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.
Author(s): Takasi Senba, Takashi Suzuki
Publisher: World Scientific Publishing Company
Year: 2004
Language: English
Pages: 393
Cover......Page 1
Applied Analysis: Mathematical Methods in Natural Science......Page 4
ISBN 9781860944406 (alk. paper)......Page 5
Preface......Page 8
Contents......Page 10
1.1 Basic Notions of Vector Analysis......Page 16
1.2 Curvature......Page 42
1.3 Extremals......Page 56
2.1 Isoperimetric Inequality......Page 68
2.2 Indirect Method......Page 77
2.3 Direct Method......Page 84
2.4 Numerical Schemes......Page 12
3.1 Hilbert Space......Page 100
3.2 Fourier Series......Page 108
3.3 Eigenvalue Problems......Page 121
3.4 Distributions......Page 142
4.1 Process of Diffusion......Page 166
4.2 Kinetic Model......Page 177
4.3 Semi-conductor Device Equation......Page 184
5.1 Well-posedness......Page 194
5.2 Fundamental Solutions......Page 200
5.3 Potential......Page 219
5.4 Regularity......Page 246
6.1 Method of Perturbation......Page 280
6.2 Method of Energy......Page 293
7.1 Story......Page 318
7.2 Well-posedness......Page 328
8.1 Catalogue of Mathematical Theories......Page 352
8.2 Commentary......Page 371
Bibliography......Page 378
Index......Page 384
