This paper presents a symbolic-numeric approach for solving linear differential equations of the fourth order in the form of generalized power series. The working program allows to find solutions to differential equations of the fourth order in the form of power series, generally, of any order, but is limited by capabilities of a given computer. Some examples of solving differential equations of the fourth order are presented, which show the efficiency of the developed progra

Belyaeva, I. N.

Bondarev, V. G.

Chekanov, N. A.

Migal, L. V.

English

DSpace at Belgorod State University

 COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI)  3182   Cite This Paper: Irina N. Belyaeva, Nikolay A.Chekanov, Larisa V.Migal, Vladimir G.Bondarev. (2019). Symbolic-numeric approach for solving linear differential equations of the fourth order, 8(6), COMPUSOFT, An International Journal of Advanced Computer Technology. PP. 3182-3186.  This work is licensed under Creative Commons Attribution 4.0 International License.    SYMBOLIC-NUMERIC APPROACH FOR SOLVING LINEAR DIFFERENTIAL EQUATIONS OF THE FOURTH ORDER  Irina N. Belyaeva*, Nikolay A. Chekanov, Larisa V. Migal, Vladimir G. Bondarev  Belgorod State University, 85, PobedyStreet, Belgorod, 308000, Russia  Abstract:  This paper presents a symbolic-numeric approach for solving linear differential equations of the fourth order in the form of generalized power series. The working program allows to find solutions to differential equations of the fourth order in the form of power series, generally, of any order, but is limited by capabilities of a given computer. Some examples of solving differential equations of the fourth order are presented, which show the efficiency of the developed program. The results are consistent with the available literature data.  Keywords: differential equations of the fourth order; generalized power series; singular regular points. I. INTRODUCTION An effective method for integrating differential equations is finding solutions in the form of power series and the subsequent finding of the coefficients of these series [1-4]. But in practice, with concrete calculations, one has to deal with a large amount of calculations when finding unknown coefficients of power series, and the complexity of finding solutions increases in cases when peculiarities appear. However, the use of modern computers with software packages for analytical calculations, such as Maple, Mathematica, Reduce, and others [5-7], allows us to quickly perform the necessary calculations to build solutions of linear differential equations in the form of series, and to very large orders. In problems of mathematical physics, linear differential equations of second and fourth orders most often arise [8]. It should also be noted that the solutions of some nonlinear differential equations can be expressed in terms of linearly independent solutions of the corresponding linear differential equations. For example, the solution of the non-linear Ermakov-Milne-Pinney equation is constructed from two independent solutions of a linear second-order ordinary differential equation. The paper presents a computational scheme for solving fourth-order ordinary linear differential equations in the form of generalized power series using the Maple computer algebra system. Also, using the compiled program, linearly independent solutions were found for a number of specific differential equations. II. METHOD Computational Scheme  )()()()()()()( 123)( xyxPxyxPxyxPxy IV0)()(0 xyxP      (1)   Available online at: https://ijact.in Date of  Submission Date of  Acceptance 03/05/2019 15/06/2019 Date of  Publication 30/06/2019 Page numbers 3182-3186(5 Pages) ISSN:2320-0790  COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI)  3183  If coefficients-functions )(3 xP , )(2 xP , )(1 xP , )(0 xP o not contain singular regular points and are holomorphic functions in the vicinity of point 0xx  , i.e. have the following forms: 00)0(0 )()(kkk xxpxP , 00)1(1 )()(kkk xxpxP  00)2(2 )()(kkk xxpxP , 00)3(3 )()(kkk xxpxP   then four linear independent solutions 1y   , 2y  , 3y    and4y    can be presented in the form of following power series:  20)1(1 )(1)(kkk xxcxy , 20)2(02 )()(kkk xxcxxxy ,  (3) 20)3(203 )(2/)()(kkk xxcxxxy , 20)4(304 )(3/)()(kkk xxcxxxy   Coefficients )1(kc  , )2(kc  , )3(kc  , )4(kc   are defined uniquely by means of substitution of series (3) in the equation (1) and equating with zero coefficients at various orders 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 0xx    then solutions (3) to be other and depending on roots of the defining equation than the fourth (see, for example, [9-11]). From the theory of the ordinary differential equations [9] 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.  )0(,)()()( 0000  cxxcxxxykkk (4)  where the indicator  is some constant number it is, enough, that this equation had an appearance  )()()()(200)2(00)3()( xyxxpxyxxpxy kkkkIV  .0)()()()( 400)0(300)1( xyxxpxyxxpkkkk  (5)  The indicator  is found from the so-called defining equation:  30)2()1()3()2()1( p  0)1( 001020  ppp     (6)  Let us assume, there are also roots 1  , 2  , 3   and 4    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  )0(,)()()( )1(000)1(011  cxxcxxxy kkk, )0(,)()()( )2(000)2(022  cxxcxxxy kkk, (7) )0(,)()()( )3(000)3(033  cxxcxxxy kkk )0(,)()()( )4(000)4(044  cxxcxxxy kkk   Coefficients 1kc ,2kc ,3kc ,4kc  , are also defined by substitution of ranks (7) in the equation (5), at the same time coefficients 10c , 20c ,30c , 40c , and remain any (further we will put their equal to unit). These 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:  121 ,,,   , 11 ,,,       (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  COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI)  3184  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  121 ,,,   ,  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 121  i correspond to a multiple root i  ;   11   jii  correspond to a multiple root  ;   correspond to a multiple root ij    , 11   kjj   etc. until a row isn't exhausted. Let's consider an indicator 1   in the first subsequence. In this case arises subsequence   of solutions:  ),()( 111 xwxy   ),(ln),()( 12112  xwxxwxy         (9) ……………………………………  2121111 ))(ln,()1())(ln,()(iii xxwixxwxy  2121111 ))(ln,()1())(ln,()(iii xxwixxwxy    where 0000 ,...)3,2,1,0(,)()(),(kskskss scxxcxxxws , the presence of111 ))(ln,(rxxw   terms  in rW   shows that all these   solutions are linearly-independent.  Let's consider ir an indicator   in the second subsequence, to it corresponds ij   solutions:   121 ))(ln,())(ln,()(iiiii xxwixxwxy ),( ii xw   …………………………………………………… (10)   22111 ))(ln,()1())(ln,()(jijij xxwjxxwxy ),(1 ij xw    Similarly, the subsequence with index j  gives jk     solutions, etc. until the all roots in subsequence is exhausted. Since the functions )(1 xy , )(2 xy , )(3 xy ,)(4 xy   are linearly independent solutions of equation (5), then with their help, we find the general solution:   )()()()()( 44332211 xyCxyCxyCxyCxy  (11)  Examples of Solving Differential Equations of the Fourth Order The developed program [12] allows to find solutions of the differential equations of the fourth order in the form of power series, generally, of any order, but is limited by capabilities of a given computer. The program was used to make symbolical and numerical calculations for some differential equations, the results of which exactly coincide with analytical solutions given.   Example 1. If the coefficients-function have values0)(3 xP , 0)(2 xP , 0)(1 xP , 4)(0 xP   equation (1) takes the following form  0)(4)()(  xyxy IV  which also has no singular points. Developed with the help of a program [12] receive the following four linearly independent solutions:  ...0817296480017484400125201611)( 1612841  xxxxxy ...00013894040161972972001226801301)( 1713952  xxxxxxy ...00025009722881136216080012268001180121)( 181410623  xxxxxxy  151173401021620600112474001630131)( xxxxxy    which exactly coincide with the analytical solutions: )sin()(1 xexyx , )cos()(2 xexyx  , )sin()(3 xexyx  , )cos()(4 xexyx .  Example 2. For a differential equation 0)(2)(5)(3)()()(  xyxyxyxyxy IV   which has no singularities, but with multiple roots   in the characteristic equation of multiplicity 3 and the fourth root 1321  kkk . Was obtained the following four 24 k linearly independent solutions:  COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI)  3185   ...60480118144012016011126019016011211)( 109876541  xxxxxxxxy  ...45360019134401134401725201401401245)( 109876542  xxxxxxxxxy  ...4032001367201403203142018016018121)( 1098765423  xxxxxxxxxy  ...75600160481201601252011180115112131)( 1098765434  xxxxxxxxxy   which exactly coincide with the analytical solutionxexy 21 )(  ,xexy )(2 ,xexxy )(3 , xexxy  24 )( .  Example 3. If the coefficients of the function have values  4)(3 xP , 6)(2 xP , 4)(1 xP , 1)(0 xP , then equation (1) will be as follows  0)()(4)(6)(4)()(  xyxyxyxyxy IV   This equation has no singular points, but with multiple roots in the defining equation are equal   of multiplicity 4. Using the developed program, we find four linearly 14321  kkkk independent solutions:  ...432001648011152125217213012411)( 109876541  xxxxxxxxy  ...1296011920133617212018161)( 109876542  xxxxxxxxxy   ...115201168012881601161612121)( 1098765423  xxxxxxxxxy ...151201216013601721181613131)( 1098765434  xxxxxxxxxy   which exactly coincide with the analytical solutions xexy )(1  ,xexxy )(2   , xexxy  23 )(  , xexxy  34 )(  .  