顶部
  1. 首页
  2. 图书产品
  3. A Practical Course in Differential Equations and Mathematical Modelling（英文版）
收藏

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

样章
作者：
Nail H. Ibragimov
定价：
68.00 元
版面字数：
255千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2009年
ISBN：
978-7-04-027603-9
物料号：
27603-00
出版时间：
2009-08-20
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 Course in Differential Equations and Mathematical Modelling is a unique blend of the traditional methods of ordinary and partial differential equations with Lie group analysis enriched by the author's own theoretical developments. The book -- which aims to present new mathematical curricula based on symmetry and invariance principles -- is tailored to develop analytic skills and 'working knowledge' in both classical and Lie's methods for solving linear and nonlinear equations. This approach helps to make courses n differential equations, mathematical modelling, distributions and fundamental solution, etc. easy to follow and interesting for students. The book is based on the author'

  • 目录
    • 前辅文
      • 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 _gures. Ellipse as an example
                • 1.1.4 Algebraic curves of the second degree
                • 1.2 Di_erential and integral calculus
                  • 1.2.1 Rules for di_erentiation
                    • 1.2.2 The mean value theorem
                      • 1.2.3 Invariance of the di_erential
                        • 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 Di_erential equations for families of curves
                                      • 1.3 Vector analysis
                                        • 1.3.1 Vector algebra
                                          • 1.3.2 Vector functions
                                            • 1.3.3 Vector _elds
                                              • 1.3.4 Three classical integral theorems
                                                • 1.3.5 The Laplace equation
                                                  • 1.3.6 Di_erentiation of determinants
                                                  • 1.4 Notation of di_erential algebra
                                                    • 1.4.1 Di_erential variables. Total di_erentiation
                                                      • 1.4.2 Higher derivatives of the product and of composite functions
                                                        • 1.4.3 Di_erential functions with several variables
                                                          • 1.4.4 The frame of di_erential 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 Di_usion 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 _nance
                                                                                                            • 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 di_erential equations: Traditional approach
                                                                                                                            • 3.1 Introduction and elementary methods
                                                                                                                              • 3.1.1 Di_erential equations. Initial value problem
                                                                                                                                • 3.1.2 Integration of the equation y(n) = f(x)
                                                                                                                                  • 3.1.3 Homogeneous equations
                                                                                                                                    • 3.1.4 Di_erent types of homogeneity
                                                                                                                                      • 3.1.5 Reduction of order
                                                                                                                                        • 3.1.6 Linearization through di_erentiation
                                                                                                                                        • 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 coe_cients
                                                                                                                                                              • 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 coe_cients
                                                                                                                                                                          • 3.4.4 Euler’s equation
                                                                                                                                                                          • 3.5 Systems of _rst-order equations
                                                                                                                                                                            • 3.5.1 General properties of systems
                                                                                                                                                                              • 3.5.2 First integrals
                                                                                                                                                                                • 3.5.3 Linear systems with constant coe_cients
                                                                                                                                                                                  • 3.5.4 Variation of parameters for systems Problems to Chapter 3
                                                                                                                                                                                • 4 First-order partial di_erential 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 di_erential equations of the second order
                                                                                                                                                                                            • 5.1 Equations with several variables
                                                                                                                                                                                              • 5.1.1 Classi_cation at a _xed point
                                                                                                                                                                                                • 5.1.2 Adjoint linear di_erential operators
                                                                                                                                                                                                • 5.2 Classi_cation 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 di_erential 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 _rst-order equations
                                                                                                                                                                                                                                          • 6.3.1 First prolongation of group generators
                                                                                                                                                                                                                                            • 6.3.2 Symmetry group: de_nition and main property
                                                                                                                                                                                                                                              • 6.3.3 Equations with a given symmetry
                                                                                                                                                                                                                                              • 6.4 Integration of _rst-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 of symmetries
                                                                                                                                                                                                                                                          • 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 particular solution
                                                                                                                                                                                                                                                                  • 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 di_erential equations
                                                                                                                                                                                                                                                                                • 7.1 Symmetries
                                                                                                                                                                                                                                                                                  • 7.1.1 De_nition 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 De_nition and examples of distributions
                                                                                                                                                                                                                                                                                                                        • 8.1.3 Representations of the _-function as a limit
                                                                                                                                                                                                                                                                                                                        • 8.2 Operations with distributions
                                                                                                                                                                                                                                                                                                                          • 8.2.1 Multiplication by a function
                                                                                                                                                                                                                                                                                                                            • 8.2.2 Di_erentiation
                                                                                                                                                                                                                                                                                                                              • 8.2.3 Direct product of distributions
                                                                                                                                                                                                                                                                                                                                • 8.2.4 Convolution
                                                                                                                                                                                                                                                                                                                              • 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 coecients
                                                                                                                                                                                                                                                                                                                                        • 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 problem from the invariance principle
                                                                                                                                                                                                                                                                                                                                                • 9.3.3 Solution of the Cauchy problem
                                                                                                                                                                                                                                                                                                                                                • 9.4 Wave equation
                                                                                                                                                                                                                                                                                                                                                  • 9.4.1 Preliminaries on di_erential forms
                                                                                                                                                                                                                                                                                                                                                    • 9.4.2 Auxiliary equations with distributions
                                                                                                                                                                                                                                                                                                                                                      • 9.4.3 Symmetries and de_nition 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 coe_cients
                                                                                                                                                                                                                                                                                                                                                            • Problems to Chapter 9
                                                                                                                                                                                                                                                                                                                                                            • Answers
                                                                                                                                                                                                                                                                                                                                                              • Bibliography
                                                                                                                                                                                                                                                                                                                                                                • Index

