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.

Czesław Mączka, PhD, senior lecturer, retired employee of the Faculty of Applied Mathematics at AGH University of Science and Technology. A long-time lecturer in various fields of mathematics, among others, delivered lectures on numerical methods for ordinary differential equations, lectures on programming in the Mathematica system — advanced level and the use of Mathematica in practical applications of dynamical systems. Co-author of over a dozen scientific papers from the above-mentioned fields of mathematics and one textbook.

Sergii Skurativskyi, leading researcher at the Department of self-organization of natural media at the Subbotin Institute of Geophysics NAS of Ukraine; senior researcher at the State Institute "The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine"; taught the course "Solving the problems of mathematical physics using the Mathematica software" as a visiting professor at the AGH University of Science and Technology (Faculty of Applied Mathematics, Kraków, Poland); author of about 40 scientific papers, two monographs and one textbook.

Vsevolod Vladimirov, professor of AGH since 2002. In the period from 2008 to 2012, the head of Department of Differential Equations at the Faculty of Applied Mathematics. In the period from 2012 to 2020, the dean of this faculty. Organizer of a series of conferences titled "Mathematics in Technical and Natural Sciences". Taught in particular the following courses: Modeling of nonlinear processes using the "Mathematica" package; Equations of mathematical physics; Theory of generalized functions; Nonlinear models of transport phenomena; Application of group theory to differential equations; Theory of dynamical systems. Authors of about 40 scientific papers in periodical issues, two monographs and one textbook. Retired since January 2022.

Chapter 1: Examples of models described in terms of ordinary differential equations. Lagrange’s formalism and its applications

Chapter 2: Qualitative methods in the study of dynamical systems

Chapter 3: Models describing nonlinear oscillations

Chapter 4: Oscillations in non-autonomous and multidimensional systems

Chapter 5: Models based on the concept of fields

Chapter 6: Methods of solving linear partial differential equations

Chapter 7: Application of numerical methods for solving partial differential equations

Chapter 8: Some completely integrable nonlinear models

Chapter 9: Techniques and methods for obtaining exact solutions of nonlinear evolutionary equations

Chapter 10: Nonlinear wave patterns described by some non-integrable models