Home/ Majors directory/Applied mathematics and computer science/Partial Differential Equations of Mathematical Physics
Partial Differential Equations of Mathematical Physics
Major: Applied mathematics and computer science
Code of subject: 6.113.00.O.027
Credits: 6.00
Department: Applied Mathematics
Lecturer: D.Sc., Professor Bogdan M. Markovych
Semester: 5 семестр
Mode of study: денна
Завдання: As a result of studying the academic discipline, students should be able to demonstrate the following learning outcomes: they should know the classification of the main types of partial differential equations, the conditions for the existence of solutions to problems, and the main properties of solutions. Students should be able to build mathematical models of certain physical phenomena, determine their type, find a solution, and analyze the obtained results. The study of an academic discipline involves forming and developing the following competencies in students.
General competencies:
ЗК2 – basic knowledge of mathematics and natural sciences to the extent necessary for mastering professionally oriented disciplines; ЗК5 – the ability to apply knowledge in practice; ЗК6 – the ability to search and analyze information from various sources; ЗК7 – have research skills; ЗК13 – the ability to communicate effectively at the professional and social levels; ЗК14 – creativity, ability to system thinking; ЗК15 – potential for further education. Professional competencies: ФК1 – basic knowledge of scientific concepts, theories, and methods necessary for understanding the principles of construction, analytical and numerical (computational experiment) research of mathematical models; ФК3 – the ability to apply and integrate knowledge and understanding of the disciplines of other specialties; ФК7 – ability to use acquired knowledge and skills for research, choice of implementation and design of mathematical models and software and information complexes for their study; ФК10 – the ability to argue the choice of research methods of mathematical models, critically evaluate the obtained results and defend the decisions made.
Learning outcomes: ЗН1 – ability to demonstrate knowledge and understanding of basic scientific and mathematical principles underlying mathematical modeling;
ЗН3 – the ability to demonstrate basic knowledge in one of the areas of mathematical modeling;
УМ1 – apply knowledge and understanding to identify and solve specialty problems, using known methods.
Required prior and related subjects: Prerequisites:
• mathematical analysis;
• algebra and geometry;
• differential equations.
Co-requisites:
• numerical methods of mathematical physics.
Summary of the subject: The educational discipline "Partial Differential Equations of Mathematical Physics" is a component of the educational and professional training program for specialists at the first level of higher education "bachelor" in the field of knowledge 11 - "Mathematics and Statistics" with the specialty 113 - "Applied Mathematics" in the educational program "Applied Mathematics and Informatics." This discipline is mandatory. It is taught in the 5th semester of the 3rd year in 180 hours. (6 ECTS credits) in particular: lectures – 45 hours, practical classes – 30 hours, independent work – 105 hours. The course includes five tests. The discipline ends with an exam. The course is dedicated to familiarizing students with the methods of building mathematical models of various physical processes, the theory of such models, and mastering the basic methods of their analysis and solution.
Опис: Chapter 1. "Classification and reduction to the canonical form of partial differential equations of the second order."
Differential equations with two independent variables. Differential equations with many independent variables. Canonical forms of linear differential equations with constant coefficients.
Chapter 2. "Basic mathematical models of physical problems."
Problems that lead to equations of the hyperbolic type (small transverse oscillations of the string, small oscillations of the membrane, small longitudinal oscillations of the rod, electrical oscillations in conductors).
Problems that lead to equations of the parabolic type (heat conduction equation for a rod, heat propagation in space, diffusion equation).
Problems that lead to elliptic-type equations (stationary temperature field, electrostatics equations, Laplace equations in an orthogonal curvilinear coordinate system).
Chapter 3. "Method of traveling waves."
The Cauchy problem for the wave equation on an infinite straight line (Dalembert's formula, the physical meaning of the Dalembert formula, continuous dependence of the solution of the Cauchy problem on the initial conditions, inhomogeneous equation, Duhamel's method).
A mixed problem for the wave equation on a half-line and on a segment (continuation method for a half-line, continuation method for a segment, non-homogeneous boundary conditions).
Problems for the equation of the second order of hyperbolic type on the plane (Cauchy problem, Riemann method, Gurs problem).
