Applied Analysis: Mathematical Methods In Natural Science (2nd Edition)

by Takashi Suzuki, Takasi Senba

Overview

This book provides a general introduction to applied analysis; vector analysis with physical motivation, calculus of variation, Fourier analysis, eigenfunction expansion, distribution, and so forth, including a catalogue of mathematical theories, such as basic analysis, topological spaces, complex function theory, real analysis, and abstract analysis. This book also uses fundamental ideas of applied mathematics to discuss recent developments in nonlinear science, such as mathematical modeling of reinforced random motion of particles, semiconductor device equation in applied physics, and chemotaxis in biology. Several tools in linear PDE theory, such as fundamental solutions, Perron's method, layer potentials, and iteration scheme, are described, as well as systematic descriptions on the recent study of the blowup of the solution.

Product Details

ISBN-13:	9781848166523
Publisher:	Imperial College Press
Publication date:	03/15/2011
Edition description:	Second Edition
Pages:	532
Product dimensions:	6.20(w) x 9.10(h) x 1.30(d)

Table of Contents

Chapter 1	Geometric Objects	1
1.1	Basic Notions of Vector Analysis	1
1.1.1	Dynamical Systems	1
1.1.2	Outer Product	5
1.1.3	Motion of Particles	8
1.1.4	Gradient	11
1.1.5	Divergence	15
1.1.6	Rotation	22
1.1.7	Motion of Fluid	24
1.2	Curvature	27
1.2.1	Quadratic Surfaces	27
1.2.2	First Fundamental Form	27
1.2.3	Curves	30
1.2.4	Second Fundamental Form	36
1.3	Extremals	41
1.3.1	Lagrange Multiplier	41
1.3.2	Implicit Function Theorem	45
1.3.3	Convex Functions	47
Chapter 2	Calculus of Variation
2.1	Isoperimetric Inequality	53
2.1.1	Analytic Proof	53
2.1.2	Geometric Proof	57
2.2	Indirect Method	62
2.2.1	Euler Equation	62
2.2.2	Lagrange Mechanics	64
2.2.3	Minimal Surfaces	66
2.3	Direct Method	69
2.3.1	Vibrating String	69
2.3.2	Minimizing Sequence	71
2.3.3	Sobolev Spaces	73
2.3.4	Lower Semi-Continuity	76
2.4	Numerical Schemes	80
2.4.1	Finite Difference Method	80
2.4.2	Finite Element Method	81
Chapter 3	Infinite Dimensional Analysis	85
3.1	Hilbert Space	85
3.1.1	Bounded Linear Operators	85
3.1.2	Representation Theorem of Riesz	87
3.1.3	Complete Ortho-Normal Systems	90
3.2	Fourier Series	93
3.2.1	Historical Note	93
3.2.2	Completeness	96
3.2.3	Uniform Convergence	100
3.2.4	Pointwise Convergence	103
3.3	Eigenvalue Problems	106
3.3.1	Vibrating Membrane	106
3.3.2	Gel'fand Triple	111
3.3.3	Self-adjoint Operator	115
3.3.4	Symmetric Bi-linear Form	117
3.3.5	Compact Operator	119
3.3.6	Eigenfunction Expansions	121
3.3.7	Mini-Max Principle	124
3.4	Distributions	127
3.4.1	Dirac's Delta Function	127
3.4.2	Locally Convex Spaces	130
3.4.3	Frechet Spaces	132
3.4.4	Inductive Limit	134
3.4.5	Bounded Sets	137
3.4.6	Definition and Examples	138
3.4.7	Fundamental Properties	141
3.4.8	Support	145
3.4.9	Convergence	147
Chapter 4	Random Motion of Particles	151
4.1	Process of Diffusion	151
4.1.1	Master Equation	151
4.1.2	Local Information Model	153
4.1.3	Barrier Model	156
4.1.4	Renormalization	157
4.2	Kinetic Model	162
4.2.1	Transport Equation	162
4.2.2	Boltzmann Equation	165
4.3	Semi-Conductor Device Equation	169
4.3.1	Modelling	169
4.3.2	Drift-Diffusion (DD) Model	173
4.3.3	Mathematical Structure	174
Chapter 5	Linear PDE Theory	179
5.1	Well-posedness	179
5.1.1	Heat Equation	179
5.1.2	Uniqueness	181
5.1.3	Existence	183
5.2	Fundamental Solutions	185
5.2.1	Fourier Transformation	185
5.2.2	Rapidly Decreasing Functions	187
5.2.3	Cauchy Problem	190
5.2.4	Gaussian Kernel	193
5.2.5	Semi-groups	196
5.2.6	Fourier Transformation of Distributions	200
5.3	Potential	204
5.3.1	Harmonic Functions	204
5.3.2	Poisson Integral	206
5.3.3	Perron Solution	211
5.3.4	Boundary Regularity	214
5.3.5	The Green's Function	216
5.3.6	Newton Potential	218
5.3.7	Layer Potentials	223
5.3.8	Fredholm Theory	230
5.4	Regularity	231
5.4.1	Poisson Equation	231
5.4.2	Schauder Estimate	233
5.4.3	Dirichlet Principle	240
5.4.4	Moser's Iteration Scheme	242
5.4.5	BMO Estimate	254
Chapter 6	Nonlinear PDE Theory	265
6.1	Method of Perturbation	265
6.1.1	Duhamel's Principle	265
6.1.2	Semilinear Heat Equation	267
6.1.3	Global Existence	271
6.1.4	Blowup	275
6.2	Method of Energy	278
6.2.1	Lyapunov Function	278
6.2.2	Solution Global in Time	283
6.2.3	Unbounded Solution	289
6.2.4	Stable and Unstable Sets	295
6.2.5	Method of Rescaling	298
Chapter 7	System of Chemotaxis	303
7.1	Story	303
7.1.1	The Keller-Segel System	303
7.1.2	Blowup Mechanism	305
7.1.3	Free Energy	308
7.2	Well-posedness	313
7.2.1	Summary	313
7.2.2	The Linearized System	316
7.2.3	Properties of F	323
7.2.4	Local Solvability	334
Chapter 8	Appendix	337
8.1	Catalogue of Mathematical Theories	337
8.1.1	Basic Analysis	337
8.1.2	Topological Spaces	340
8.1.3	Complex Function Theory	344
8.1.4	Real Analysis	348
8.1.5	Abstract Analysis	354
8.2	Commentary	356
8.2.1	Elliptic and Parabolic Equations	356
8.2.2	Systems of Self-interacting Particles	358
	Bibliography	363
	Index	369