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4ЧИСЛЕННОЕ ИНТЕГРИРОВАНИЕ ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ
УРАВНЕНИЙ
Аскарова А.Ж.
кандидат физико-математических наук, доцент кафедры Высшей математики Казахский агротехнический университет имени Сакена Сейфуллина
г. Астана, Казахстан Грипп Е.А.
магистр, старший преподаватель кафедры Высшей математики, Казахский агротехнический университет имени Сакена Сейфуллина
г. Астана, Казахстан Елеусизова Г.Р. магистр, старший преподаватель кафедры Высшей математики, Казахский агротехнический университет имени Сакена Сейфуллина
г. Астана, Казахстан
NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS
Askarova A.,
PhD in Physics and Mathematics, Associate Professor of the Department of Higher mathematics Saken Seifullin Kazakh Agrotechnical University,
Astana, Kazakhstan Gripp Ye.,
Master's degree, Senior lecturer of the Department of Higher mathematics, Saken Seifullin Kazakh Agrotechnical University,
Astana, Kazakhstan Yeleussizova G. Master's degree, Senior lecturer of the Department of Higher mathematics, Saken Seifullin Kazakh Agrotechnical University,
Astana, Kazakhstan DOI: 10.5281/zenodo.7467608
Аннотация
Для многих прикладных задач получение точного решения дифференциальных уравнений практически невозможно. В этих случаях применяют методы приближенного решения дифференциальных уравнений. В статье рассматривается решение дифференциального уравнения различными численными методами.
Abstract
For many applied problems it is practically impossible to obtain the exact solution of differential equations. In these cases, methods of approximate solution of differential equations are used. This article considers the solution of a differential equation by various numerical methods.
Ключевые слова: математика, дифференциальное уравнение, метод Эйлера, метод Адамса, численные методы.
Keywords: mathematics, differential equation, Euler method, Adams method, numerical methods.
Methods for obtaining the exact solution of differential equations are possible only for a relatively small part of the equations encountered in practice.
Therefore, methods of approximate solution of differential equations are of great importance, which, depending on the form of representation of the solution, can be separated into two groups:
1) analytical methods, which give an approximate solution of a differential equation in the form of an analytical expression (application of the Taylor formula);
2) numerical methods giving an approximate solution in the form of a table. These methods are widely developed with great application of the computers and its capabilities (methods of Euler, Adams, etc.)[1], [3].
Nowadays, the amount of work involved in training to solve differential equations on a computers is not so large and does not exceed the amount of work involved in writing the solution. With the help of the computers it is possible to obtain a graph or its image on the screen. As a result, there is no urgent need to
study theoretical ways of integrating differential equations [2].
Recently, a large number of different software products (MathCAD, MathLAB, etc.) with which, specifying only the input data, it is possible to solve many different tasks of differential equations, algebra and other sections of mathematics have appeared [5].
The use of such programs significantly reduces the time to solve important problems. Nevertheless, using
y(x) = yo + xy'o +
these programs without analyzing the method by which the problem is solved cannot guarantee that the problem is solved correctly. Therefore, to fully understand how various kinds of differential equations and their systems are calculated, it is necessary to study and analyze numerical methods of solution [4].
One of the developed methods for approximate solution of differential equations is the method of decomposition of required solution in power series
2 3
where y0, y'0, y''0, ... are determined from the initial condition of the problem.
For example, solve the differential equation y' = y - x under initial conditions x0 = 0,y0 = 1,5.
To determine the values of y0, y'0, y''0, ... we use this equation. So y'' = y' - 1,y''' = y'',y(/^) = y''' =
y'', ..., y(") = y(n-1) = ... = y''
When x0 = 0, y„ = 1,5, y' = 1,5 - 0 = 1,5, y'' = 1,5 - 1 = 0,5, y'" = 0,5, = 0,5,..., y(n) = 0,5. Substituting these values into formula (1), we obtain
2 3 4
x2 x3 x4 y(x) = 1,5 + 1,5x + —— + —-- + —- + . y 2^2! 2 • 3! 2 • 4!
By setting values of x = 0; 0,25; 0,5; 0,75; 1; 1,25; 1,5, we find
y(0) = 1,5; y(0,25) = 1,8919; y(0,5) = 2,3138; y(0,75) = 2,8025; y(1) = 3,3541; y(1,25) = 3,8165; y(1,5) = 4,6693.
Let us solve the same problem under the same initial conditions by numerical Euler method, dividing the interval [0; 1,5] into six parts with step length h = 0,25.
