: Nikos I. Kavallaris, Takashi Suzuki
: Non-Local Partial Differential Equations for Engineering and Biology Mathematical Modeling and Analysis
: Springer-Verlag
: 9783319679440
: 1
: CHF 114.00
:
: Wahrscheinlichkeitstheorie, Stochastik, Mathematische Statistik
: English
: 310
: Wasserzeichen/DRM
: PC/MAC/eReader/Tablet
: PDF
This book presents new developments  in non-local mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermo-dynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bio-science and material science to demonstrate applications, and provide recent advanced studies, both on deterministic non-local partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and non-local partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, self-organization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electro-magnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blow-up analysis, and energy methods are indispensable in understanding and analyzing these phenomena.
This book aims for researchers and upper grade students in mathematics, engineering, physics, economics, and biology.
Preface7
References13
Acknowledgements15
Contents16
Part I Applications in Engineering19
1 Micro-Electro-Mechanical-Systems (MEMS)20
1.1 Derivation of the Basic Model and Its Variations20
1.1.1 The Elastic Problem21
1.1.2 The Electric Problem23
1.1.3 An Uncoupled Local Model24
1.1.4 An Uncoupled Non-local Model26
1.2 Mathematical Analysis29
1.2.1 A Non-local Parabolic Problem29
1.2.2 A Non-local Hyperbolic Problem64
References78
2 Ohmic Heating Phenomena81
2.1 Ohmic Heating of Foods81
2.1.1 Derivation of the Basic Model and Its Variations81
2.1.2 Local Existence and Monotonicity85
2.1.3 Stationary Problem89
2.1.4 Stability95
2.1.5 Finite-Time Blow-Up101
2.2 A Non-local Thermistor Problem108
2.2.1 Neumann Problem108
2.2.2 Robin Problem111
2.2.3 Dirichlet Problem114
References122
3 Linear Friction Welding125
3.1 Derivation of the Model125
3.2 The Exponential Case130
3.3 Numerical Results140
3.3.1 The Soft Material Case140
3.3.2 The Hard Material Case142
References144
4 Resistance Spot Welding146
4.1 Derivation of the Non-local Model146
4.2 The Mathematical Problem151
4.3 The Numerical Scheme152
4.4 Stability154
4.5 Error Estimates158
4.6 Numerical Experiments165
References173
Part II Applications in Biology175
5 Gierer--Meinhardt System176
5.1 Derivation of the Non-local Model176
5.2 Mathematical Analysis181
5.2.1 Global-in-time Existence181
5.2.2 ODE Type Blow-Up189
5.2.3 Diffusion Driven Blow-Up192
5.2.4 Blow-Up Rate and Blow-Up Pattern202
References205
6 A Non-local Model Illustrating Replicator Dynamics207
6.1 Derivation of the Non-local Model207
6.2 Mathematical Analysis212
6.2.1 Local Existence and Extendability of Weak Solutions213
6.2.2 Global Existence Versus Blow-Up232
References237
7 A Non-local Model Arising in Chemotaxis240
7.1 Derivation of the Non-local Model240
7.2 Mathematical Analysis242
7.2.1 Preliminaries242
7.2.2 Blow-Up Results244
7.3 An Associated Competition-Diffusion System251
7.4 Miscellanea257
References259
8 A Non-local Reaction-Diffusion System Illustrating Cell Dynamics261
8.1 Derivation of the Non-local Reaction-Diffusion System261
8.2 Mathematical Analysis265
8.2.1 Preliminary Results265
8.2.2 Phase Separation269
8.2.3 Long-Time Behavior276
8.2.4 Decay Rate Towards the Steady States281
References299
Appendix Appendix301
A.1 Kirchhoff Equation301
A.2 Equilibrium and Relaxation States of Point Vortices302
A.3 Normalized Ricci Flow on Surfaces305
References307
Index309