                                                                                                                                                                                                                                                                                                                                                                相关图书

                                                                                                                                                                                                                                                                                                                                                                Approximate and Renormgroup Symmetries 逼近与重整化群对称

                                                                                                                                                                                                                                                                                                                                                                N.H. Ibragimov，V. F. Kovalev

                                                                                                                                                                                                                                                                                                                                                                39.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                Approximate and Renormgroup Symmetries

                                                                                                                                                                                                                                                                                                                                                                N.H. Ibragimov，V. F. Kovalev

                                                                                                                                                                                                                                                                                                                                                                39.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                Partial Differential Equations and Solitary Waves Theory 偏微分方程与孤波理论

                                                                                                                                                                                                                                                                                                                                                                Abdul-Majid Wazwaz

                                                                                                                                                                                                                                                                                                                                                                96.00 元 收藏

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

                                                                                                                                                                                                                                                                                                                                                                Nail H. Ibragimov

                                                                                                                                                                                                                                                                                                                                                                0.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                Trends in Partial Differential Equations（偏微分方程进展）

                                                                                                                                                                                                                                                                                                                                                                边保军 李胜宏 汪徐家

                                                                                                                                                                                                                                                                                                                                                                88.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                同伦分析方法与非线性微分方程（英文版）

                                                                                                                                                                                                                                                                                                                                                                廖世俊

                                                                                                                                                                                                                                                                                                                                                                98.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                物理学与偏微分方程(上册)(英文版)

                                                                                                                                                                                                                                                                                                                                                                李大潜、秦铁虎 著，李亚纯 译

                                                                                                                                                                                                                                                                                                                                                                69.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                Lectures on the Analysis of Nonlinear Partial Differential Equations Vol. 3

                                                                                                                                                                                                                                                                                                                                                                林芳华 张平

                                                                                                                                                                                                                                                                                                                                                                89.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                微分方程群性质理论讲义 （Lectures on the Theory of Group Properties of Differential Equation

                                                                                                                                                                                                                                                                                                                                                                L.V. Ovsyannikov, N.H. Ibragimov, E.D. Avdonina

                                                                                                                                                                                                                                                                                                                                                                49.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                微分方程与数学物理问题(中文校订版)

                                                                                                                                                                                                                                                                                                                                                                Nail H. Ibragimov

                                                                                                                                                                                                                                                                                                                                                                49.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                微分方程与数学物理问题(中文校订版)

                                                                                                                                                                                                                                                                                                                                                                Nail H. Ibragimov

                                                                                                                                                                                                                                                                                                                                                                68.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                Lectures on the Analysis of Nonlinear Partial Differential Equations Vol. 4

                                                                                                                                                                                                                                                                                                                                                                Jean-Yves Chemin 林芳华 张平

                                                                                                                                                                                                                                                                                                                                                                128.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                Lectures on the Analysis of Nonlinear Partial Differential Equations Vol. 5

                                                                                                                                                                                                                                                                                                                                                                Jean-Yves Chemin, Fanghua Lin, Ping Zhang

                                                                                                                                                                                                                                                                                                                                                                128.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                Lectures on Differential Equations and Differential Geometry 微分方程和微分几何 （英文版）

                                                                                                                                                                                                                                                                                                                                                                Louis Nirenberg （路易斯·尼伦伯格 ）

                                                                                                                                                                                                                                                                                                                                                                89.00 元 收藏

                                                                                                                                                                                                                                                                                                                                                                神经科学的数学基础

                                                                                                                                                                                                                                                                                                                                                                G. Bard Ermentrout， David H. Terman 著；吴莹 刘深泉 译

                                                                                                                                                                                                                                                                                                                                                                79.00 元 收藏