Some Topics of Calculus, Differential Equations and Nonlinear Dynamics

2024/2025

Academic Year

ENG

Instruction in English

6

ECTS credits

Delivered at:

Institute for Cognitive Neuroscience

Course type:

Elective course

When:

1 year, 1, 2 module

Instructor

Zakharov, Denis

Course Syllabus

Abstract

The bridging course “Advanced Calculus” has an aim to train the master students to be ready for the courses devoted to Bayesian Statistics, Qualitative and Quantitative Research Methods in Psychology, Computational Neuroscience, Digital Signal Processing and some others. In the framework of this course the students study the real- and complex-valued functions, theory of derivatives and integrals, differential equations and dynamical systems, as well as Taylor, Fourier and Laplace series.

Learning Objectives

• • Gain understanding of the concept of functions, continuous functions and different kind of discontinuous functions
• • Gain skills in evaluating derivatives and integrals.
• • Gain skills in expansion of a function into a Taylor series and finding its convergence interval.
• • Gain skills in solving of homogeneous and nonhomogeneous linear differential equations.
• • Gain understanding of Fourier series, Fourier transforms and their applications.
• • Gain understanding of a Laplace transform and its applications for solving of differential equations
• • Gain understanding of dynamical systems analysis.

Expected Learning Outcomes

• Be able to analyze one dimensional and two dimensional flows
• Know basic facts about Fourier series, Fourier transforms and their applications
• Know basic facts about Laplace transforms and their applications in solving of ordinary differential equations
• Know basic methods for solving of linear differential equations
• Know main operations, rules and properties of infinite series, functional series as well as Taylor series
• Know man rules and properties of derivatives and integrals
• • Know main operations, rules and properties of sets, real numbers, functions, continued functionsю

Course Contents

• Special elementary functions.
• Infinite and Functional series.
• Fourier Analysis.
• Linear homogenous and non-homogeneous differential equations.
• Laplace transform.
• Introduction to dynamical systems.

Assessment Elements

• written exam
• Tests

Interim Assessment

• 2024/2025 2nd module
  0.2 * Tests + 0.2 * Tests + 0.6 * written exam

Bibliography

Recommended Core Bibliography

• Anton, H., Bivens, I. C., & Davis, S. (2016). Calculus (Vol. 11th ed). New York: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1639210
• Gorain, G. C. (2014). Introductory Course on Differential Equations. New Delhi: Alpha Science Internation Limited. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1878058
• Strogatz, S. H. (2000). Nonlinear Dynamics and Chaos : With Applications to Physics, Biology, Chemistry, and Engineering (Vol. 1st pbk. print). Cambridge, MA: Westview Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421098

Recommended Additional Bibliography

• Nekorkin, V. I. (2015). Introduction to Nonlinear Oscillations. Weinheim: Wiley-VCH. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1099772

Authors

• ZAKHAROV DENIS GENNADEVICH