Propagation of waves in space (partial solutions of the uniform wave equation, averaging method, inhomogeneous wave equation, Kirchhoff's formula, inhomogeneous plane wave equation and descent method, physical content of solutions of the wave equation in space and on the plane, continuation method).
Chapter 4. "The method of separating variables. Schedule by eigenfunctions of the Sturm-Liouville problem".
Uniform boundary conditions. Heterogeneous boundary conditions. Integral Fourier and Laplace transforms.
Chapter 5. "Boundary-value problems for the equation of oscillations."
Theorem on the uniqueness of the solution. Equation of free oscillations of a string with fixed ends, physical interpretation of the solution. The inhomogeneous equation of oscillations. A case of force localized at a point.
Chapter 6. "The equation of thermal conductivity."
The maximum principle for solutions of the heat conduction equation. Theorem on the uniqueness of the solution.
Equation of thermal conductivity on a segment. Equation of thermal conductivity on a straight line. Heat conduction equation on a half-line (homogeneous and non-homogeneous boundary conditions). The equation of thermal conductivity in space and on a plane.
Chapter 7. "Boundary-value problems for Laplace and Poisson equations."
Setting of boundary value problems. Fundamental solutions of the Laplace equation in space and on the plane. Inversion transformation.
Green's formulas. Basic properties of harmonic functions. The maximum principle and its consequences.
Properties of solutions of the Dirichlet problem. Properties of solutions of the second and third boundary value problems.
Chapter 8. "The method of separation of variables for the Laplace equation."
Dirichlet's problem in a circle. Poisson's integral.
Chapter 9. "The Green's function of the Laplace operator."
The Green's function of the Dirichlet problem. Green's function of the third boundary value problem. Green's function of the Neumann problem.
The method of electrostatic images. Representation of the Green's function in the form of a series. Method of conformal mapping.
Chapter 10. "Potential theory."
Bulk potential and its properties. Lyapunov surfaces and curves. Double layer potential and its properties. The potential of a simple layer and its properties.
Application of potentials to solving boundary value problems.
Assessment methods and criteria: • practical classes, oral interviews, tests (30%)
• final control (70%): combined written and oral exam
Критерії оцінювання результатів навчання: 100–88 points – ("excellent") is awarded for a high level of knowledge (some inaccuracies are allowed) of the educational material of the component contained in the primary and additional recommended literary sources, the ability to analyze the phenomena being studied in their interrelationship and development, clearly, succinctly, logically, consistently answer the questions, the ability to apply theoretical provisions when solving practical problems; 87–71 points – ("good") is awarded for a generally correct understanding of the educational material of the component, including calculations, reasoned answers to the questions posed, which, however, contain certain (insignificant) shortcomings, for the ability to apply theoretical provisions when solving practical tasks; 70 – 50 points – ("satisfactory") awarded for weak knowledge of the component's educational material, inaccurate or poorly reasoned answers, with a violation of the sequence of presentation, for soft application of theoretical provisions when solving practical problems; 49-26 points - ("not certified" with the possibility of retaking the semester control) is awarded for ignorance of a significant part of the educational material of the component, significant errors in answering questions, inability to apply theoretical provisions when solving practical problems; 25-00 points - ("unsatisfactory" with mandatory re-study) is awarded for ignorance of a significant part of the educational material of the component, significant errors in answering questions, inability to navigate when solving practical problems, ignorance of the main fundamental provisions.
Порядок та критерії виставляння балів та оцінок: The educational discipline ends with a semester control, the form of which is provided by the curriculum with a semester assessment. The semester grade consists of the points provided for current management and examination control. The teacher proves this information to the students in the first lesson on academic discipline.
Recommended books: 1. Markovych B. M. Rivnyannya matematychnoyi fizyky. –L'viv: Vydavnytstvo Natsional'noho universytetu «L'vivs'ka politekhnika», 2010. –384 s.
2. Tikhonov A. N., Samarskiy A. A. Uravneniya matematicheskoy fiziki. –M.: Nauka, 1977. –736 s.
3. Arsenyi V. Ya. Matematicheskaya fizika. Osnovnye uravneniya i spetsial'nye funktsii. –M.: Nauka, 1966. –368 s.
4. Perestyuk M. O., Marynets' V. V. Teoriya rivnyan' matematychnoyi fizyky. –K.: Lybid', 2001. –336 s.
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