DETERMINATION OF PARTICULAR SOLUTIONS OF NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS BY DISCRETE DECONVOLUTION

Transcription

1 U.P.B. ci. Bull. eries A Vol. 69 No. 7 IN 3-77 DETERMINATION OF PARTIULAR OLUTION OF NONHOMOGENEOU LINEAR DIFFERENTIAL EQUATION BY DIRETE DEONVOLUTION M. I. ÎRNU e preziă o ouă meoă e eermiare a soluţiilor pariculare ale ecuaţiilor ifereţiale liiare eomogee cu coeficieţi cosaţi şi pare reapă uzuală. Nouaea cosă î uilizarea ecovoluţiei iscree la calculul coeficieţilor polioamelor care apar î formulele soluţiilor. Meoa poae fi uşor implemeaă pe calculaor. A ew meho o eermie he paricular soluios for ohomogeeous liear iffereial euaios wih cosa coefficies a usual righ pars is presee. The ovely cosiss i he use of he iscree ecovoluio for he compuaio of he coefficies of he polyomials ha appear i he soluios formulae. The meho ca easy be implemee o compuer. Keywors: ohomogeeous liear iffereial euaios wih cosa coefficies paricular soluios iscree covoluio a ecovoluio. Mahemaics ubec lassificio: 34A344A35. Iroucio I a earlier paper [4] he covoluio bewee wo seueces was use o eermiae he cosas ha appear i he soluio of he auchy problem for a homogeeous liear iffereial euaios wih cosa coefficies a paricularly he elemeary soluio of he euaio was euce. I he case of a ohomogeeous euaio of his ype a paricular soluio was obaie by he iegral of covoluio bewee he righ par of he euaio a he elemeary soluio of he associae homogeeous euaio. Usig he ecovoluio of he seueces of he same fiie legh ( see [3] ) i he prese paper we give a irec meho o obai paricular soluios for ohomogeeous liear iffereial euaios wih cosa coefficies a saar righ pars wihou he use of oher ype of calculi as woul be he eucio by he ieificaio of he coefficies of liear algebraic sysems a heir solvig by recurrece he eermiaio of he roos of he characerisic euaio or he compuaio of iegrals. Prof. Deparme of Mahemaics III Uiversiy Poliehica of Buchares ROMANIA

2 4 M. îru The ecovoluio meho presee here resuls from he usual meho of he ieificaio of he coefficies (ueermiae coefficies meho). The riagular sysems of liear algebraic euaios ha are obaie for he eermiaio of he cosas ha appear i he paricular soluios of he ohomogeeous euaios are solve by iscree ecovoluio. I accor wih he ype of he righ par of he euaio a is resoace a umerical ecovoluio formula o obai a paricular soluio is give for every case. The ecovoluio meho is very suiable o be implemee o compuer. Oher applicaios of he iscree covoluio a ecovoluio o soluio of boh iiial a boue value problems for liear ifferece a iffereial euaios a for compuaio he roos of polyomials are give i he papers [] a [].. Discree covoluio a ecovoluio Le be a ( a a ;...; ) a b ( b b ;...; ) ; a ; b wo fiie seueces of real or comple umbers havig he same legh. We call ( see [3] ) heir iscree covoluio ( auchy prouc ) he fiie seuece c ( c ; c ;...; c ) a b () havig he compoes give by he formulae c ab c ab + ab... c k akb k. () The covoluio is associaive commuaive isribuive relaig o aiio a i has δ ( ;;...;) as ui. The eermiig of he seuece b c / a ha saisfies he relaio () for wo give seueces a a c is calle he iscree ecovoluio of c by a (see also [3] ). I ca be performe if a he compoes of he fiie seuece b beig give by he relaios ( ) b c a b c ab a b ( c a b ) a k k k (3) herefore by he algorihm c c c c a a a a b a b b b b / c a b c a b c c a b a b a b a b

3 Deermiaio of paricular soluios of ohomogeeous liear iffereial euaios 5 / c a c a b b If a a ; a ; ; a ) is a fiie seuece wih a he seuece ( a δ / a is calle he iverse of he seuece a wih respec o he covoluio a we have c / a c a. 3. Liear iffereial euaios wih cosa coefficies a polyomial righ par wihou resoace / We cosier he problem of fiig a paricular soluio of he liear iffereial euaio of orer ( k ( () ) P ( D) ( ) a ) () f () L (4) k o k havig a real variable a comple cosa coefficies a k wih a o. The mai case o which oher siuaios are also reuce is he oe whe he righ par f () of he euaio is a polyomial wih comple coefficies of a arbirary egree f () Q () b k. (5) If a i which case w is o a roo of he characerisic euaio P ( w) k a w k k (6) we say ha he euaio (4) wih he righ par (5) has o resoace. I his case we shall eermie a paricular soluio of he euaio of he form () () c (7) he coefficies of he polyomial beig eermie by he coiio ha he fucio () give by (7) saisfies he euaio (4).The

