Jornal of Kerbala University, Vol. 5 No.4 Scientific.Decembar 2007

Transcription

1 Joral of Kerbala Uiversiy, Vol. No. Scieific.Decembar 7 Soluio of Delay Fracioal Differeial Equaios by Usig Liear Mulise Mehod حل الوعادالث التفاضل ت الكسز ت التباطؤ ت باستخذام طز قت هتعذد الخطىاث الخط ت Basim K. AL-Salai Dearme of Comuer, College of Sciece, Kerbala Uiversiy, Iraq (basim78@yahoo.com Absrac: The objecive of his aer o fid he soluio of delay fracioal differeial equaios as well as, heir umerical soluio. The modificaio was mode by iroducig he ew mehod which is liear mulise mehods, which is solvig umerically ordiary differeial equaios, delay differeial equaios ad delay fracioal differeial equaios. Furhermore his aer reses he saeme ad roof of he fudameal heorem of coverge ad sabiliy of delay fracioal differeial equaios, his had bee doe wih some modificaio. الخالصة الهذف هي هذا البحث هى إل جاد ح ل الوعادالث التفاضل ت الكسىر ت التباطؤ ت باإلضافت إلى ح ل هن العذدي. التعذ ل ك اى وطا بت قذ ن الطز قت الجذ ذة الت ه طزق هتعذدة الخطىاث الخط ت الت ت ح ل هعادالث تفاضل ت عاد ت بشكل عذدي الوعادالث التفاضل ت التباطؤ ت والوعادالث التفاضل ت الكسز ت التباطؤ ت. اضافت اى هذا البحث ب ي و بزهي ال ظز ت األساس ت الوتقارب ه وإستقزارالوعادالث التفاضل ت الكسز ت التباطؤ ت هذا هع ع و ل ب ع ض التعذ ل عل ها..Iroducio The delay fracioal differeial equaios lay a imora role i he heory of fucioal differeial equaios.i rece year, he heory of his class of equaios had become a ideede red ad lieraure o his subjec comrises over iles. delay fracioal differeial equaios have bee sudied durig he las wo ceuries. The sigificace of hese equaios lies i heir abiliy o describe rocesses wih rearded ime. The imorace of hese equaios i various braches of echology, ecoomics, biology, medical sciece, hysics ad social sciece has bee recogized recely ad has caused mahemaics o sudy hem wih icreasig ieres.oe of he mos imora braches of fracioal differeial equaios is he so called delay fracioal differeial equaios, which is a fracioal differeial equaio cosis of oe or some is argumes evaluaed a a ime which is differ by ay fixed umber or variable which is called he ime or lag. Also, heory of delay fracioal differeial equaio ca be cosidered as a geeralizaio o he heory of fracioal differeial equaios ad usig Liear Mulise Mehods (LMM's o solve delay fracioal differeial equaios of he rearded ad derivig some of he well ow umerical mehods o solve delay fracioal differeial equaios..mehod of soluio delay fracioal differeial equaios Because of he iiial codiio which a fucio give for ime se ierval wih legh equals, mus fid his soluio for divided i o ses wih legh also. Hece, we mus fid he soluio for each ime se. The bes well ow mehod for solvig delay fracioal differeial equaios is he mehod of ses (soluio of fracioal differeial equaios, [Driver, 977],[Halaay, 966], [El'sgol'c ad Nori, 96] [Basim ], which is for simliciy wih ou los of geeraliy ad illusraio urose of solvig delay fracioal differeial equaios of he form: x q ( = f(, x(, x(-, x ' (-...( for each equaio he soluio is cosruced se by se as follows: Give ha a fucio ( coiues o [ -, ], herefore oe ca form he soluio i he ex ime se ierval [, +], by solvig he followig equaio: x q ( = f(, x(, (-, ' (-, for +, q< 7

