Mathematical Modelling for Earth Sciences

by Yang, Xin-She

• ISBN13: 9781903765920
• ISBN10: 1903765927
• Format: Paperback
• Copyright: 2008-04-17
• Publisher: Dunedin Academic Press

Summary

Mathematical modelling and computer simulations are an essential part of the analytical toolset used by earth scientists. In this textbook, designed for use by mathematicians and non-mathematicians encounteringg the application of mathematics to the earth sciences for the first time, Dr Yang has carefully selected topics which will be of most value to students. The book is designed to be 'theorem-free¿ and yet to balance formality and practicality.

Author Biography

Xin-She Yang is currently a research fellow at the University of Cambridge. He is in the Structure Group in the Civil, Structural and Environmental Engineering Division

Table of Contents

Preface	p. vii
Mathematical Methods	p. 1
Mathematical Modelling	p. 3
Introduction	p. 3
Mathematical Modelling	p. 3
Model Formulation	p. 5
Parameter Estimation	p. 8
Mathematical Models	p. 11
Differential Equations	p. 11
Functional and Integral Equations	p. 16
Statistical Models	p. 16
Numerical Methods	p. 17
Numerical Integration	p. 17
Numerical Solutions of PDEs	p. 19
Topics in This Book	p. 20
Calculus and Complex Variables	p. 23
Calculus	p. 23
Set Theory	p. 23
Differentiation and Integration	p. 26
Partial Differentiation	p. 33
Multiple Integrals	p. 35
Jacobian	p. 36
Complex Variables	p. 38
Complex Numbers and Functions	p. 39
Analytic Functions	p. 40
Complex Integrals	p. 41
Cauchy's Integral Theorem	p. 42
Residue Theorem	p. 43
Vectors and Matrices	p. 45
Vectors	p. 45
Dot Product and Norm	p. 45
Cross Product	p. 47
Differentiation of Vectors	p. 48
Line Integral	p. 49
Three Basic Operators	p. 49
Some Important Theorems	p. 51
Matrix Algebra	p. 51
Matrix	p. 51
Determinant	p. 53
Inverse	p. 54
Matrix Exponential	p. 54
Solution of linear systems	p. 55
Gauss-Seidel Iteration	p. 57
Tensors	p. 58
Notations	p. 58
Tensors	p. 59
ODEs and Integral Transforms	p. 61
Ordinary Differential Equations	p. 61
First-Order ODEs	p. 62
Higher-Order ODEs	p. 64
Linear System	p. 65
Sturm-Liouville Equation	p. 66
Integral Transforms	p. 68
Fourier Series	p. 69
Fourier Integral	p. 73
Fourier Transforms	p. 74
Laplace Transforms	p. 75
Wavelets	p. 77
PDEs and Solution Techniques	p. 79
Partial Differential Equations	p. 79
First-Order PDEs	p. 80
Classification of Second-Order PDEs	p. 80
Classic Mathematical Models	p. 81
Laplace's and Poisson's Equation	p. 81
Parabolic Equation	p. 82
Wave Equation	p. 82
Other Mathematical Models	p. 82
Elastic Wave Equation	p. 83
Reaction-Diffusion Equation	p. 83
Navier-Stokes Equations	p. 83
Groundwater Flow	p. 84
Solution Techniques	p. 84
Separation of Variables	p. 84
Laplace Transform	p. 87
Fourier Transform	p. 87
Similarity Solution	p. 88
Change of Variables	p. 89
Calculus of Variations	p. 91
Euler-Lagrange Equation	p. 91
Curvature	p. 91
Euler-Lagrange Equation	p. 93
Variations with Constraints	p. 99
Variations for Multiple Variables	p. 103
Integral Equations	p. 104
Fredholm Integral Equations	p. 104
Volterra Integral Equation	p. 105
Solution of Integral Equations	p. 105
Separable Kernels	p. 105
Volterra Equation	p. 106
Probability	p. 109
Randomness and Probability	p. 109
Conditional Probability	p. 115
Random Variables and Moments	p. 116