4 6 M. îru ( k ) ( k () ) () c ( k ) ( k () ) k k > k k ( i + k) ck i ( k) i i for k wih he chage of he ie i k. Imposig he coveio a k k < (8) if () is he polyomial give by he relaio (7) he euaio (4) becomes a k ( k ) ( () k a ) () k ak ck k k i i ( i + k) i k k i akck i i k i i b i i ( i + k) Ieifyig he coefficies a performig he chage of he ie of summaio i he followig relaios resul ( ( k) ) b ( ) a kc k (9) k which represe a riagular liear algebraic sysem havig as ukows he coefficies c. This sysem is compaible sice a. I he seuel we shall resolve he algebraic sysem (9) by iscree ecovoluio of he umerical seueces of he same fiie legh a i he followig secios we shall prese he chages ha have o mae o he meho i orer o apply i boh i he case of resoace a for more geeral righ pars f (). Reurig o he eermiaio of a soluio of he form (7) for he euaio (4) wih he righ par (5) i he case a a if he coiio (8) is saisfie we cosier he seuece of he coefficies ( a :... ) of he euaio (4) a ( b :... ) respecively ( c :... ) of he polyomial (5) respecively (7) a he ormalizaios ( b ( ) b :... ) respecively ( c ( ) c :... ). Wih hese oaios he relaio (9) akes he form i i i

5 Deermiaio of paricular soluios of ohomogeeous liear iffereial euaios 7 ( a )* ( c ) ( b ). Thus he coefficies of he soluio (7) ca be eermie by he followig ecovoluio formula c : b : / a : () ( ) ( ) ( ) Eample. Fi a paricular soluio of he iffereial euaio f ( ) () ( ) ( ) ( ) ( ) ( ) 3 where a) () f ; b) f () oluio. a) We have 4 3 he euaio havig o resoace 3 hece we eermie a paricular soluio of he form c + c + c3 + c4. Also ( a : ) ( a4 : 3) ; a ; 3;5; 7 a ( : 3 ) ( 4; 4;;5) ( a 4 ; a3 ; a ) ( ) : 3 3 4; 4;; 5 ( ) ( ) ( 4; 8;;5) b be obaie by he followig ecovoluio algorihm: b hece. The seuece c ca Oe obais c ( ;4; 3;5). Hece c ; ; ; 3 ;7; 3;5 herefore a paricular soluio of he euaio is ( ) b) We eermie a paricular soluio of he form c + c + c + c + c + c +. We have () c6 ( a :... ) ( a : 6) ; 3;5; 7;3;; accorig o he coveio (8) b ( ; 4;; 64; 8;6; ) b ( 6 ; 54; 4; 3 64; 8;6; ) ( )

6 8 M. îru ( 44; 68; 88; 384;56;6; ) By ecovoluio we obai c b ( a ) ( 7; 4;;; 4;;) hece c 7; 4;; ; 4;; ( ;;;;;;) 6 5. oseuely a paricular soluio is 6 5 () + +. Remark. A varia of he above presee meho useful especially whe we have o solve he same euaio for several righ pars cosiss i he eermiaio by ecovoluio of he iverse of he seuece ( a :... ) a i fiig he seuece ( c ) by he covoluio bewee his iverse a he seuece ( b ). For eample i he case of he above cosiere euaio we have ( a ) δ ( a ) ( ;3 4 ; 8 ; 5 6 ;55 3 ;5 6 ; 3 8) Performig he covoluio bewee his iverse a he seueces ( ) b from pois a) a b) we obai agai he above calculae paricular soluios of he euaio. 4. Euaios wih polyomial righ pars a resoace If a a am+ am he w is a roo of mulipliciy m of he characerisic euaio (6) a we say ha he iffereial euaio (4) has a resoace of orer m. I his case he euaio (4) has he form k m If we cosier he ew ukow fucio y ( k ) () f () a k () ( m ) () he euaio akes he form m a mk y k ( k ) () f (). (3)

