Journal of differential equations. International student scientific newsletter

Differential Equations (magazine)

"Differential Equations"- a monthly mathematical magazine devoted to differential equations and related integro-differential, integral, and finite difference equations. Published since 1965. Included in the list of scientific journals of the Higher Attestation Commission. Name of the English version of the journal: Differential Equations.

Editorial board: A. V. Arutyunov, F. P. Vasiliev, I. V. Gaishun, A. V. Gulin, S. V. Emelyanov, N. A. Izobov, S. K. Korovin (deputy editor-in-chief) , I.K. Lifanov, E.F. Mishchenko, E.I. Moiseev, Yu.S. Osipov, S.I. Pokhozhaev (deputy editor-in-chief), N.H. Rozov, V.G. Romanov, V. A. Sadovnichy, V. A. Solonnikov, F. L. Chernousko, T. K. Shemyakina (deputy editor-in-chief, executive secretary)

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I Differential equations are equations containing the required functions, their derivatives of various orders and independent variables. Theory D. u. arose at the end of the 17th century. influenced by the needs of mechanics and other natural science disciplines,... ... Great Soviet Encyclopedia

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A review and systematization are provided, as well as methods for solving problems in mathematical physics using differential equations of the first and second orders, and the classification of differential equations are considered. This approach made it possible to obtain the necessary optimality conditions. Mathematical models of natural science phenomena and processes often represent problems containing first- and second-order partial differential equations. Differential equations are essential for physics; mechanics and technology are called differential equations of mathematical physics. A quasilinear partial differential equation of the first order is considered. A linear second order partial differential equation with two independent variables is considered. To obtain a general solution to the equation, a characteristic system of ordinary differential equations is considered. An example of the application of differential equations to the solution of various applied, including engineering, problems is given.

solution methods

mathematical physics

differential equations

1. Bondarenko V.A., Mamaev I.I. Professional orientation in teaching mathematics to students of biological faculties // Bulletin of the AIC of Stavropol. – 2014. – No. 1 (13). – P. 6–9.

2. Bondarenko V.A., Tsyplakova O.N. Problems with economic content in differential calculus classes // Current issues in the theory and practice of accounting, analysis and audit: annual 75th scientific and practical conference / Editorial Board: V.Z. Mazloev, A.V. Tkach, I.S. Sandu, I.Yu. Sklyarov, E.I. Kostyukova; resp. per issue A.N. Bobryshev. – 2011. – P. 124–127.

3. Bondarenko V.A., Tsyplakova O.N. Some aspects of an integrated approach to the study of mathematical analysis // Accounting, analytical and financial-economic problems of regional development: annual 76th scientific and practical conference of the Stavropol State Agrarian University “Agricultural science for the North Caucasus region”. – 2012. – P. 280–283.

4. Litvin D.B., Gulay T.A., Dolgopolova A.F. Application of operational calculus in modeling economic systems // Agrarian science, creativity, growth. 2013.

5. Prospective appearance of fault-tolerant digital control systems for maneuverable aircraft / V.V. Kosyanchuk, S.V. Konstantinov, T.A. Kolodyazhnaya, P.G. Redko, I.P. Kuznetsov // Flight: All-Russian scientific and technical journal. – 2010. – No. 2. – P. 20–27.

6. Popova S.V., Smirnova N.B. Elements of algorithmization in the process of teaching mathematics in higher school // Modern problems of development of economics and social sphere: collection. materials International scientific-practical conference dedicated to the 75th anniversary of the Stavropol State Agrarian University. – 2005. – P. 526–531.

The basic equations of mathematical physics for the case when the desired function u depends on two independent variables are the following second-order partial differential equations.

I. Wave equation

This equation is the simplest second-order partial differential equation of hyperbolic type. Problems about transverse vibrations of a string and longitudinal vibrations of rods, sound and electromagnetic vibrations, gas vibrations, etc. are reduced to solving such an equation.

II. Wave equation

This equation is the simplest parabolic type equation. Problems of heat propagation in a homogeneous medium, filtration of liquids and gases, some questions of probability theory, etc. are reduced to solving such an equation.

III. Laplace's equation

representing the simplest equation of elliptic type. Problems about the properties of stationary electric and magnetic fields, about the stationary distribution of heat in a homogeneous body, problems of hydrodynamics, diffusion, etc. are reduced to solving this equation.

