A review and systematization are given, as well as methods for solving problems in mathematical physics by means of differential equations of the first and second orders, and a 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 are often problems containing differential equations with partial derivatives of the first and second orders. Differential equations essential for physics, mechanics of 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 and technical problems, is given.
solution methods
mathematical physics
differential equations
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The basic equations of mathematical physics for the case when the desired function u depends on two independent variables are the following partial differential equations of the second order.
I. Wave equation
This equation is the simplest partial differential equation of the second order of hyperbolic type. The problems of 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. The problems of heat propagation in a homogeneous medium, the filtration of liquids and gases, some problems of probability theory, etc., are reduced to solving such an equation.
III. Laplace 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 the research problem, it should be taken into account that a physical phenomenon can be one-dimensional, two-dimensional and three-dimensional, and also be stationary (not changing in time).
The two-dimensional wave equation has the form:
which describes the vibrations of the membrane and the surface of an incompressible fluid.
In specific problems that reduce to the equations of mathematical physics, not a general, but a particular solution of the equation is always sought, which satisfies some additional specific conditions arising from physical considerations and features of the given problem.
These additional conditions are:
a) initial conditions, usually related to the initial moment of time (), from which the study of this phenomenon begins;
b) boundary conditions, that is, the conditions specified on the boundary of the medium (region) under consideration, inside which the solution of the given differential equation compiled by them is located.
The set of initial and boundary conditions is called boundary conditions.
The task of finding a particular solution of equations under initial conditions is called the Cauchy problem.
The problem of mathematical physics, in which both initial and boundary conditions are taken into account, is called a mixed problem (general Cauchy problem).
To solve the 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 a 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 described process. It is necessary to set the initial and boundary conditions for concretization.
Differential equations of mathematical physics of the second order. 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 and including the second order.
In order to bring equation (3) to the canonical form, it is necessary to write the so-called characteristic equation (4):
from which there are two equations:
;
and find their common 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, thermal processes that depend on time.
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 based on reduction
2. Equations in partial derivatives to ordinary or system of ordinary equations;
3. Numerical methods of solution (with the help of a computer).
Example: Find the function w=w(x,t), as a solution to the equation , where a>0, a=const, with the initial condition
.
The solution is an equation (transfer equation) in partial derivatives:
The characteristic equation for (1.1) has the form
where C is an arbitrary constant. The general solution of equation (1.1) has the form of a traveling wave:
It can be seen from (1.3) that a is the transfer rate. Since a > 0, the wave runs from left to right. Substituting the initial condition, we get:
. (1.4)
We get:
Answer: Function 
, is the solution of the transport equation for a given initial condition.
Bibliographic link
Kalanchuk I.V., Popov N.I. DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS // International Student Scientific Bulletin. - 2018. - No. 3-1 .;URL: http://eduherald.ru/ru/article/view?id=18212 (date of access: 09/10/2019). We bring to your attention the journals published by the publishing house "Academy of Natural History"
Scientometric indicators
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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 a journal's contextual citation impact by weighting the citations in each subject group. The contribution of each individual citation is the higher in each specific subject category, the less likely (for reasons of subject content) that such a citation will occur.
- Q2 Quartile: Mathematics (miscellaneous) 2018
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- 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 rating of citing journals.
- 25 Hirsch 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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Differential Equations (journal)
"Differential Equations"- a monthly mathematical magazine dedicated to differential equations and related integro-differential, integral equations, as well as equations in finite differences. Published from 1965. Included in list of scientific journals VAK. 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 chief editor), I. K. Lifanov, E. F. Mishchenko , E. I. Moiseev , Yu. S. Osipov, S. I. Pokhozhaev (deputy chief editor), N. Kh. Rozov, V. G. Romanov, V. A. Sadovnichiy, V. A. Solonnikov, F. L. Chernousko, T. K. Shemyakina (deputy editor-in-chief, secretary)
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