POWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS

Transcription

1 Joural of Applied Mathematics ad Computatioal Mechaics 4 3(3) 3-8 POWER SERIES SOLUION OF FIRS ORDER MARIX DIFFERENIAL EQUAIONS Staisław Kukla Izabela Zamorska Istitute of Mathematics Czestochowa Uiversity of echology Częstochowa Polad staislaw.kukla@im.pcz.pl izabela.zamorska@im.pcz.pl Abstract. I the paper the solutio of secod order differetial equatios with various coefficiets is preseted. he cocerig equatios are writte as first order matri differetial equatios ad solved with the use of the power series method. Eamples of applicatio of the proposed method to the equatios occurrig i the techical problems are preseted. Keywords: power series differetial equatios Itroductio he goal of may researchers work is fidig ew ad improvig eistig various methods for solvig ordiary differetial equatios. I various types of differetial equatios (liear ad o-liear with costat or variable coefficiets) occurrig for eample i techical specificatio umber of issues it is possible to desigate the eact solutio aalysis ad i the others however it is ecessary to approimate methods. Oe of the classical methods kow sice the seveteeth cetury i solvig differetial equatios is the method of power series ivolvig the appoitmet of a solutio i the form of a ifiite aylor series (series coefficiets are obtaied from recursive equatios). his method was applied e.g. by Eiseberger i [] to solve the secod order differetial equatio describig rods vibratio problems ad by Quaisi i [] for the o-liear free vibratio of beams with restraied eds. I tur Zhou i [3] proposed a method of differetial trasformatio - a improved method of power series differig the way of determiig the coefficiets of the series. A power series method with domai partitio i a implemeted matri formulatio is aother method alterative to other techiques of power series [4]. I this paper the method of solvig secod order ordiary differetial equatio will be preseted by trasformig this equatio i the system of differetial equatios of the first order the presetig it i matri otatio ad solvig with the use of power series method. As eamples there will be preseted solutios of two classic equatios: the Airy equatio ad equatio occurrig i the descriptio of the rod s vibratio.

2 4 S. Kukla I. Zamorska. Formulatio ad solutio of the problem Let us cosider the first order matri differetial equatio ( ) ( ) ( ) ( ) = B + F () where ( ) y ( ) y ( ) L y ( ) F( ) f ( ) f ( ) L f ( ) ad B ( ) = bi j( ) = m i j m = m. he equatio is completed by iitial coditio ( ) = () We are lookig o solutio of the equatio () i the form of power series ( ) = where = [ y y L ym]. Assumig that B( ) = B =! =! ad F( ) = F substitutig ito equatio () we obtai: =! = B + F (3) =! =! =! =! After trasformatios i equatio (3) we get the followig recursive relatio he coefficiets may be writte as: = F+ B + = F + Bk k =... (4) k= k = Φ + =... (5) where Φ = =E (the uit matri) ad Φ + = F + B kφk k= k + = kφk k= k B. Fially a solutio of a ihomogeeous matri differetial equatio () ca be epressed as a sum ( ) = Φ + = Φ ( ) + ( ) (6) =! =!

3 Power series solutio of first order matri differetial equatios 5. Eamples of method s applicatio Let the ordiary differetial equatio of secod order with variable coefficiets ( ) d y d ( ) dy + a( ) + a( ) y( ) = f( ) (7) d with the coditios: ( ) ( ) l y = y y l = y (8) Itroducig fuctios y ( ) = y( ) y ( ) y ( ) a system y ( ) = y( ) ( ) ( ) ( ) ( ) = equatio (7) ca be writte as y = f a y a y (9) = or i the matri form as equatio () where ( ) y ( ) y ( ) F ( ) f( ) ad ( ) = B a( ) a(. ) Fuctios occurrig i equatio (7) are C class ad may be writte as i ( ) = i i( ) =! y y = i (i = ) ( ) =! a a Iitial ad boudary equatios (8) are as follows: ( ) ( ) y y where y ( ) = y y ( l) = y y ( ) ad y ( ) from the relatios l y ( ) = y = y y + y l l = ( l) ( l) = () =! f f y l () y l ca be respectively determied l! ( ) l = +. () =! y l y Airy equatio Schrödiger equatio oe of the fudametal equatios of o-relativistic quatum mechaics uder certai assumptios ca be reduced to the Airy equatio

4 6 S. Kukla I. Zamorska which has a well-kow aalytical solutio i the form: y ( ) + y( ) = (3) 3 3 y( ) C J + CJ (4) where J( ) is a Bessel fuctio of the first kid ad costat values C C deped o the selected iitial coditios. Usig the matri otatio we have F( ) = ( ) = = + B B B B (5) =! where B B B= B3= K =. Recursive relatios for series coefficiets + are B B =... (6) = B + = + Solutio of Airy equatio (3) i a matri form may be writte as where the first few values of are: =E = B (7) ( ) = =! = B + B 3 3= B+ BB+ BB 4 4= B+ BB+ BBB+ 3BB + 3B = B+ BB+ BBB + 3BBB + 3BB+ 4BB+ 4BBB+ 8B B By usig desigatios: M M M M 3 M 4 (B = M B = M 3 ) it ca be see that sets of matrices {Ο M M M M 3 M 4 } with sets multiplicatio create a semigroup. he algebraic properties of pair ({Ο M M M M 3 M 4 } ) describes the multiplicatio table (able ).

5 Power series solutio of first order matri differetial equatios 7 Multiplicatio table able M M M M 3 M 4 M M M M M 3 M 4 M M M M Ο Ο M M Ο Ο M M M 3 M 3 M 3 M 4 Ο Ο M 4 M 4 Ο Ο M 3 M 4 d dy + = d d Equatio p( ) k p( ) y (8) his type of equatio occurs frequetly i problems of mechaics for eample i the vibratio s descriptio of rods or strigs. he fuctios occurrig i the equatio have a physical iterpretatio: y() is a fuctio of deflectio p() is a cross-sectio area ad k is a parameter characterized vibratio frequecy of the mechaical system uder cosideratio. Suppose that p( ) = e α where α is a costat. he geeral solutio of homogeeous equatio (8) is [5] α y( ) = e C cos 4k α + C si 4k α (9) Matrices B ad F eistig i the equatio () are as follows: B( ) p ( ) k = k α B B = =... p( ) Series coefficiets of the solutio of (8) are: F( ) = () = B = B... = B () ad the solutio may be writte i the form: ( ) = B () =!

6 8 S. Kukla I. Zamorska Coclusios he method of solvig secod order liear differetial equatios preseted i the work may be used for solvig -th order liear equatios. Ay of these equatios ca be represeted by a system of first order liear differetial equatios with more tha oe depedet variable ad as the result by oe first order differetial matri equatio. Although this method should t be applied idiscrimiately it is suitable to solve particularly equatios of arbitrary order with variable coefficiets which typically arise i vibratio or heat trasfer problems. Refereces [] Eiseberger M. Eact logitudial vibratio frequecies of a variable cross-sectio rods Applied Acoustics [] Quaisi M.I. A power series solutio for the o-liear vibratio of beams Joural of Soud ad Vibratio (4) [3] Zhou K. Differetial rasformatio ad Its Applicatio to Electrical Circuits Wuha People Republic of Chia Huazhog Uiversity Press 986 (i Chiese). [4] Iaudi J.A. Matusevich A.E. Domai-partitio power series i vibratio aalysis of variable cross-sectio rods Joural of Soud ad Vibratio [5] Zamorska I. Logitudial vibratios of a o-uiform rods coupled by double sprig-mass systems Scietific Research of the Istitute of Mathematics ad Computer Sciece 7 (6)