 189 193 263 366 Local discretisation error, 274 Local discretisation error, Lax - Wendroff scheme 369 Local discretisation error, upwind scheme 366 Local discretisation error, vector 98 Logarithmic switchback 481 Longitudinal 96 Longitudinal method of lines 265 Lorentz force 125 Lubrication flow 496 Mach number 150 Matched asymptotic expansions (MAEs) 475 477 Matching 477 483 Material surface 111 Material volume 111 Mathematising 130 Matrix defect 641 Matrix method 276 Matrix pencil 328 Matrix pencil, regular 328 Matrix pencil, singular 328 Matrix, similarity transformation 640 Matrix, stable 641 Matrix, trace 639 Maxwell equations 123 Maxwell model 122 Median 2-norm 98 Method of lines (MOL) 96 Method of Massau 333 Method of strained coordinates 474 486 Midpoint formula 75 Midpoint rule 89 Minmod slope 407 Mixed boundary conditions 282 MODE 147 Model 131 Model, asymptotic 131 Model, building block 132 Model, canonical 135 Model, characteristic 135 Model, constructing 132 Model, lumped parameter 132 Model, quintessential 135 Model, reducing 131 Model, systematic 131 Modified equation 421
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