7 Deermiaio of paricular soluios of ohomogeeous liear iffereial euaios 9 If f () is he polyomial give by (5) i accorace wih hose above meioe he euaio (3) has he paricular soluio () c y (4) he coefficies beig eermie wih he help of he relaio ( c ) ( b )/( a : ) m. (5) oseuely a paricular soluio of euaio () will be he fucio () y(). m m c where m+ c ( m + ) m () () (7) he coefficies of he polyomial beig give by he formula c c. (8) ( m + ) ( m + ) Eample. Fi a paricular soluio of he iffereial euaio f wih he righ pars a) f 4 ; ( ) () ( ) () () 4 3 b) f () oluio. a) We have 4 m 3 (riple resoace) a we 3 eermie a paricular soluio of he form ( c + c ) a : ( a : ) ( a ) ( ; ) b b ( 4; ) ( ) m Usig he ecovoluio algorihm. We have a. ;

8 M. îru 3 c c c. From (8) c hece ( 3 + ) 3 c c a oe obais he paricular oe obais ( ; ) ; m soluio () () ( 3) 3 m b) We have 3 m 3 4 a we eermie a paricular soluio of he form () ( c + c + c + c3 + c4 ). We have ( a m : ) ( a : 3 4) ( ;;;;) i coformiy wih he coveio (8) b ( 7;6; ; ;) b ( 4 7;3 6; ; ;) ( 68;36; ; ;). From he ecovoluio algorihm oe obais ( 84; 4;;;) c From (8) we have hece c c 4 ( 3 + )

9 Deermiaio of paricular soluios of ohomogeeous liear iffereial euaios 4 c soluio () ( + ) c 3 4 a he paricular is obaie Euaios wih ep-polyomial righ par The case i which he righ par of he euaio (4) has he form f e Q (9) () ( ) wih z a comple umber a Q he polyomial give by (5) is reuce o he previous case by chage of he ukow fucio () e y. () Usig he Leibiz rule for he iffereiaio of a prouc of fucios he euaio (4) becomes L ( k ( () ) a ) k () ak k ( k ) k k k z k e y ( ) () () e Q. () k Because z >k a z euaio () urs as follows L y() ak z z k k ( k ) z k ( ) ( ) ( z y ) () k z k... k he P ( ) ( ) ( ) ( z y () A y ) () Q (). () The euaio () is of he ype cosiere i he pars a 3 a i has he characerisic euaio of he form P ( w) A w (3) wih ( A ) P ( z).... (4)

10 M. îru The umber z is a roo of mulipliciy m of he characerisic euaio (6) ( m if P ( ) ( ) ) z P z P ( z) a ( + P m ) ( z). I his case he umber w is a roo of mulipliciy m of he characerisic euaio (3). We cosier also m if P ( z) hece if w is o a roo of he characerisic euaio (3). I accorace wih hose above presee a pars a 3 a soluio of he m euaio () is give by he formula y hece he iffereial euaio (4) wih he righ par (9) has he paricular soluio m e (5) () ( ) he coefficies of he polyomial beig eermie by he relaio () if m respecively by (5) a (8) if m wih he observaio ha i he prese case he seuece ( a m :... ) of he coefficies of euaio (4) mus be replace by he seuece ( A m :... ) of he coefficies of he euaio (). Eample 3. Fi a paricular soluio of he iffereial euaio L( () ) ( ) e L f wih oluio. We cosier he euaio ( ( )) ( ) () 4 f e f respecively ( ) ( ). I he case of he firs euaio 3 m because he umber w is o a roo of he 3 P 3 w w 4w + 4 hece he firs iffereial euaio is o i resoace a we eermie is paricular soluio of he form () c + c + c We have ( a3 : ) ( 4; 4; ) b ( ; 4; ) b ( ; 4; ) ( 4; 4; ). From he ecovoluio algorihm characerisic euaio ( w) i resuls ha c ( ;;) c ;; ;; hece a soluio for he firs euaio is (). For he seco euaio we have 3 m

