A Practical Course in Differential Equations and Mathematical Modelling: Classical and New Methods, Nonlinear Mathematical Models, Symmetry and Invariance Principles

by Ibragimov, Nail H.

• ISBN13: 9789814291941
• ISBN10: 9814291943
• eBook ISBN(s): 9789813107762
• Format: Hardcover
• Copyright: 2010-01-30
• Publisher: World Scientific Pub Co Inc
• More Book Details
• Purchase Benefits

Summary

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 in differential equations, mathematical modelling, distributions and fundamental solution, etc. easy to follow and interesting for students. The book is based on the author’s extensive teaching experience at Novosibirsk and Moscow universities in Russia, Collège de France, Georgia Tech and Stanford University in the United States, universities in South Africa, Cyprus, Turkey, and Blekinge Institute of Technology (BTH) in Sweden. The new curriculum prepares students for solving modern nonlinear problems and will essentially be more appealing to students compared to the traditional way of teaching mathematics.

Author Biography

Dr. Nail H. Ibragimov is a professor at the Department of Mathematics and Science, Blekinge Institute of Technology in Sweden. He is widely regarded as one of the world's foremost experts in the field of differential equations, group analysis and mathematical modelling.

Table of Contents

Preface	p. 1
Selected topics from analysis	p. 1
Elementary mathematics	p. 1
Numbers, variables and elementary functions	p. 1
Quadratic and cubic equations	p. 5
Areas of similar figures. Ellipse as an example	p. 8
Algebraic curves of the second degree	p. 9
Differential and integral calculus	p. 14
Rules for differentiation	p. 14
The mean value theorem	p. 15
Invariance of the differential	p. 15
Rules for integration	p. 16
The Taylor series	p. 17
Complex variables	p. 19
Approximate representation of functions	p. 20
Jacobian. Functional independence. Change of variables in multiple integrals	p. 21
Linear independence of functions. Wronskian	p. 22
Integration by quadrature	p. 23
Differential equations for families of curves	p. 24
Vector analysis	p. 26
Vector algebra	p. 26
Vector functions	p. 28
Vector fields	p. 29
Three classical integral theorems	p. 30
The Laplace equation	p. 31
Differentiation of determinants	p. 32
Notation of differential algebra	p. 32
Differential variables. Total differentiation	p. 32
Higher derivatives of the product and of composite functions	p. 33
Differential functions with several variables	p. 34
The frame of differential equations	p. 35
Transformation of derivatives	p. 36
Variational calculus	p. 38
Principle of least action	p. 38
Euler-Lagrange equations with several variables	p. 39
Problems to Chapter 1	p. 40
Mathematical models	p. 45
Introduction	p. 45
Natural phenomena	p. 46
Population models	p. 46
Ecology: Radioactive waste products	p. 47
Kepler's laws. Newton's gravitation law	p. 48
Free fall of a body near the earth	p. 50
Meteoroid	p. 50
A model of rainfall	p. 52
Physics and engineering sciences	p. 53
Newton's model of cooling	p. 53
Mechanical vibrations. Pendulum	p. 60
Collapse of driving shafts	p. 64
The van der Pol equation	p. 66
Telegraph equation	p. 67
Electrodynamics	p. 68
The Dirac equation	p. 69
Fluid dynamics	p. 69
The Navier-Stokes equations	p. 71
A model of an irrigation system	p. 71
Magnetohydrodynamics	p. 71
Diffusion phenomena	p. 72
Linear heat equation	p. 72
Nonlinear heat equation	p. 74
The Burgers and Korteweg-de Vries equations	p. 75
Mathematical modelling in finance	p. 75
Biomathematics	p. 76
Smart mushrooms	p. 76
A tumour growth model	p. 78
Wave phenomena	p. 79
Small vibrations of a string	p. 79
Vibrating membrane	p. 82
Minimal surfaces	p. 84
Vibrating slender rods and plates	p. 84
Nonlinear waves	p. 86
The Chaplygin and Tricomi equations	p. 88
Problems to Chapter 2	p. 88
Ordinary differential equations: Traditional approach	p. 91
Introduction and elementary methods	p. 91
Differential equations. Initial value problem	p. 91
Integration of the equation y(n) =f(x)	p. 93
Homogeneous equations	p. 93
Different types of homogeneity	p. 96
Reduction of order	p. 97
Linearization through differentiation	p. 98
First-order equations	p. 99
Separable equations	p. 99
Exact equations	p. 99
Integrating factor (A. Clairaut, 1739)	p. 101
The Riccati equation	p. 102
The Bernoulli equation	p. 106
Homogeneous linear equations	p. 106
Non-homogeneous linear equations. Variation of the parameter	p. 107
Second-order linear equations	p. 108
Homogeneous equation: Superposition	p. 109
Homogeneous equation: Equivalence properties	p. 110
Homogeneous equation: Constant coefficients	p. 113
Non-homogeneous equation: Variation of parameters	p. 115
Bessel's equation and the Bessel functions	p. 119
Hypergeometric equation	p. 119
Higher-order linear equations	p. 120
Homogeneous equations. Fundamental system	p. 121
Non-homogeneous equations. Variation of parameters	p. 121
Equations with constant coefficients	p. 122
Euler's equation	p. 123
Systems of first-order equations	p. 124
General properties of systems	p. 124
First integrals	p. 125
Linear systems with constant coefficients	p. 129
Variation of parameters for systems	p. 130
Problems to Chapter 3	p. 133
