Mathematical Physics Equations

Description: Students master methods for solving partial differential equations of various types (elliptic, hyperbolic, and parabolic equations). A large place in the course is occupied by the presentation of the method of separation of variables, the solution of initial-boundary value problems

Amount of credits: 5

Пререквизиты:

• Differential Equations

Course Workload:

Component: Component by selection

Cycle: Base disciplines

Goal

• Mastering the necessary knowledge and skills for setting, solving and analyzing the results of solving problems of partial differential equations that arise when modeling physical objects and processes.

Objective

• consideration of the main types of equations of mathematical physics and methods of solution,
• - instilling in the student the skills of constructing mathematical models of practical problems and the skills of choosing an adequate mathematical apparatus for their research;
• - development of the skill to analyze and practical interpretation of the obtained mathematical results of the study of a real problem;

Learning outcome: knowledge and understanding

• Know: existing mathematical concepts, methods and models used in the analysis of partial differential equations;
• -analytical methods for solving equations of mathematical physics

Learning outcome: applying knowledge and understanding

• the ability to solve problems of a mechanical, applied and physical nature using the mathematical apparatus of the course being studied;
• the development of logical and algorithmic thinking, independent thinking skills, mathematical culture and mathematical intuition, necessary in further work in the study and solution of problems of mechanics, physics, natural science and technology.

Learning outcome: formation of judgments

• 1.Analyze the behavior of solutions of partial differential equations, based on the results obtained as a result of the study
• 2. for differential equations, realize selection of classical physics problems and analytical methods for solving them.

Learning outcome: communicative abilities

• The ability to work in a team in the process of solving practical problems of mechanics, physics, natural science and technology, to express and correctly defend their point of view in controversial issues.

Learning outcome: learning skills or learning abilities

• strive for professional and personal growth by mastering techniques and skills for solving specific problems from different areas of the discipline, helping to further solve engineering, production and scientific problems

Teaching methods

interactive technologies (with active forms of learning: executive (supervised) conversation; moderation; brainstorming; motivational speech);

independent research work of students during the educational process;

solving educational problems.

Assessment of the student's knowledge

Teacher oversees various tasks related to ongoing assessment and determines students' current performance twice during each academic period. Ratings 1 and 2 are formulated based on the outcomes of this ongoing assessment. The student's learning achievements are assessed using a 100-point scale, and the final grades P1 and P2 are calculated as the average of their ongoing performance evaluations. The teacher evaluates the student's work throughout the academic period in alignment with the assignment submission schedule for the discipline. The assessment system may incorporate a mix of written and oral, group and individual formats.

The evaluating policy of learning outcomes by work type

Evaluation form

The student's final grade in the course is calculated on a 100 point grading scale, it includes:

• 40% of the examination result;
• 60% of current control result.

The final grade is calculated by the formula:

 

Where Midterm 1, Midterm 2are digital equivalents of the grades of Midterm 1 and 2;

E is a digital equivalent of the exam grade.

Final alphabetical grade and its equivalent in points:

The letter grading system for students' academic achievements, corresponding to the numerical equivalent on a four-point scale:

Topics of lectures

• Statement of the problem of mathematical physics
• Classification and reduction to the canonical form of second-order partial differential equations
• The Cauchy problem for the equation of string vibrations
• The Cauchy problem for the wave equation
• A mixed problem for the equation of string vibrations
• Фурье әдісінің жалпы схемасы
• The first boundary value problem for the heat equation
• Cauchy problems for the heat equation Statement of the problem
• Integral representation of doubly differentiable functions Green's formula
• Integral representation
• Basic boundary value problems for the Laplace equation
• Solving the internal and external Dirichlet problem for a circle
• Green's function method
• TFinding the Green's function by the method of electrostatic images
• The definition of the potentials

Key reading

• А.Н.Тихонов, А.А.Самарский «Уравнения математической физики». Москва 2006 г.
• С.Л.Соболев, «Уравнения математической физики». Москва 2010 г.
• Болсун, А. И. Методы математической физики Минск : Вышэйш. шк., 2008
• Мукашева Р.У. Уравнения математической физики. Конспект лекций. ВКГТУ, 2011
• Чудесенко В.Ф. Сборник заданий по специальным курсам высшей математики. М., «Высшая школа»,2009.
• Будак Б.М. Сборник задач по математической физике./ Б.М. Будак, А.А. Самарский, А. Н. Тихонов, Гостехиздат; 2006

Further reading

• Смирнов М.М. Дифференциальные уравнения в частых производных второго порядка, Наука, 1964.
• Арсенин, В. Я. Методы математической физики и специальные функции. М.: "Наука", 1974.
• Болсун, А. И. Методы математической физики Минск : Вышэйш. шк., 1988