1. SylabUZ
2. Wydział Nauk Ścisłych i Przyrodniczych
3. semestr zimowy 2024/2025
4. WFiA - oferta ERASMUS - Program Erasmus
5. Differential equations in physics

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Differential equations in physics - opis przedmiotu

Cel przedmiotu

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.

Wymagania wstępne

Zapisz zmiany

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

Zakres tematyczny

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.

 

 

Metody kształcenia

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.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

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.

 

Literatura podstawowa

[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.

Literatura uzupełniająca

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

Uwagi

Zmodyfikowane przez dr Marcin Kośmider (ostatnia modyfikacja: 28-04-2024 14:52)