Algorithm of symbolic-numeric integration of the linear differential equation of four degree in the form of power series

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Algorithm of symbolic-numeric integration of the linear differential equation of four degree in the form of power series

I.N. Belyaeva

The developed algorithm and program in MAPLE for the solution ordinary differential equations of IV order, in general, in the form of generalized power series. The differential equation could consist the singular regular points. Some examples of the solution differential equations IV order are presented, that show the efficiency of the developed program.

Keywords: differential equations IV order, generalized power series, singular regular points.

differential equation singular regular

The most of the differential equations don't admit the decision in an explicit form. Therefore there is a need for development of effective methods of search reshening solutions of differential equations [1,2]. Recently the perspective direction are the methods combining symbolical transformations with the subsequent if necessary numerical calculation with application of modern mathematical packages, for example, Maple, Mathematica, Reduce, MACSYMA, etc.[3,4].

In the present work the algorithm is developed and the program for symbolical and numerical integration of the linear differential equation of the fourth order in the form, in general, of the generalized series with use of system of computer algebra of Maple is made. By means of the made program all have been found linearly - independent decisions for a number of the concrete equations for which in the current literature there are exact solutions.

Let's consider the differential equation of the fourth order

(1)

In case if coefficients-functions , , , do no't contain regular singular points and are analytic functions in the neighborhood of point , i.e. have the following forms

,

,

,

(2)

then four linear independent solutions , , and can be present in form of following power series:

,

,

,

. (3)

Coefficients , , , are defined uniquely by means of substitution of series (3) in the equation (1) and equating with zero coefficients at various degrees of an independent variable in the left part of the received equality.

In the presence of poles the type that not be higher than the fourth order in a point then solutions (3) to be other and depending on roots of the defining equation than the fourth (see, for example, [5-7]). From the theory of the ordinary differential equations [5] it is known that in order that the equation, in particular, of a look (1) had in the neighborhood of a special point at least if only one partial solution in the form of the generalized power series

, (4)

where the indicator is some constant number it is, enough, that this equation had an appearance

(5)

The indicator is found from the so-called defining equation:

. (6)

Let, there are also roots , , and of the equation (6). Then, if roots of the defining equation, are also independent, and any two of them don't differ on an integer, then to each number there corresponds a certain sequence of coefficients, and all four independent solutions forming fundamental system turn out are equal to

,

,

(7)

Coefficients, , , , are also defined by substitution of ranks (7) in the equation (5), at the same time coefficients , , , , and remain any (further we will put their equal to unit). This last coefficients are defined by initial conditions.

If the founded four values are such that two or several differ on an integer, then they can be located in the form of the following independent subsequences:

,

(8)

so that values in each sequence of the various were only integers, and the real part of the subsequences would be a non-increasing sequence. Only the first member of each sequence gives the solution (4), since, for example, any member of the sequence is equal to or less it than on the positive integer.

Let's consider one of the sequences of indicators of the differential equation, for example, the sequence , which is so located that if , then a positive integer or zero. As these indicators aren't surely equal, they can be divided into subsequences so that members of each subsequence were equal among themselves. So, suppose, that correspond to a multiple root ; correspond to a multiple root ; correspond to a multiple root , etc. until a row isn't exhausted.

Let's consider an indicator in the first subsequence. In this case arises subsequence of solutions:

,

,

(9)

where , the presence of terms in shows that all these solutions are linearly-independent.

Let's consider an indicator in the second subsequence, to it corresponds soutions:

(10)

Similarly, the subsequence with index gives solutions, etc. until the all roots in subsequence is exhausted. Since the functions , , , are linearly independent solutions of equation (5), then with their help, we find the general solution:

(11)

The developed algorithm of symbolical and numerical integration of the linear differential equation of the fourth order in the form of the, in general, generalized power series is given below.

1. Input of four coefficients functions, , , , that define the given differential equation, and also a desirable maximum order of power series.

2. Purpose of a flag of potencial:

if potencial=1, then the equation doesn't contain features in a point ;

if potencial=0, then the equation has a pole not above the fourth order in a point .