First-order partial differential equations	p. 135
Introduction	p. 135
Homogeneous linear equation	p. 136
Particular solutions of non-homogeneous equations	p. 138
Quasi-linear equations	p. 139
Systems of homogeneous equations	p. 142
Problems to Chapter 4	p. 145
Linear partial differential equations of the second order	p. 149
Equations with several variables	p. 149
Classification at a fixed point	p. 149
Adjoint linear differential operators	p. 151
Classification of equations in two independent variables	p. 153
Characteristics. Three types of equations	p. 153
The standard form of the hyperbolic equations	p. 155
The standard form of the parabolic equations	p. 156
The standard form of the elliptic equations	p. 157
Equations of a mixed type	p. 158
The type of nonlinear equations	p. 158
Integration of hyperbolic equations in two variables	p. 159
d'Alembert's solution	p. 159
Equations reducible to the wave equation	p. 160
Euler's method	p. 165
Laplace's cascade method	p. 167
The initial value problem	p. 169
The wave equation	p. 169
Non-homogeneous wave equation	p. 170
Mixed problem. Separation of variables	p. 172
Vibration of a string tied at its ends	p. 172
Mixed problem for the heat equation	p. 175
Problems to Chapter 5	p. 177
Nonlinear ordinary differential equations	p. 179
Introduction	p. 179
Transformation groups	p. 180
One-parameter groups on the plane	p. 180
Group generator and the Lie equations	p. 181
Exponential map	p. 183
Invariants and invariant equations	p. 184
Canonical variables	p. 187
Symmetric of first-order equations	p. 188
First prolongation of group generators	p. 188
Symmetry group: definition and main property	p. 189
Equations with a given symmetry	p. 191
Integration of first-order equations using symmetries	p. 193
Lie's integrating factor	p. 193
Integration using canonical variables	p. 196
Invariant solutions	p. 199
General solution provided by invariant solutions	p. 200
Second-order equations	p. 201
Second prolongation of group generators Calculation of symmetries	p. 201
Lie algebras	p. 204
Standard forms of two-dimensional Lie algebras	p. 205
Lie's integration method	p. 206
Integration of linear equations with a known particular solution	p. 212
Lie's linearization test	p. 214
Higher-order equations	p. 218
Invariant solutions. Derivation of Euler's ansatz	p. 218
Integrating factor (N.H. Ibragimov, 2006)	p. 220
Linearization of third-order equations	p. 228
Nonlinear superposition	p. 235
Introduction	p. 235
Main theorem on nonlinear superposition	p. 236
Examples of nonlinear superposition	p. 241
Integration of systems using nonlinear superposition	p. 249
Problems to-Chapter 6	p. 251
Nonlinear partial differential equations	p. 255
Symmetries	p. 255
Definition and calculation of symmetry groups	p. 256
Group transformations of solutions	p. 260
Group invariant solutions	p. 262
Introduction	p. 262
The Burgers equation	p. 263
A nonlinear boundary-value problem	p. 266
Invariant solutions for an irrigation system	p. 268
Invariant solutions for a tumour growth model	p. 270
An example from nonlinear optics	p. 272
Invariance and conservation laws	p. 274
Introduction	p. 274
Preliminaries	p. 277
Noether's theorem	p. 278
Higher-order Lagrangians	p. 278
Conservation theorems for ODEs	p. 279
Generalization of Noether's theorem	p. 280
Examples from classical mechanics	p. 281
Derivation of Einstein's formula for energy	p. 284
Conservation laws for the Dirac equations	p. 285
Problems to Chapter 7	p. 286
Generalized functions or distributions	p. 291
Introduction of generalized functions	p. 291
Heuristic considerations	p. 292
Definition and examples of distributions	p. 293
Representations of the ¿-function as a limit	p. 294
Operations with distributions	p. 295
Multiplication by a function	p. 295
Differentiation	p. 296
Direct product of distributions	p. 296
Convolution	p. 297
The distribution ¿(r2-n)	p. 298
The mean value over the sphere	p. 298
Solution of the Laplace equation ¿v(r) = 0	p. 298
Evaluation of the distribution ¿(r2-n)	p. 299
Transformations of distributions	p. 301
Motivation by linear transformations	p. 301
Change of variables in the ¿-function	p. 302
Arbitrary group transformations	p. 302
Infinitesimal transformation of distributions	p. 304
Problems to Chapter 8	p. 304
Invariance principle and fundamental solutions	p. 307
Introduction	p. 307
The invariance principle	p. 308
Formulation of the invariance principle	p. 308
Fundamental solution of linear equations with constant coefficients	p. 308
Application to the Laplace equation	p. 309
Application to the heat equation	p. 311
Cauchy's problem for the heat equation	p. 313
Fundamental solution for the Cauchy problem	p. 313
Derivation of the fundamental solution for the Cauchy problem from the invariance principle	p. 313
Solution of the Cauchy problem	p. 315
Wave equation	p. 316
Preliminaries on differential forms	p. 316
Auxiliary equations with distributions	p. 320
Symmetries and definition of fundamental solutions for the wave equation	p. 321
Derivation of the fundamental solution	p. 323
Solution of the Cauchy problem	p. 324
Equations with variable coefficients	p. 325
Problems to Chapter 9	p. 325
Answers	p. 327
Bibliography	p. 337
Index	p. 341
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