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Applying Power Series to Differential Equations
Applying Power Series to Differential EquationsНазвание: Applying Power Series to Differential Equations: An Exploration through Questions and Projects
Автор: James Sochacki, Anthony Tongen
Издательство: Springer
Серия: Problem Books in Mathematics
Год: 2023
Страниц: 220
Язык: английский
Формат: pdf (true), epub
Размер: 24.8 MB

This book is aimed to undergraduate STEM majors and to researchers using ordinary differential equations. It covers a wide range of STEM-oriented differential equation problems that can be solved using computational power series methods. Many examples are illustrated with figures and each chapter ends with discovery/research questions most of which are accessible to undergraduate students, and almost all of which may be extended to graduate level research. Methodologies implemented may also be useful for researchers to solve their differential equations analytically or numerically. The textbook can be used as supplementary for undergraduate coursework, graduate research, and for independent study.

This text assumes that students have taken an elementary ODE course covering the basics mentioned above. This text could be used as a supplemental text in the upper-level undergraduate course discussed above. The material is intended for exploration and insight for these students, their instructors and, also, for scientists and engineers who encounter ODE dynamical systems in their work. This text can be used by instructors who desire interesting examples for these courses that highlight common ODE topics and demonstrate the strong relationship between ODEs, polynomials and power series solutions to these ODEs. The goal is to elucidate the importance of understanding and using polynomials in ODE models, especially nonlinear ODEs, while demonstrating how numerical approximations of solutions to initial value ODEs (IV ODE) are sensitive to different types of numerical methods. This text is unique in that it presents 12 ODEs that collectively demonstrate all of the above concepts.

This text highlights the beauty and complexity of the solutions of ODEs using ODEs that are amenable to student investigation, and many have been used by the authors as undergraduate research projects. In fact, most of the chapters in this text are focused on a nonlinear ODE that highlights the properties of the ODEs through polynomials and their solutions through power series. The examples demonstrate the power of polynomials in ODEs to generate trajectories and phase portraits demonstrating properties like stability, continuous dependence, sensitivity to initial conditions, periodic solutions, limit cycles, Poincaré-Bendixson theory, chaos, heteroclinic and homoclinic orbits and delay differential equations. All the examples can be assigned to students as projects that include visualization and discovery learning.

This text begins with an overview of linear ODEs and their relationship with polynomials, especially power series. This relationship for linear ODEs extends to almost all nonlinear ODEs that students encounter. The polynomial ideas presented in the text are shown through interesting and exciting examples, but it is straightforward to see how these ideas can be extended to most nonlinear ODE dynamical systems. The text is not a theorem/proof text, but rather a text that encourages exploration and discovery through examples. Interesting student projects and research questions are presented in the examples. The authors and many of their colleagues have used some of these examples to get undergraduate students excited about performing research in ODEs. Many of these students are now using some of these ideas in their professional research.

Most of the ODEs presented in this text are common ODEs that are examined in a different manner than in most ODE texts. We develop enough of the ODE and/or its applications that the reader can get a general understanding of the problem and see where further problem-solving and research can take them. The text is not an exercise textbook, but rather an exploratory and discovery text. However, we do ask interesting questions for each ODE presented.
Every chapter and appendix start with a brief description of the material that will be presented and the mathematics that will be used and ends with exercises that lead to questions and/or projects. Each question can lead to a project depending on how far the reader, the student, the teacher or the researcher wants to take exploring the exercise.

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