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:

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  Models Described in Terms of Ordinary Differential Equations and Their Discrete Analogs:

  
  
  
  
  • Examples of Models Described in Terms of Ordinary Differential Equations. Lagrange's Formalism and Its Applications
  
  
  • Qualitative Methods in the Study of Dynamical Systems
  
  
  • Models Describing Nonlinear Oscillations
  
  
  • Oscillations in Non-Autonomous and Multidimensional Systems
  
  
  
  


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  Models Described in Terms of Partial Differential Equation:

  
  
  
  
  • Models Based on the Concept of Fields
  
  
  • Methods of Solving Linear Partial Differential Equations
  
  
  • Application of Numerical Methods for Solving Partial Differential Equations
  
  
  • Some Completely Integrable Nonlinear Models
  
  
  • Techniques and Methods for Obtaining Exact Solutions of Nonlinear Evolutionary Equations
  
  
  • Nonlinear Wave Patterns Described by Some Non-Integrable Models
  
  
  
  


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  Appendices:

  
  
  
  
  • Elements of Calculus of Functions of Complex Variables
  
  
  • Certain Statements Justifying the Use of Integral Transformations in Solving Differential Equations
  
  
  • An Introduction into the Theory of Special Functions
  
  
  
  

Readership: Graduate students and researchers interested in mathematical modeling of nonlinear systems; advanced undergraduate and graduate students in applied mathematics, physics, engineering and technology; researchers in nonlinear science, engineering and technology; libraries.

Key Features:

• Educational material for a full two-semester course has been collected. The presentation is carried out according to the principle of transition from simple to complex, therefore the first part covering one-semester course is entirely based on the classical material including various aspects of nonlinear mechanics and the theory of oscillations, while the second part, devoted to modeling physical fields, combines classical sections with material reflecting modern aspects of research


• The book includes an extensive section devoted to linear models of mathematical physics and methods for their solutions. From the point of view of the advantages provided by the use of software packages such as "Mathematica", linear models represent the most fertile field for application, because they solve, as a rule, in a matter of seconds those problem that, being calculated manually, take a lot of time and require significant dexterity


• The book contains various applications of the classical Hirota method, known from theory of solitons, for obtaining exact solutions to a large number of nonlinear models in no way related to soliton topics


• Progress in the wide and successful application of the Hirota method and its modifications was due precisely to the possibility of using special software packages, as this is illustrated in the relevant sections. The methods presented, as well as the corresponding programs, can be easily adapted to solve the problems of this kind that are of interest to a potential reader


• The authors paid special attention to balancing the levels of physical and mathematical presentation of the material so that the book be understandable to the widest possible range of readers' background


• The book combines well-known facts and the recent research, which helps lecturers to present modern achievements of nonlinear science, and students to get an idea of the actual work of scientists


• Together with the book, computer programs are provided, which not only visualize the material of the book, but can also be the basis for students to develop and analyze their own models


• The examples discussed in the book are multidisciplinary, which makes the manuscript interesting for a wide audience


• The book presents a number of new results of the authors, in particular, the search for exact solutions of Burgers-type equations, the study of the structure of non-analytical solutions of evolution equations, the construction of exact self-similar solutions to various nonlinear models. Special attention is paid to the study of localized wave modes (solitons, compactons), their formation and evolution within the models taking into account properties of the carrier medium, such as the Rosenau-Hyman K(m, n) hierarchy, Nesterenko's equation and hydrodynamic-type model of relaxing media. It is important that the book collects in one place the mentioned nonlinear models and describes in detail the numerical schemes, which are scattered among the various scientific papers