Example 4. Consider the equation  0)()()( 2)(4  xyxxyxxyx IV    This equation has a singular point 00 x  . In this case, the defining equation has roots equal 2321     ,04    . Using the developed program [12], obtained the four linearly independent solutions:21 )( xxy  , 222 )ln()( xxxxy  ,22223 xxx2xxxy  )ln()(ln)( , 1)(4 xy  .  Example 5. Consider the equation  0)()()()(2)( 23)(4  xyxyxxyxxyxxyx IV   This equation has a singular point 00 x   . In this case the defining equation has roots equal 14321    . Using the developed program [12], obtained the four linearly independent solutions: xxy )(1 , xxxxy  )ln()(2 , xxxxxxy  )ln(2)(ln)( 23 , xxxxxxxxy  )ln(2)(ln3)(ln)( 234   With the help of the developed program, solutions of two problems from the theory of resistance of materials were found.  Example 6. The basic equation of stability for a flat bend shape of an I-beam, to which a bending moment and external force is applied, is a fourth-order linear differential equation [4].  0)(1)(1)(42)(  xQdxQaxQ IV   where a  , d   - geometric and strength characteristics of the beam , )(xQ   - node twisting the beam along its length. In this equation, the coefficients are functions0)(3 xP    , 221)(axP   , 0)(1 xP   , 401)(dxP   , have no particular points.  Using the developed program [12], we obtain four linearly independent solutions: 12128844810864488444642441 479001600)3(3628800)2(40320)(72012411)( xdaadadxdaadxdaadxdaxdxQ  11864498444742542 39916800)2(362880)(504011201)( xdaadxdaadxdaxdxxQ  1312884486227020800)3(xdaadad    10888448846446444442233628800)3(40320)2(720)(2412)( xdaadadxdaadxdaadxaxxQ 128108448479001600)34(xdaadad   COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI)  3186    118884489464474444523419958400)3(181440)2(2520)(6013)( xdaadadxdaadxdaadxaxxQ 1381084483113510400)34(xdaadad   III. CONCLUSION This paper presents a computational scheme for solving fourth-order ordinary differential equations in the form of generalized power series. The Maple program was composed based on the developed algorithm and some problems from the material resistance theory were solved. Moreover, some examples of solving differential equations of the fourth order are presented, which show the efficiency of the developed program. The results are consistent with the available literature data.  IV. REFERENCES  [1] Bahvalov, N.S., Dhidkov, N.P., Kobelkov, G.M., 2011. Numerical methods (Chislennyemetody). Binom. Laboratory of knowledge. Moscow, p 640. (in Russian).  [2] Kontorovich, L.V., V.I. Krylov, 1962. Approximate methods of higher analysis. FIZMATGIZ, Moscow, 1962, 5th ed; Engtransl of 3rded, Interscience, New York and Noordhoff, Groniongen 1958.       [3] Collatz, L.: EigenwertaufgabenmittechnischenAnwendungen. Leipzig: AkademischeVerlagsgesellschaftGeest u. Portig K.-G. 1949. 466 S.  [4] Föppl, August. Drang und zwang. Vol. 2. R. Oldenbourg, 1928.  [5] Forst, W., Hoffmann, D., 2012. Explore function theory with Maple (Funktionentheorieerkundenmit Maple). Springer-Verlag Berlin Heidelberg”. [6] Easayan, A.P., Chybarikov, V.N., Dobrovolskii, N.M., Martynyuk, Yu.M., 2007. Control structures and data structures in Maple. Tula: Izd-vogos.ped. yn-ta im. L.N. Tolstogo. (in Russian) [7] Franco Vivaldi, 2018. Experimental Mathematics with Maple. Rome: CRC Press. [8] Kamke, E., 1965.  Manual of ordinary differential equations. Moscow: Nauka. (In Russian) [9] Ince, E.L., 1939. Differential equations. London: University Press. [10] Sansone, J., 1948. Ordinary differential equations. V.1. Rome: CRC Press. [11] Knesckke, A. (1961), F. G. Tricomi, Differential Equations. X + 273 S. London 1961. Blackie & Son Ltd. Preisgeb. 50 s. Z. angew. Math. Mech., 41: 470-470.( Accessed from https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.19610411027) [12] Belyaeva, I.N., Chekanov, N.A., Chekanova, N.N., 2016. Program of symbol-numeric integration of linear differential equation of four order. Patent of RU, Program for ECM, №2016611952. (in Russian)   

Symbolic-numeric approach for solving linear differential equations of the fourth order

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Symbolic-numeric approach for solving linear differential equations of the fourth order

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