11 Deermiaio of paricular soluios of ohomogeeous liear iffereial euaios 3 because he umber w z is a simple roo of he characerisic euaio P w he iffereial euaio beig ow wih simple resoace a we ( ) 3 c + c + c e. I eermie is paricular soluio of he form ( ) ( ) accorace o (4) we have A... A ( ) ( : ) ( A ; A ; A ) ( 4;5;) m : From he ecovoluio algorihm c ; ;. I accorace o (8) we have i resuls ha ( c c c ) ( ;;) ( + ) c hece c c c Therefore a soluio for he seco euaio is () e a paricular 6 soluio for he seco iffereial euaio beig 3 () () + () + e Euaios wih rig-ep-polyomial righ pars If he iffereial euaio (4) has real coefficies a is righ par is of he form f [ ] α () e Q cos ( β ) + R si( β ) (6) α are real umbers a Q ( ) ( ) were β R r polyomials of egree respecively r havig real coefficies he he euaio ca be reuce o he above case a a paricular soluio ca be eermie wih he help of he iscree ecovoluio. Firsly we observe ha by compleio wih ull erms he polyomials Q a P ca be cosiere as havig he same egree ma( r). We will fi some paricular soluios of he euaio (4) for ay righ par r

12 4 M. îru f f α α () e Q cos( β ) f e R cos( β ) α α () e Q () si( β ) f () e R () si( β ). (7) I his case he fucios () () + i () () + i are paricular soluios of he euaio (4) for he righ pars () f () + i f () e Q f f + i f e R f (8) respecively where z α + iβ. The iffereial euaio (4) wih he righ ha pars of he form (8) is of he ype cosiere i he secio 4. I accorace wih hose presee here i resuls ha he iffereial euaio (4) wih he righ pars give by he relaio (8) has a paricular soluio of he form m m () e e T (9) respecively where m is he mulipliciy of he umber z as a roo of he characerisic euaio (6) paricularly m if z is o a roo of his euaio. Here () T are polyomials of egree heir coefficies beig eermie by ecovoluio wih he umerical seuece ( A m :... ) as i was presee i he above secio. ice f () f () + f Re f + Im f a paricular soluio of he euaio (4) wih he righ par (6) is he fucio L m () () + () + Im Re e m ( ) + Im( e T ()) Re m α e ( + { Re[( cos( β ) + i si( β )) Re + i Im ( ))] [( cos( β ) + i si( β ) )( ReT + i ImT ( ))]} + Im [( Re () + ImT ) cos( β ) + ( ReT Im ) si( β )] m α e. (3) Eample 4. Fi a paricular soluio of he iffereial euaio e + 7 cos + 3 si () () () () [( ) ( ) ( ) ( )] oluio. We cosier he euaios L f e ( + 7 ) a

13 Deermiaio of paricular soluios of ohomogeeous liear iffereial euaios 5 L() f () ( 3 e ) for z + i ha is o a roo of he characerisic euaio P ( w) w 4w + 3 hece he wo cosiere euaios have o resoace. By he chage of he ukow fucio e y a by he oaio Ly() A y () + A y + A y y + ( + 4i) y + ( 6 4i) y( ) sice A P ( z) A P ( z) z A i ( z) z 4z i P we obai he iffereial euaios L ( y() ) Q + 7 a ( y ) R 3 L. The fiie seueces of he coefficie of polyomials from he righ par of hese euaios beig b ( ;7;) b ( 4;7; ) b ( 3;; ) b ( 6;; ) a he fiie seuece of he coefficies of he lef par of he euaios beig ( A : ) ( A ; A ; A ) ( 6 4 i; + 4 i;) oe obais by ecovoluio b ( c ; c ; ) ( ) c ( 3 + i) i i ; ; A b ( c ; c ; ) ( ) c 3( 3 + i) 3( 9 + i) 8 79 i ; ; A hece he coefficies 3 + i c c i c c i c c i c c i c c 8 79 i c c are obaie. I resuls ha he wo euaios have respecively he paricular soluios () e () e T where e e [ cos + i si ] 3 + i i i () i i 8 79i T () + +. Therefore Re () e + + cos + si

14 6 M. îru Im () e + cos si hece he iiial iffereial euaio has he paricular soluio () Re () + Im () e [ cos + si ]. R E F E R E N E. M.I.îru olvig ifferece a iffereial euaios by iscree ecovoluio U. P. B. ci Bull eries A Vol. 69 No. pp M.I.îru Approimae compuaio by iscree ecovoluio of he polyomials roos (o appear) 3. M.A.ueo Iroucio a l aalyse impulsioelle. Pricipe e applicaio Duo Paris G.Hall A.E.Taylor Algorihm for eermiaio of cosas i soluios of he liear iffereial euaios Amer. Mah. Mohly 55 (948) 6-.