Preface
1 Selected topics from analysis
1.1 Elementary mathematics
1.1.1 Numbers, variables and elementary functions
1.1.2 Quadratic and cubic equations
1.1.3 Areas of similar figures. Ellipse as an example
1.1.4 Algebraic curves of the second degree
1.2 Differential and integral calculus
1.2.1 Rules for differentiation
1.2.2 The mean value theorem
1.2.3 Invariance of the differential
1.2.4 Rules for integration
1.2.5 The Taylor series
1.2.6 Complex variables
1.2.7 Approximate representation of functions
1.2.8 Jacobian. Functional independence. Change of variables in multiple integrals
1.2.9 Linear independence of functions. Wronskian
1.2.10 Integration by quadrature
1.2.11 Differential equations for families of curves
1.3 Vector analysis
1.3.1 Vector algebra
1.3.2 Vector functions
1.3.3 Vector fields
1.3.4 Three classical integral theorems
1.3.5 The Laplace equation
1.3.6 Differentiation of determinants
1.4 Notation of differential algebra
1.4.1 Differential variables. Total differentiation
1.4.2 Higher derivatives of the product and of composite functions
1.4.3 Differential functions with several variables
1.4.4 The frame of differential equations
1.4.5 Transformation of derivatives
1.5 Variational calculus
1.5.1 Principle of least action
1.5.2 Euler-Lagrange equations with several variables
Problems to Chapter 1
2 Mathematical models
2.1 Introduction
2.2 Natural phenomena
2.2.1 Population models
2.2.2 Ecology: Radioactive waste products
2.2.3 Kepler s laws. Newton s gravitation law
2.2.4 Free fall of a body near the earth
2.2.5 Meteoroid
2.2.6 A model of rainfall
2.3 Physics and engineering sciences
2.3.1 Newton s model of cooling
2.3.2 Mechanical vibrations. Pendulum
2.3.3 Collapse of driving shafts
2.3.4 The van der Pol equation
2.3.5 Telegraph equation
2.3.6 Electrodynamics
2.3.7 The Dirac equation
2.3.8 Fluid dynamics
2.3.9 The Navier-Stokes equations
2.3.10 A model of an irrigation system
2.3.11 Magnetohydrodynamics
2.4 Diffusion phenomena
2.4.1 Linear heat equation
2.4.2 Nonlinear heat equation
2.4.3 The Burgers and Korteweg-de Vries equations.
2.4.4 Mathematical modelling in finance
2.5 Biomathematics
2.5.1 Smart mushrooms
2.5.2 A tumour growth model
2.6 Wave phenomena
2.6.1 Small vibrations of a string
2.6.2 Vibrating membrane
2.6.3 Minimal surfaces
2.6.4 Vibrating slender rods and plates
2.6.5 Nonlinear waves
2.6.6 The Chaplygin and Tricomi equations
Problems to Chapter 2
3 Ordinary differential equations: Traditional approach
3.1 Introduction and elementary methods
3.1.1 Differential equations. Initial value problem
3.1.2 Integration of the equation y(n) = f(x)
3.1.3 Homogeneous equations
3.1.4 Different types of homogeneity
3.1.5 Reduction of order
3.1.6 Linearization through differentiation
3.2 First-order equations
3.2.1 Separable equations
3.2.2 Exact equations
3.2.3 Integrating factor (A. Clairaut, 1739)
3.2.4 The Riccati equation
3.2.5 The Bernoulli equation
3.2.6 Homogeneous linear equations
3.2.7 Non-homogeneous linear equations. Variation of the parameter
3.3 Second-order linear equations
3.3.1 Homogeneous equation: Superposition
3.3.2 Homogeneous equation: Equivalence properties
3.3.3 Homogeneous equation: Constant coefficients
3.3.4 Non-homogeneous equation: Variation of parameters
3.3.5 Bessel s equation and the...