2 Joral of Kerbala Uiversiy, Vol. No. Scieific.Decembar 7 wih iiial codiio x( =(. If we suose ha ( is he exisig by virue of coiuiy hyoheses. Similarly if ( is defied o he whole segme [, +], hece by formig he ew equaio: x q ( = f(, x(, (-, ' (-, for + +, q< oe ca fid he soluio of his equaio ( o he ime se [ +, +], wih he iiial codiio x( += ( +. I geeral, by assumig ha - ( is defied o he ierval [ +(-, +(-], =,,,, he followig equaio is cosruced: x q ( = f(, x(, - (-, ' - (-, for +(- +, q< ad cosider a soluio of his equaio wih he iiial codiio: x( +(- = - ( +(-, which is deoed by ( for all =,. Now, cosider he examle o exlai he mehod of ses. Examle (.: Cosider he rearded delay fracioal differeial equaios: x ( x(...( wih iiial codiio x(= (=, for -, ad o fid he soluio i he firs se ierval [, ], we have o solve he followig equaio: x he, we have: x ( (, for...( (, for...( by usig he soluio of fracioal differeial equaios, [Basim, ], alyig d o boh sides of d equaio (, we ge: x( c, ( 7 ( i order o fid he soluio i he secod se ierval, suose ha: ( x( c, ( 7 ( is he iiial codiio. Sice ( is defied o he whole segme [, ], hece by formig a ew equaio: x wih iiial codiio, ( (, for...( ( c, for ( 7 ( oe ca fid he soluio i he ex se ierval [, ], ad solvig equaio (, for x( we have: x ( ( ( ( c(,...(6 ( 7 ( by usig fracioal differeial equaio alyig d d o boh sides of equaio (6, we ge: 8

3 Joral of Kerbala Uiversiy, Vol. No. Scieific.Decembar 7 x( ( ( ( ( ( 7 ( 9 c ( c ( ( c...(7 for all,. herefore x( is he desired secod se soluio. similarly, we roceed o he ex iervals..the Exisece ad Uiqueess Theorem As i ordiary ad fracioal differeial equaios he exisece ad uiqueess delay fracioal differeial equaios could also be esablished for his urose, cosider he delay fracioal differeial equaios: y q (=f(, y(, y(-w( wih iiial codiio, y(=(, -w( ad suose ha R deoed he recagular regio defied by: R={(, y: -a-w +a, y-y b}.. Exisece heorem cosider he iiial value roblem of delay fracioal differeial equaios: y q (=f(, y(, y(-w(...(8 wih iiial codiio, y(=(, -w( f f ad suose ha f, ad are coiuous o a regio R, ad f saisfies Lischiz y( y( w( cosa codiio. The equaio (8 has a soluio o R. oe hyohesis of he exisece i he recagle R. I is follows ha f mus be boded i R. le M> be umber such ha f(, y(, y(-w( M, for every oi i R. if we ow ae = mi{a, M b }, ad defie he recagle R o R ={(, y:-, y-y b} clearly R is a subse of R. If we ow cosider he sequece of fucios: ( f (s, (s, hus, we have he sequece of soluios will coverge. (s w(sds.. The uiqueess heorem The soluio of he delay fracioal differeial equaio: y q (=f(, y(, y(-w( wih iiial codiio y(=(, -w(, uique. Proof: Suose ha ( ad ( are ay wo soluio, he: [f (s, (s, (s w(s f (s, (s, (s w(s]ds ad suose ha K is he Lischiz cosa codiio of he fucio f i a comac subse R coaiig (, (, (-w( ad (, (, (-w(, for each R. he: [f (s, (s, (s w(s f (s, 8 (s, (s w(s]ds