In the equation replace the derivative by the finite difference ratio
Ay
— = /(x;y)
Hence
or
Ax
yfc -yfc-i = /(^fc-i;yfc-i)Ax yfc = yfc-i + /(^fc-i;yfc-i)Ax
When x0 = 0, y0 = 1,5; if x1 = 0,25, then
y1 = y0 + (y0 - *0)h = 1,5 + (1,5 - 0) • 0,25 = 1,875; if x2 = 0,5, then
y2 = y1 + (y1 - ^1)h = 1,875 + (1,875 - 0,25) • 0,25 = 2,2313; if x3 = 0,75, then
y3 = y2 + (y2 - *2)h = 2,2313 + (2,2313 - 0,5) • 0,25 = 2,6641; if x4 = 1, then
y4 = y3 + (y3 - x3)h = 2,6641 + (2,6641 - 0,75) • 0,25 = 3,1426; if x5 = 1,25, then
y5 = y4 + (y4 - x4)h = 3,1426 + (3,1426 - 1) • 0,25 = 3,6782; if x6 = 1,5, then
y6 = y5 + (y5 - x5)h = 3,6782 + (3,6782 - 1,25) • 0,25 = 4,2853.
Now find the solution of the equation y' = y - x with initial condition x0 = 0, y0 = 1,5 on the interval [0; 1,5] by Adams method with step h = 0,25.
Using the Adams method, find the first two values of the solution y1 and y2 by the below formulas
h2 h3 y! = yo + h y'o+^y''o+^j-y'''o+ "•
, (2h)2 „ (2h)3 ,„ y2=y1+2h y o + Jj y o + Jj y o +—
where
y'o = /Oo; yo) = Cy - OU=o = 1,5
y"o = /"Oo yo) = (y'- OU=o = 0,5 y"'o = /"'Oo; yo) = Cy''- OU=o = 0,5
Next, determine the values of y3 and y4 by the formula
5ft _
yfc+i = yfc + ft y ft + ftAyfc_i + — A2yfc-2 + -So, for example, when k = 2
ft2 5ft
= y2 + ft(ji - xi) + y (y2 - yi) + — (y2 - 2yi + Уo),
ft2 5ft
y4 = y3 + ft(j2 - ^2) + y (J3 - y2) + (J3 - 2y2 + yi),
ft2 52
ys = y4 + ft(j3 - + y (J4 - y3) ^ (y4 - 2y3 + y2), ft22 552ft y6 = ys + ft(J4 - *4) + y (ys - y^ + ^(ys - 2y4 + y3).
When x0 = 0, y0 = 5,5; when x1 = 0,25, y1 = 5,8920; when x2 = 0,5,
y2 = 2,3243; when x3 = 0,75, y3 = 2,8084; when x4 = 5, y4 = 3,3585; when x5 = 5,25, y5 = 3,9944; when x6 = 5,5, y6 = 4,7344.
Now let us put the obtained results of solving the equation y' = y - x by different methods into the table
y,-
according to Taylor Euler's method Adams method analytical solution 5 y = x + 5 + — ex
Xo = 0 5,5 5,5 5,5 5,5
X-, = 0,25 5,8959 5,875 5,8920 5,892
x2 = 0,5 2,3538 2,2353 2,3243 2,3243
x3 = 0,75 2,8075 2,6645 2,8084 2,8782
T-H II 3 X 3,3545 3,5426 3,3585 3,359
x5 = 5,25 3,8565 3,6782 3,9944 3,995
x6 = 5,5 4,6693 4,2853 4,7344 4,7404
The results most similar to the analytical solution were obtained when solving the equation by the Adams method.
Numerical methods allow us to solve equations of higher orders and systems of differential equations and obtain solutions close to analytical ones.
References
1. N.S. Bakhvalov. Numerical methods. M. Nauka, 1975, 631 pp.
2. Danilina N.I., Dubrovskaya N.S., Kvasha O.P., Smirnov T.L., Feklisova I.I. Numerical methods.
Textbook for technical high schools. Moscow, Vyssh. shkola, 1976, 367 pp.
3. N.S.Piskunov. Differential and Integral Calcu-lus.vol.2:Tutorial for High Schools. Moscow: Nauka, 1985, 560 pp.
4. A. Mеirmanov, N. Erygina, S. Mukha-metzhanov. Mathematical model of a liquid filtration from reservoirs. Electronic Journal of Dierential Equations, Vol. 2014 (2014), No. 49, pp. 1-13.
5. Maruani, E; Grabisch, M; Rusinowska, A. "A study of the influence through differential equations", Insurance: Rairo-operations research. Volume 46, JAN 2012, pages 83-106
