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A Practical Course in Differential Equations and Mathematical Modelling（英文版）

作者：
Nail H. Ibragimov
定价：
0.00元
ISBN：
978-7-04-027603-9
版面字数：
255.000千字
开本：
暂无
全书页数：
364页
装帧形式：
暂无
重点项目：
暂无
出版时间：
2009-08-01
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
偏微分方程

A Practical Coursein Differential Equations and Mathematical Modelling is a uniqueblend of the traditional methods of ordinary and partialdifferential equations with Lie group analysis enriched by theauthor's own theoretical developments. The book -- which aims topresent new mathematical curricula based on symmetry and invarianceprinciples -- is tailored to develop analytic skills and 'workingknowledge' in both classical and Lie's methods for solving linearand nonlinear equations. This approach helps to make courses ndifferential equations, mathematical modelling, distributions andfundamental solution, etc. easy to follow and interesting forstudents. The book is based on the author's extensive teachingexperience at Novosibirsk and Moscow universities in Russia,College de France, Georgia Tech and Stanford University in theUnited States, universities in South Africa, Cyprus, Turkey, andBlekinge Institute of Technology (BTH) in Sweden. The newcurriculum prepares students for solving modern nonlinear problemsand will essentially be more appealing to students pared to thetraditional way of teaching mathematics. The book can be used as amain textbook by undergraduate and graduate students and universitylecturers in applied mathematics, physics andengineering.

  • 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 inmultiple 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 positefunctions
      • 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 Magohydrodynamics
    • 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 Trii 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 theparameter
    • 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 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 Euler's 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 d'Alembert's solution
      • 5.3.2 Equations reducible to the wave equation
      • 5.3.3 Euler's method
      • 5.3.4 Laplace's 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 Lie's 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 ofsymmetries
      • 6.5.2 Lie algebras
      • 6.5.3 Standard forms of two-dimensional Lie algebras
      • 6.5.4 Lie's integration method
      • 6.5.5 Integration of linear equations with a known particularsolution
      • 6.5.6 Lie's linearization test
    • 6.6 Higher-order equations
      • 6.6.1 Invariant solutions. Derivation of Euler's 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 Noether's theorem
      • 7.3.4 Higher-order Lagrangians
      • 7.3.5 Conservation theorems for ODEs
      • 7.3.6 Generalization of Noether's theorem
      • 7.3.7 Examples from classical mechanics
      • 7.3.8 Derivation of Einstein's 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 constantcoefficients
      • 9.2.3 Application to the Laplace equation
      • 9.2.4 Application to the heat equation
    • 9.3 Cauchy's 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 problemfrom 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 thewave 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

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