3. Finding of four linearly independent decisions, , , and according to formulas (4) if coefficients functions , , , , don't contain regular special points and are holomorphic functions in the neighborhood of a point.

4. In the presence of poles isn't higher than the fourth the equation has an appearance (5).

5. We find roots , , and of the defining equation (6).

6. If roots of the defining equation , , and , are also independent, and any two of them don't differ on an integer, then coefficients , , and , and are defined by substitution of ranks (7) in the equation (5).

7. If the found four values are that that two or several differ on an integer, then linearly independent solutions equation (1) are built according to formulas (8), (9).

8. Construction general solution of the equation (1) cording to expression (11).

The developed program allows to find solutions of the differential equations of the fourth order in the form of power series, generally, of any degree, but limited to opportunities of the concrete computer. By means of this program test symbolical and numerical calculations for some differential equations which results coincide with exact analytical decisions have been carried out.

Example 1. If the coefficients of the function have values , , , , e.i. the given differential equation has no singular points, then the equation (1) will be as follows

.

Using the developed program [8], obtained the four linearly independent solutions of the first members of which is below given:

,

,

,

,

which exactly coincide with the known analytical solution , , ,

Example 2. Consider the differential equation

.

This equation without singular points, but with multiple roots in the characteristic equation multiple equal to 2 and has other roots: , .

Using the developed program [8] was found linearly independent solutions:

,

,

,

,

which exactly coincide with the analytical solution , , , .

Example 3. Consider the equation

.

This equation has a singular point . In this case, the defining equation has roots equal , , , . Using the developed program [8], obtained the four linearly independent solutions:

, , , ,

which exactly coincide with the analytical solution [9].

Example 4. Consider the equation

.

This equation has a singular point . In this case the defining equation has roots equal , . Using the developed program, obtained the four linearly independent solutions:

,

,

,

,

which exactly coincide with the analytical solution [9].

Example 5. Consider the equation

.

This equation has a singular point . In this case the defining equation has roots equal , , . Using the developed program [8], obtained the four linearly independent solutions:

,

,

,

.

Example 6. Consider the equation

.

This equation has a singular point . In this case, the defining equation has roots equal , . Using the developed program [8], obtained the four linearly independent solutions:

, ,

, .

Example 7. Consider the equation

.

This equation has a singular point . In this case the defining equation has roots equal . Using the developed program [8], obtained the four linearly independent solutions:

,

,

, .

This paper presents the algorithm and program for symbol-numeric calculation of linear-independent solutions for the ordinary differential equation of fourth order, which can contain regular singular points.

References

1. Bahvalov N.S., Dhidkov N.P., Kobelkov G.M. Numerical methods. M.: BINOM. Laboratory of knowledge, 2008. - 636 c.

2. Kontorovich L.V., Krylov V.I. Approximate methods of higher analysis. Л. Fizmatgiz, 1962. - 708с.

3. Denverport Dj., Sire I., Tyrne E. Computer algebra: Пер. с франц. - M.: Mir, 1991. - 352с.

4. Easayan A.P. Control structures and data structures in Maple / A.P Easayan, V.N. Chybarikov, N.M. Dobrovolskii, Yu.M. Martynyuk - Tula: Izd-vo gos.ped. yn-ta im. L.N. Tolstogo, 2007. - 316 c.

5. Ince E.L. Differential equations. London: University Press, 1939. - 700 p.

6. Sansone Dj. Ordinary differential equations / Дж. Сансоне. - Т.1 - M.: Izd-vo IL, 1953. -346 с.

7. Tricomi F. Differential equations. Turin: Blackie & son limited, 1961. - 348 p.

8. Belyaeva I.N., Chekanov N.A., Chekanova N.N. Program of symbol-numeric integration of linear differential equation of four order. Patent of RU, Program for ECM, №2016611952 - 2016.

9. Kamke E. Spravocnik po ordinary differential equations. - M.: Nayka, 1965. - 703 c.

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