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“Equations of Mathematical Physics” - course 2800 rub. from MSU, training 15 weeks. (4 months), Date: November 30, 2023.
Miscellaneous / / December 02, 2023
The course is aimed at bachelors, masters and specialists specializing in mathematical, engineering or natural science disciplines, as well as university teachers. The purpose of the course is to introduce the student to a range of classical issues in the field of equations with mathematical physics and to teach the student the basic methods of studying such equations. The course covers classical material on the equations of mathematical physics (partial differential equations) within one semester of study. The sections “Linear and quasilinear equations of the first order”, “Classification of linear equations”, “Wave equation”, “Parabolic equation”, “Fundamental solutions”, “Laplace’s equation”. We will get acquainted with the classical formulations of problems - the Cauchy problem, boundary problem. Let's master the basic methods of studying equations - direct integration, the method of continuation of solutions, the Fourier method, the method of fundamental solutions, the method of potentials. We will often recall the derivation of these equations in problems of mathematical physics and the limits of applicability of our models.
Form of study
Correspondence courses using distance learning technologies
Admission Requirements
Availability of VO or SPO
Savchuk Artem Markovich
2
courseDoctor of Physical and Mathematical Sciences, Professor Position: Professor of the Department of Fundamental and Applied Mathematics, Faculty of Space Research, Moscow State University named after M.V. Lomonosov
1. First meeting.
Introductory word. Basic principles of working with equations of mathematical physics. Examples of simple equations. Classification. Solving simple equations by reducing them to ordinary differential equations. Replacing variables in an equation.
2. First order equations – linear and quasilinear.
Linear equations. Finding a suitable replacement - compiling and solving a system of first-order ordinary differential equations. First integrals of the system. Characteristics. Quasilinear equations. Finding a solution in an implicit form.
3. Cauchy problem. Classification of linear second order equations.
Statement of the Cauchy problem. Theorem on the existence and uniqueness of a solution to the Cauchy problem. Classification of second order linear equations with constant coefficients. Reduction to canonical form.
4. Hyperbolic, parabolic and elliptic equations.
Classification of second order linear equations with variable coefficients on the plane. Hyperbolic, parabolic and elliptic type. Solving hyperbolic equations. Problems with initial and boundary conditions.
5. String equation.
One-dimensional wave equation on the entire axis. Forward and backward wave. d'Alembert's formula. Duhamel integral. Boundary conditions for the equation on the semi-axis. Basic types of boundary conditions. Continuation of the solution. The case of a finite segment.
6. Fourier method using the string equation as an example.
The idea of the Fourier method. The first step is to find a basis. The second step is to obtain ordinary differential equations for the Fourier coefficients. The third step is taking into account the initial data. Convergence of series.
7. Diffusion equation (finite segment).
Derivation of the equation. Statement of problems (initial and boundary conditions). Fourier method. Taking into account the right-hand side and inhomogeneity in boundary conditions. Convergence of series.
8. Diffusion equation (whole axis).
Fourier transform, inversion formula. Solving the equation using the Fourier transform. Theorem – justification of the method (two cases). Poisson's formula. The case of an equation with the right side.
9. Generalized functions.
Writing Poisson's formula as a convolution. Recording in the form of a convolution of the solution to the heat equation on a finite segment. Schwartz class. Examples of functions from the class. Definition of generalized functions, connection with classical functions. Multiplication of a generalized function by a basic function, differentiation. Convergence of generalized functions. Examples of generic functions.
10. Working with generic functions.
Solving ordinary differential equations in generalized functions. Fourier transform of generalized functions. Convolution. Direct product. The carrier of a generalized function. Solving the inhomogeneous one-dimensional heat equation using the fundamental solution. Fundamental solution of an ordinary differential operator on an interval.
11. Fundamental solutions.
Derivation of the Poisson formula for the multidimensional heat equation. Derivation of Kirkhoff's formula. Derivation of Poisson's formula for the wave equation. Solving problems using the method of separation of variables, the method of superposition.
12. Laplace's equation.
Derivation of Laplace's equation. Vector field – potential, flow through a surface. Volume potential. Simple layer potential. Double layer potential. Logarithmic potential.
13. Dirichlet problem, Neumann problem and Green's function.
Harmonic functions. Weak extremum principle. Harnack's theorem. Strict maximum principle. Uniqueness theorem. Mean value theorem. Endless smoothness. Liouville's theorem. Green's formula. Green's function, its properties. Solution of the Poisson problem with Dirichlet conditions using the Green's function. Other boundary value problems. Construction of the Green's function by the reflection method.
14.Multidimensional Fourier method.
Solving problems using the Fourier method. Various boundary conditions. Bessel functions. Legendre polynomial. Review of the completed course. Summarizing.
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