4 Joral of Kerbala Uiversiy, Vol. No. Scieific.Decembar 7 K [f (s, (s, (s (s w(s f (s, (s (s, (s w(s (s (s w(s]ds w(s ds K Su (s (s (s w(s (s w(s asb aig < K <, he we have: [ (s (s (s w(s ad hece: (- ( = herefore ( = (, foe each R. (s w(s ].Numerical Soluio of Delay Fracioal Differeial Equaios Usig Liear Mulise Mehods (LMM's. The rese secio cosiss of a geeralizaio ad modificaio of he geeral LMM o solve DFDE. For his urose, well cosider, for simliciy ad wihou lose of geeraliy he rearded DFDE., which has he form : y (q ( = f(, y(, y(-,...(9 wih iiial codiio: y( = (, -, q <, where is a fixed umber ad y( =y =(... Geeral LMM's i DFDE's: Cosider IVP give be equaio (9, where y( = (, -, ad y( =y =(. Hece he geeral form of LMM for solvig equaio (9 is give by: j y j h jf y y j...( j j where x +j =x + jh, y +j = y +jh ad y +j- =y +jh- ad j, j are cosa. Hece: j y j h jf y j j j exaded his formulae, gives: y + y y + = h[ f, y, - + f +, y +, f( x +, y +, +- ] which is geeral Delay Fracioal Liear Mulise Mehod (DFLMM. As a classificaio o he DFLMM, we say ha he DFLMM is of exlici ye if = ad i is imlici ye if. Also i some cases, imlici mehod may reduce o exlici mehods deedig o he fucio f wheher i is liear or o-liear. The local rucaio error of DFLMM is defied o be, T jy jh h jf jh, y jh, y jh... ( j j or 88

5 Joral of Kerbala Uiversiy, Vol. No. Scieific.Decembar 7 T (q jy jh h jy jh j j afer exadig he fucio y +jh ad i's by usig soluio fracioal differeial equaio alyig d d ad q q y y (q o boh sides, ad collocaio erms i equaio (. y y jh y jh x!! y y y jh (jh...!! jh y (q y (q! jh y (q ad so o. by subsiuig i he local rucaio error, we ge: T jy j h jf y y j j j j j y j h jf y j j! (jh jh x... ax q... = y + y y + - h[ f, y, - + f +, y +, f( x +, y +, +- ] (q (... y h( ( y (q h ( ( y (q h (! (!... 8!!!... (q q! y... ax herefore T =C y (q- + C hy (q +C h y (q+ + +C h y (q+ + +ax q-... ( where C, C,,C,a are cosas, ad a simle calculaio yields he followig formula for he cosas C i 's, i=,,,, i erms of he coefficies j ad j : C... C ( ( C C! (! C! ( (... 8!...! 88!... ( (! hese formula ca be used o derived a cerai DFLMM of give srucure.!...

6 Joral of Kerbala Uiversiy, Vol. No. Scieific.Decembar 7. Illusraive examle Examle (.: Cosider he Delay fracioal differeial equaio: x ( x(, wih iiial codiio: x(=(=, for -, ad suose ha we wa o fid he umerical soluio i firs ime se, wih se legh h=. give by: y + =y +hf, y, y -, by usig he comuer rogram o fid he soluio o he firs ime se, ge he resuls reseed i able (. ad i's comariso wih he exac soluio. Table (.. x i Numerical soluio Exac soluio Absolue Error Refereces:. Al-Dayee, K. K., O Soluio of Delay Boudary Value Problems, M.sc. heses, dearme of mahemaics, Al_ahrai uiversiy, Baghdad, Iraq,.. Basim K., Soluio of Fracioal Differeial Equaios Usig Variaioal Aroach, Alahrai uiversiy, Baghdad, Iraq,.. Driver R. D. E., Ordiary ad Delay Differeial Equaios, Surigier-Verlag Ic., New Yor,977.. El'sgol'z, L.E. Qualiaive Mehod i Mahemaical Aalysis, Tras. Mah. Moo., Vol., Amer. Mah Soc.,96.. Halaay A., Differeial Equaios; Sabiliy, Oscillaios ad Time Lages, Academic ress, New Yor ad Lodo Ic.,