1. SylabUZ
2. Faculty of Physics and Astronomy
3. winter term 2020/2021
4. Physics - First-cycle studies leading to Bachelor's degree
5. Differential equations in physics

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Differential equations in physics - course description

Aim of the course

Learning students of basic concepts, facts and methods of ordinary and partial differential equations. Obtaining the ability to solve certain types of ordinary differential equations, systems of ordinary and partial differential equations. Preparation for courses in which physical phenomena are modeled by differential equations.

Prerequisites

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Mathematical analysis I and II and algebraic and geometric methods in physics

Scope

1. Reminder of basic notions of ordinary differential equations, definition, types of ordinary differential equations, general and particular solutions, initial problem, geometric interpretation. Equations solvable in elementary way, homogeneous, with separable variables, equation with integrating factor, Bernoulli equation, Riccati equation.
2. Basic properties of solutions of linear first order differential equations: linear space of homogeneous solutions, its dimension, base - fundamental system, Wronski matrix and its determinant, solving systems of homogeneous linear equations with constant coefficients.
3. Solving the higher order linear equations with analytical coefficients using the power series - some special functions.
4. Basic concepts of partial differential equations: definition, examples, order; linear, semi-linear, quasi-linear, nonlinear differential equations.
5. First-order partial differential equations: relationship with ordinary equations, method of characteristics.
6. Classification of partial differential equations of order two of two independent variables.

7. Laplace and Poisson equations.
8. Fourier method of variable separation. Initial problem of thermal conductivity equation with periodic boundary conditions
9. Wave equation.
10. Soliton equations: dispersion and nonlinear wave equations, various forms of KdV equations, various types of solutions and their properties, infinitely many conservation laws and integrability of KdV.

 

 

Teaching methods

Conventional lecture  illustrated with examples of the use of equations in physics solved analytically and with the help of software for symbolic and numerical calculations.

During classes students analyse and solve exercises illustrating the content of the lecture.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Lecture: Positive passing of exam (written). Obtaining a positive grade requires at least 55% of correct answers to the questions and tasks asked.
Classes: Passing condition - positive grades of two written tests on the basis of obtaining at least 55% of points on each of them.
Before taking the exam a student must gain positive grade during the class.

 

Recommended reading

[1] Gewert M., Skoczylas Z., "Równania różniczkowe zwyczajne. Teoria, przykłady, zadania.", wyd. Wrocław, 2002r

[2] W. Krysicki, L. Włodarski, Analiza matematyczna w zadaniach, tom 2., Wydawnictwo Naukowe PWN, Warszawa

[3] W. Walter, Ordinary differential equations. Springer-Verlag, Berlin, 1998

[4] D.W. Jordan, P. Smith, Nonlinear ordinary differential equations, Oxford University Press, Oxford, 2011

[5] H. Marcinkowska, Wstęp do teorii równań różniczkowych cząstkowych, PWN, Warszawa 1986,

[6] L. C. Evans, Równania różniczkowe cząstkowe, Wydawnictwo Naukowe PWN, Warszawa 2002.

[7] J.D. Logan, An introduction to nonlinear partial differential equations, Wiley-Interscience, John Wiley & Sons, Inc., Hoboken, 2008

[8] P.V. ONeil, Advanced engineering mathematics, International Student Edition, Thomson, Canada, 2007

[9] L. C. Evans, Partial Differential Equations, AMS, 1998.

[10]  Materials made available by the lecturers.

Further reading

P. Olver, Introduction to partial differential equations, Springer-Verlag, New York, 2014

Notes

Modified by dr hab. Piotr Lubiński, prof. UZ (last modification: 03-06-2020 16:57)