Remark 1. In general, when setting up a research problem, it should be taken into account that a physical phenomenon can be one-dimensional, two-dimensional and three-dimensional in nature, and also be stationary (not changing in time).

The two-dimensional wave equation is:

which describes the vibrations of the membrane and the surface of an incompressible fluid.

In specific problems that can be reduced to equations of mathematical physics, one always seeks not a general, but a particular solution to the equation that satisfies some additional specific conditions arising from physical considerations and the peculiarities of the given problem.

These additional conditions are:

a) initial conditions, usually relating to the initial moment of time () from which the study of a given phenomenon begins;

b) boundary conditions, that is, conditions specified on the boundary of the medium (region) under consideration, within which the solution to the given differential equation they compiled is located.

The set of initial and boundary conditions is called boundary conditions.

The problem of finding a particular solution to equations under initial conditions is called the Cauchy problem.

A problem of mathematical physics in which both initial and boundary conditions are taken into account is called a mixed problem (Cauchy problem of general form).

To solve equations of mathematical physics, the following are usually used:

a) d’Alembert’s method (method of characteristics),

b) Fourier method (method of separation of variables).

Consider the quasilinear partial differential equation of the first order:

. (1)

To obtain a general solution to equation (1), consider the characteristic system of ordinary differential equations:

If c = 0, then the system is reduced to one equation

If the general integral of the equation, then

Common decision.

The differential equation itself contains only the most general information about the process being described. It is necessary to set initial and boundary conditions for specification.

Differential equations of second order mathematical physics. A large number of processes and phenomena in physics are described using second-order partial differential equations; this is due to the fact that the fundamental laws of physics - conservation laws - are written in terms of second derivatives.

Consider a second-order linear partial differential equation with two independent variables:

(3)

where a, b, c are some functions of x, y that have continuous derivatives up to the second order inclusive.

In order to bring equation (3) to canonical form, it is necessary to write the so-called characteristic equation (4):

from which two equations emerge:

;

and find their general integrals.

In general, a second-order linear partial differential equation of parabolic type with n independent variables can be written as:

,

Parabolic type equations describe unsteady diffusion, time-dependent thermal processes.

Methods for solving equations of mathematical physics

All methods for solving these equations can be divided into two groups:

1. Analytical methods for solving equations that are based on reduction

2. Partial differential equations to ordinary or a system of ordinary equations;

3. Numerical methods of solution (using a computer).

Example: Find the function w=w(x,t), as a solution to the equation, where a>0, a=const, under the initial condition

.

The solution is the partial differential equation (transfer equation):

The characteristic equation for (1.1) has the form

where C is an arbitrary constant. The general solution to equation (1.1) has the form of a traveling wave:

From (1.3) it is clear that a is the transfer speed. Since a >0, the wave runs from left to right. Substituting the initial condition, we get:

. (1.4)

We get:

Answer: Function , is a solution to the transport equation for a given initial condition.

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Scientometric indicators

Usage

• 10274 Download full texts 2018

  Springer measures the number of full text downloads from the SpringerLink platform according to the COUNTER (Counting Online Usage of NeTworked Electronic Resources) standards.

• 21 Usage factor 2017/2018

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Influence

• 0.659 Impact factor 2018

  Impact factor published by Clarivate Analytics in Journal Citation Reports. Impact factors refer to the previous year.

• 1.02 Source Normalized Impact per Paper (SNIP) 2018

  Source Normalized Impact per Paper (SNIP) measures the contextual citation impact of a journal by weighting citations within each subject group. The contribution of each individual citation is higher in each specific subject category, the less likely (from considerations of subject content) that such a citation will occur.

• Q2 Quartile: Mathematics (miscellaneous) 2018

  A set of journals from the same subject category are ranked according to their SJR and divided into 4 groups called quartiles. Q1 (green) unites journals with the highest scores, Q2 (yellow) - the next ones, Q3 (orange) - the third group by SJR value, Q4 (red) - journals with the lowest scores.

• 0.47 SCImago Journal Rank (SJR) 2018

  SCImago Journal Rank (SJR) is a measure of a journal's scientific impact that takes into account the number of citations a journal receives and the ranking of citing journals.

• 25 H-Index 2018

SCOPE

Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.

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