Study Of Linear And Nonlinear Models With "mathematica"

The book is devoted to the problems of modeling physical systems and fields using the tools and capabilities of the "Mathematica" software package. In the process of teaching classical courses in mechanics and mathematical physics, one often has to overcome significant difficulties associated with the cumbersomeness of the mathematical apparatus, which more than once distracts from the essence of the problems under consideration. The use of the "Mathematica" package, which has a rich set of analytical and graphic tools, makes the presentation of classic issues related to modeling and interpretation of physical processes much more transparent. This package enables the visualization of both analytical solutions of nonlinear differential equations and solutions obtained in the form of infinite series or special functions. The textbook consists of two parts that can be studied independently of each other. The first part deals with the issues of nonlinear mechanics and the theory of oscillations. The second part covers linear problems of classical mathematical physics and nonlinear evolution models describing, inter alia, transport phenomena and propagation of waves. The book contains the codes of programs written in the "Mathematica" package environment. Supplementary materials of programs illustrating and often complementing the presented material are available on the publisher's website. Contents Preface Models described in terms of ordinary differential equations and their discrete analogs 1. Examples of models described in terms of ordinary differential equations. Lagrange's formalism and its applications 1.1 Malthusian model and logistic model 1.2 Discrete analogs of the logistic equation 1.3 Model of a pendulum moving in a gravitational field 1.4 Lagrange's formalism 1.5 Examples of obtaining the governing equations using the Lagrange formalism 1.5.1 Double pendulum 1.5.2 Flat pendulum with a movable suspension point 1.5.3 A system of two masses: one mass is suspended in a gravitational field, and the other can move along a horizontal surface without friction 2. Qualitative methods in the study of dynamical systems 2.1 Basic concepts. Linearization. Classification of stationary points on a plane 2.2 Conservative systems with one degree of freedom 2.3 Dynamical systems in R2 having non-analytical solutions 2.4 Dynamial systems in Rn. Invariant manifolds and subspaces 2.5 Discrete maps generated by the phase flows of dynamical systems 2.6 Parametric resonance 2.7 Kapitza's pendulum 2.8 Modification of the problem posed by Kapitza 3. Models describing nonlinear oscillations 3.1 Introduction 3.2 Predator-prey model 3.3 Van der Pol's equation 3.4 Conditions for the existence of periodic trajectories 3.5 Andronov-Hopf bifurcation 3.5.1 Introduction 3.5.2 Central manifold 3.5.3 Perturbations of the center 3.5.4 Reduction of the dynamical system to the canonical form 3.5.5 Normal form. Criterion of the appearance of the limit cycle 3.5.6 Examples of appearance of limit cycles 3.5.7 The homoclinic bifurcation 4. Oscillations in non-autonomous and multidimensional systems 4.1 Introduction 4.2 Bifurcations in the Duffing equation with the periodic inhomogeneous part 4.3 Methods of investigations of chaotic solutions 4.3.1 Lyapunov index 4.3.2 Rössler system and the Lorenz map 4.3.3 Logistic map 4.3.4 Period doubling bifurcations in the logistic map Models described in terms of partial differential equations 5. Models based on the concept of fields 5.1 Introductory remarks 5.2 (Gas-) Hydro-dynamic equations 5.2.1 Balance equation for mass 5.2.2 Balance of momentum equation. Complete systems of hydrodynamic-type equations 5.2.3 Sub-models and their self-similar solutions 5.3 Transport equations 5.3.1 Derivation of the heat transport equation 5.3.2 Thermal explosion problem for linear transport equation 5.3.3 Thermal explosion problem for nonlinear transport equation 5.3.4 Generalized transport models 5.4 Classification of the second order partial differential equations 6. Methods of solving linear partial differential equations 6.1 Method of solving Cauchy problem based on the Fourier transform 6.2 Laplace transform 6.3 The method of separating variables 6.3.1 One dimensional case 6.3.2 Method of separating variables: evolutionary equations with two spatial variables 6.3.3 Solving linear evolutionary equations for three spatial variables 7. Application of numerical methods for solving partial differential equations 7.1 Finite-difference method 7.2 The method of lines 7.3 Galerkin method 7.4 Finite elements method 8. Some completely integrable nonlinear models 8.1 Introduction 8.2 The simplest wave equations. Inuence of nonlinearity and dispersion on the evolution of perturbations 8.3 Further examples of generalized solutions of the Hopf equation 8.4 The Burgers equation 8.4.1 Traveling wave solution to the Burgers equation, describing viscous shock wave 8.4.2 Exact solutions of the Burgers equation 8.5 Korteweg-de Vries (KdV) equation and its modifications 8.5.1 Introductory remarks 8.5.2 Scott Russell's discovery. Formal derivation of the KdV equation 8.5.3 Conservation laws of the KdV equation 8.5.4 Complete integrability of the KdV equation 8.5.5 Numerical scheme for the KdV equation and its implementation for solving the Cauchy problem 9. Techniques and methods for obtaining exact solutions of nonlinear evolutionary equations 9.1 Hirota method and multi-soliton solutions to the Korteweg-de Vries equation 9.2 Application of the Hirota method and its modifications for solving the nonlinear evolutionary equations 9.2.1 Traveling wave solution to the Burgers-like equation describing active media 9.2.2 Traveling wave solutions supported by the hyperbolic modification of the Burgers equation 9.3 Bi-kink solutions of the convection-reaction-diffusion equation and its hyperbolic modification 9.3.1 Introduction 9.3.2 Solutions of the convection-reaction-diffusion equation describing interacting wave fronts 9.3.3 Bi-kink solutions of the modified convection-reaction-diffusion equation 10. Nonlinear wave patterns described by some non-integrable models 10.1 Introduction 10.2 Rosenau-Hyman equation: evolution of compactons 10.3 Evolutionary PDEs associated with a chain of pre-stressed granules 10.3.1 Introduction 10.3.2 Evolutionary PDEs associated with the granular pre-stressed chains 10.3.3 Numerical simulations for dynamics of compactons 10.3.4 Conclusions and discussion 10.4 Relaxing hydrodynamic-type model and its qualitative investigations 10.4.1 Numerical study of the relaxing hydrodynamic-type model Appendix A Elements of calculus of functions of complex variables A.1 Holomorphic functions and their properties A.2 Integration of the complex-valued functions A.3 Functions holomorphic in the ring. Laurent series expansion and classification of singular points Appendix B Certain statements justifying the use of integral transformations in solving differential equations B.1 The space of tempered distributions B.2 Fourier transform of the tempered distributions B.3 Conditions enabling to reduce the search for the inverse Laplace transform to calculation of residues Appendix C An introduction into the theory of special functions C.1 Introduction C.2 Cylindrical functions C.3 Legendre polynomials Bibliography Index "A textbook for modeling nonlinear mechanics and the theory of oscillations, as well as linear problems of classical mathematical physics and nonlinear evolution models using the tools and capabilities of the "Mathematica" software package, containing codes of programs in the book"-- Provided by publisher