Introduction to Mathematical Modeling and Computer Simulations

• Main Authors: Mityushev, Vladimir, Nawalaniec, Wojciech, Rylko, Natalia
• Format: eBook Book
• Language: English
• Published: Boca Raton CRC Press 2018; CRC Press LLC; Chapman & Hall
• Edition: 1
• Subjects: asymptotic methods; calculus; Computational Numerical Analysis; Computer simulation; Mathematical Modeling; Mathematical models; MATHnetBASE; Natalia Rylko; numeric and symbolic computations; SCI-TECHnetBASE; STMnetBASE; stochastic models; vector analysis; Wojciech Nawalaniec; Lotka Volterra Model; Dirichlet Problem; Finite Difference Method; Voronoi Diagrams; Vice Versa; Default Step Size; Heat Equation; Green's Function; Chosen Confidence Interval; Introductory Mathematical Courses; Approximate Polynomial; Standard Orthogonal Basis; Banach Fixed Point Theorem; Cauchy's Problem; Effective Conductivity Tensor; Linear Homogeneous Ode; Macroscopic Energy Equation; Neighbor Balls; Ordinary Differential Equation; Einstein Summation Convention; Heat Flux; Effective Conductivity; Voronoi Regions; Delaunay Graph; Vector Field
• ISBN: 9781138197657; 1138197653; 9781032095752; 103209575X
• DOI: 10.1201/9781315277240

Introduction to Mathematical Modeling and Computer Simulations is written as a textbook for readers who want to understand the main principles of Modeling and Simulations in settings that are important for the applications, without using the profound mathematical tools required by most advanced text...

Table of Contents:

• Cover -- Half Title -- Title Page -- Copyright Page -- Table of Contents -- List of Figures -- List of Tables -- Preface -- I: General Principles and Methods -- 1: Principles of Mathematical Modeling -- 1.1 How to Develop a Mathematical Model -- 1.1.1 A Simple Mathematical Model -- 1.1.2 Use of a Computer -- 1.1.3 Development of Mathematical Models -- 1.2 Types of Models -- 1.3 Stability of Models -- 1.4 Dimension, Units, and Scaling -- 1.4.1 Dimensional Analysis -- 1.4.2 Scaling -- Exercises -- 2: Numerical and Symbolic Computations -- 2.1 Numerical and Symbolic Computations of Derivatives and Integrals -- 2.2 Iterative Methods -- 2.3 Newton's Method -- 2.4 Method of Successive Approximations -- 2.5 Banach Fixed Point Theorem -- 2.6 Why Is It Difficult to Numerically Solve Some Equations? -- Exercises -- II: Basic Applications -- 3: Application of Calculus to Classic Mechanics -- 3.1 Mechanical Meaning of the Derivative -- 3.2 Interpolation -- 3.3 Integrals -- 3.4 Potential Energy -- Exercises -- 4: Ordinary Differential Equations and Their Applications -- 4.1 Principle of Transition for Ode -- 4.2 Radioactive Decay -- 4.3 Logistic Differential Equation and Its Modifications -- 4.3.1 Logistic Differential Equation -- 4.3.2 Modified Logistic Equation -- 4.3.3 Stability Analysis -- 4.3.4 Bifurcation -- 4.4 Time Delay -- 4.5 Approximate Solution to Differential Equations -- 4.5.1 Taylor Approximations -- 4.5.2 Padé Approximations -- 4.6 Harmonic Oscillation -- 4.6.1 Simple Harmonic Motion -- 4.6.2 Harmonic Oscillator with Friction and Exterior Forces -- 4.6.3 Resonance -- 4.7 Lotka-volterra Model -- 4.8 Linearization -- Exercises -- 5: Stochastic Models -- 5.1 Method of Least Squares -- 5.2 Fitting -- 5.3 Method of Monte Carlo -- 5.4 Random Walk -- Exercises -- 6: One-Dimensional Stationary Problems -- 6.1 1D Geometry -- 6.2 Second Order Equations
• 6.3 1D Green's Function -- 6.4 Green's Function as a Source -- 6.5 The δ-function -- Exercises -- III: Advanced Applications -- 7: Vector Analysis -- 7.1 Euclidean Space R3 -- 7.1.1 Polar Coordinates -- 7.1.2 Cylindrical Coordinates -- 7.1.3 Spherical Coordinates -- 7.2 Scalar, Vector and Mixed Products -- 7.3 Rotation of Bodies -- 7.4 Scalar, Vector and Mixed Product in Mathematica -- 7.5 Tensors -- 7.6 Scalar and Vector Fields -- 7.6.1 Gradient -- 7.6.2 Divergence -- 7.6.3 Curl -- 7.6.4 Formulae for Gradient, Divergence and Curl -- 7.7 Integral Theorems -- Exercises -- 8: Heat Equations -- 8.1 Heat Conduction Equations -- 8.2 Initial and Boundary Value Problems -- 8.3 Green's Function for the 1d Heat Equation -- 8.4 Fourier Series -- 8.5 Separation of Variables -- 8.6 Discrete Approximations of Pde -- 8.6.1 Finite-Difference Method -- 8.6.2 1D Finite Element Method -- 8.6.3 Finite Element Method in R2 -- 8.7 Universality in Mathematical Modeling. Table -- Exercises -- 9: Asymptotic Methods in Composites -- 9.1 Effective Properties of Composites -- 9.1.1 General Introduction -- 9.1.2 Strategy of Investigations -- 9.2 Maxwell's Approach -- 9.2.1 Single-Inclusion Problem -- 9.2.2 Self Consistent Approximation -- 9.3 Densely Packed Balls -- 9.3.1 Cubic Array -- 9.3.2 Densely Packed Balls and Voronoi Diagrams -- 9.3.3 Optimal Random Packing -- Exercises -- Bibliography -- Index