Random Variables	p. 116
Mean and Variance	p. 117
Moments and Generating Functions	p. 118
Binomial and Poisson Distributions	p. 119
Binomial Distribution	p. 119
Poisson Distribution	p. 120
Gaussian Distribution	p. 121
Other Distributions	p. 123
The Central Limit Theorem	p. 124
Weibull Distribution	p. 126
Geostatistics	p. 131
Sample Mean and Variance	p. 131
Method of Least Squares	p. 133
Maximum Likelihood	p. 133
Linear Regression	p. 133
Correlation Coefficient	p. 136
Hypothesis Testing	p. 137
Confidence Interval	p. 137
Student's t-distribution	p. 138
Student's t-test	p. 140
Data Interpolation	p. 142
Spline Interpolation	p. 142
Lagrange Interpolating Polynomials	p. 149
Bezier Curve	p. 150
Kriging	p. 151
Numerical Algorithms	p. 159
Numerical Integration	p. 161
Root-Finding Algorithms	p. 161
Bisection Method	p. 162
Newton's Method	p. 164
Iteration Method	p. 166
Numerical Integration	p. 168
Trapezium Rule	p. 168
Order Notation	p. 170
Simpson's Rule	p. 171
Gaussian Integration	p. 173
Optimisation	p. 177
Unconstrained Optimisation	p. 177
Newton's Method	p. 178
Steepest Descent Method	p. 179
Constrained Optimisation	p. 182
Finite Difference Method	p. 185
Integration of ODEs	p. 185
Euler Scheme	p. 186
Leap-Frog Method	p. 188
Runge-Kutta Method	p. 188
Hyperbolic Equations	p. 189
First-Order Hyperbolic Equation	p. 189
Second-Order Wave Equation	p. 190
Parabolic Equation	p. 191
Elliptical Equation	p. 193
Finite Volume Method	p. 195
Introduction	p. 195
Elliptic Equations	p. 196
Hyperbolic Equations	p. 197
Parabolic Equations	p. 198
Finite Element Method	p. 201
Concept of Elements	p. 202
Simple Spring Systems	p. 202
Bar Elements	p. 206
Finite Element Formulation	p. 209
Weak Formulation	p. 209
Galerkin Method	p. 210
Shape Functions	p. 211
Estimating Derivatives and Integrals	p. 215
Heat Transfer	p. 216
Basic Formulation	p. 216
Element-by-Element Assembly	p. 218
Application of Boundary Conditions	p. 219
Transient Problems	p. 221
The Time Dimension	p. 221
Time-Stepping Schemes	p. 223
Travelling Waves	p. 223
Applications to Earth Sciences	p. 225
Reaction-Diffusion System	p. 227
Mineral Reactions	p. 227
Travelling Wave	p. 229
Pattern Formation	p. 230
Reaction-Diffusion System	p. 231
Elasticity and Poroelasticity	p. 235
Hooke's Law and Elasticity	p. 235
Shear Stress	p. 240
Equations of Motion	p. 241
Euler-Bernoulli Beam Theory	p. 246
Airy Stress Functions	p. 249
Fracture Mechanics	p. 252
Biot's Theory	p. 257
Biot's Poroelasticity	p. 257
Effective Stress	p. 259
Linear Poroelasticity	p. 259
Poroelasticity	p. 259
Equation of Motion	p. 262
Flow in Porous Media	p. 263
Groundwater Flow	p. 263
Porosity	p. 263
Darcy's Law	p. 263
Flow Equations	p. 265
Pollutant Transport	p. 269
Theory of Consolidation	p. 272
Viscous Creep	p. 277
Power-Law Creep	p. 277
Derivation of creep law	p. 278
Hydrofracture	p. 283
Hydrofracture	p. 283
Diagenesis	p. 284
Dyke and Diapir Propagation	p. 285
Mathematical Formulae	p. 291
Differentiation and Integration	p. 291
Differentiation	p. 291
Integration	p. 291
Power Series	p. 292
Complex Numbers	p. 292
Vectors and Matrices	p. 292
Asymptotic Expansions	p. 293
Matlab and Octave Programs	p. 295
Gaussian Quadrature	p. 295
Newton's Method	p. 297
Pattern Formation	p. 299
Wave Equation	p. 301
Bibliography	p. 303
Index	p. 307
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