Modern mathematics has over 300 years of history. From the very beginning，it was focused on differential equations as a major tool for mathematical mod-elling. Most of mathematical models in physics， engineering sciences， biomath-ematics， etc. lead to nonlinear differential equations.

　　Todays engineering and science students and researchers routinely confrontproblems in mathematical modelling involving solution techniques for differen-tial equations. Sometimes these solutions can be obtained analytically by nu-merous traditional ad hoc methods appropriate for integrating particular typesof equations. More often， however， the solutions cannot be obtained by thesemethods， in spite of the fact that， e.g. over 400 types of integrable second-orderordinary differential equations were accumulated due to ad hoc approaches andsummarized in voluminous catalogues.

　　On the other hand， the fundamental natural laws and technological prob-lems formulated in terms of differential equations can be successfully treatedand solved by Lie group methods. For example， Lie group analysis reducesthe classical 400 types of equations to 4 types only！ Development of groupanalysis furnished ample evidence that the theory provides a universal tool fortackling considerable numbers of differential equations even when other meansof integration fail. In fact， group analysis is the only universal and effectivemethod for solving nonlinear differential equations analytically. The old inte-gration methods rely essentially on linearity as well as on constant coefficients.Group analysis deals equally easily with linear and nonlinear equations， as wellas with constant and variable coefficients. For example， from the traditionalpoint of view， the linear equation

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 Keplers laws. Newtons 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 Newtons 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 Bessels equation and the Bessel functions

3.3.6 Hypergeometric equation

3.4 Higher-order linear equations

3.4.1 Homogeneous equations. Fundamental system

3.4.2 Non-homogeneous equations. Variation of parameters

3.4.3 Equations with constant coefficients

3.4.4 Eulers equation

3.5 Systems of first-order equations

3.5.1 General properties of systems

3.5.2 First integrals

3.5.3 Linear systems with constant coefficients

3.5.4 Variation of parameters for systems

Problems to Chapter 3

4 First-order partial differential equations

4.1 Introduction

4.2 Homogeneous linear equation

4.3 Particular solutions of non-homogeneous equations

4.4 Quasi-linear equations

4.5 Systems of homogeneous equations

Problems to Chapter 4

5 Linear partial differential equations of the second order

5.1 Equations with several variables

5.1.1 Classification at a fixed point

5.1.2 Adjoint linear differential operators

5.2 Classification of equations in two independent variables

5.2.1 Characteristics. Three types of equations

5.2.2 The standard form of the hyperbolic equations

5.2.3 The standard form of the parabolic equations

5.2.4 The standard form of the elliptic equations

5.2.5 Equations of a mixed type

5.2.6 The type of nonlinear equations

5.3 Integration of hyperbolic equations in two variables

5.3.1 dAlemberts solution

5.3.2 Equations reducible to the wave equation

5.3.3 Eulers method

5.3.4 Laplaces cascade method

5.4 The initial value problem

5.4.1 The wave equation

5.4.2 Non-homogeneous wave equation

5.5 Mixed problem. Separation of variables

5.5.1 Vibration of a string tied at its ends

5.5.2 Mixed problem for the heat equation

Problems to Chapter 5

6 Nonlinear ordinary differential equations

6.1 Introduction

6.2 Transformation groups

6.2.1 One-parameter groups on the plane

6.2.2 Group generator and the Lie equations

6.2.3 Exponential map

6.2.4 Invariants and invariant equations

6.2.5 Canonical variables

6.3 Symmetries of first-order equations

6.3.1 First prolongation of group generators

6.3.2 Symmetry group: definition and main property

6.3.3 Equations with a given symmetry

6.4 Integration of first-order equations using symmetries

6.4.1 Lies integrating factor

6.4.2 Integration using canonical variables

6.4.3 Invariant solutions

6.4.4 General solution provided by invariant solutions

6.5 Second-order equations

6.5.1 Second prolongation of group generators Calculation of symmetries

6.5.2 Lie algebras

6.5.3 Standard forms of two-dimensional Lie algebras

6.5.4 Lies integration method

6.5.5 Integration of linear equations with a known particular solution

6.5.6 Lies linearization test

6.6 Higher-order equations

6.6.1 Invariant solutions. Derivation of Eulers ansatz

6.6.2 Integrating factor (N.H. Ibragimov, 2006)

6.6.3 Linearization of third-order equations

6.7 Nonlinear superposition

6.7.1 Introduction

6.7.2 Main theorem on nonlinear superposition

6.7.3 Examples of nonlinear superposition

6.7.4 Integration of systems using nonlinear superposition

Problems to Chapter 6

7 Nonlinear partial differential equations

7.1 Symmetries

7.1.1 Definition and calculation of symmetry groups

7.1.2 Group transformations of solutions

7.2 Group invariant solutions

7.2.1 Introduction

7.2.2 The Burgers equation

7.2.3 A nonlinear boundary-value problem

7.2.4 Invariant solutions for an irrigation system

7.2.5 Invariant solutions for a tumour growth model

7.2.6 An example from nonlinear optics

7.3 Invariance and conservation laws

7.3.1 Introduction

7.3.2 Preliminaries

7.3.3 Noethers theorem

7.3.4 Higher-order Lagrangians

7.3.5 Conservation theorems for ODEs

7.3.6 Generalization of Noethers theorem

7.3.7 Examples from classical mechanics

7.3.8 Derivation of Einsteins formula for energy

7.3.9 Conservation laws for the Dirac equations

Problems to Chapter 7

8 Generalized functions or distributions

8.1 Introduction of generalized functions

8.1.1 Heuristic considerations

8.1.2 Definition and examples of distributions

8.1.3 Representations of the δ-function as a limit

8.2 Operations with distributions

8.2.1 Multiplication by a function

8.2.2 Differentiation

8.2.3 Direct product of distributions

8.2.4 Convolution

8.3 The distribution △(r2-n)

8.3.1 The mean value over the sphere

8.3.2 Solution of the Laplace equation △v(r)=0

8.3.3 Evaluation of the distribution △(r2-n)

8.4 Transformations of distributions

8.4.1 Motivation by linear transformations

8.4.2 Change of variables in the d-function

8.4.3 Arbitrary group transformations

8.4.4 Infinitesimal transformation of distributions

Problems to Chapter 8

9 Invariance principle and fundamental solutions

9.1 Introduction

9.2 The invariance principle

9.2.1 Formulation of the invariance principle

9.2.2 Fundamental solution of linear equations with constant coefficients

9.2.3 Application to the Laplace equation

9.2.4 Application to the heat equation

9.3 Cauchys problem for the heat equation

9.3.1 Fundamental solution for the Cauchy problem

9.3.2 Derivation of the fundamental solution for the Cauchy problem from the invariance principle

9.3.3 Solution of the Cauchy problem

9.4 Wave equation

9.4.1 Preliminaries on differential forms

9.4.2 Auxiliary equations with distributions

9.4.3 Symmetries and definition of fundamental solutions for the wave equation

9.4.4 Derivation of the fundamental solution

9.4.5 Solution of the Cauchy problem

9.5 Equations with variable coefficients

Problems to Chapter 9

Answers

Bibliography

Index