VIM for Determining Unknown Source Parameter in Parabolic Equations

Transcription

1 ISSN , Eglad, UK Joural of Iformaio ad Compuig Sciece Vol. 11, No., 16, pp VIM for Deermiig Uko Source Parameer i Parabolic Equaios V. Eskadari *ad M. Hedavad Educaio ad Traiig, Dourod, Ira, veskadari.se@gmail.com (Received Augus 6, 15, acceped December 11, 15) Absrac. I his paper, a applicaio of he variaioal ieraio mehod (VIM) is preseed. This echique provides a sequece of fucio hich coverges o he eac soluio of he problem. The mai propery of he mehod is i is fleibiliy ad abiliy o solve oliear equaio accuraely ad coveiely. For solvig he discussed iverse problem, a firs e rasform i io a oliear direc problem he use he proposed mehod. Numerical eamples are eamied o sho he efficiecy of he echique. Keyords: VIM; iverse parabolic problem; uko source parameer; addiioal codiio. 1. Iroducio I his paper, VIM is preseed as a aleraive mehod for simulaeously fidig he ime-depede source parameer ad he emperaure disribuio i oe-dimesioal hea equaio. Cosider he parabolic equaio: u u a( ) u f (, );, T, (1) ih uko coefficie. Impose he iiial ad boudary codiio: u(,) ( );, () u(,) g (); T, (3) a () a () ad he addiioal codiio: u (,) E (); T, (4) here T is fial ime ad, f, g ad E are ko fucios. If u is a emperaure he (1)-(4) ca be regarded as a corol problem fidig he corol such ha he ieral cosrai is saisfied. If is ko he direc iiial-boudary value problem (1)-(4) has a uique smooh soluio u (,) [4]. For he eisece ad uiqueess of soluios of hese iverse problems ad also more applicaios, he reader ca refer o [3, 4, 6, 1, 13, 17]. The VIM is a poerful ool o searchig for approimae soluios of oliear equaio ihou requireme of liearizaio or perurbaio. This mehod, hich as firs proposed by He [7, 8] i 1998, has bee proved by may auhors o be a poerful mahemaical ool for various kids of oliear problems [1,, 15, 19]. The ieresed reader ca see [9, 1, 14] for some oher applicaios of he mehod. The res of his paper is orgaized as follos: I Secio, he variaioal ieraio mehod is revieed. I Secio 3, applicaio of he VIM is preseed o solve he discussed iverse problem. I Secio 4, several umerical eamples are preseed o cofirm he accuracy ad efficiecy of he e mehod ad fially a coclusio is preseed i Secio 5. a (). Basic idea of he variaioal ieraio mehod To illusrae is basic coceps of VIM, e cosider he folloig geeral oliear differeial equaio: Lu( ) Nu( ) f ( ), (5) here L ad N are liear ad oliear operaors, respecively ad f is source or sik erm. Accordig o VIM [1,, 7-1, 14, 15, 19], e ca rie do folloig correcio fucioal: u 1( ) u ( ) (, ){ Lu ( ) Nu ( ) f ( )} d;, (6) Published by World Academic Press, World Academic Uio

2 94 V. Eskadari e al.: VIM for Deermiig Uko Source Parameer i Parabolic Equaios here is geeral Lagrage muliplier [11], hich ca be ideified opimally via he variaioal heory [7, 8]. The subscrip deoes he h order approimaio ad is cosidered as resriced variaio [7, 8] hich meas. I is required firs o deermie he Lagrage muliplier. Employig he resriced variaio i correcio u fucioal ad usig iegraio by par makes i easy o compue he Lagrage muliplier, see for isace [8]. For liear problems, is eac soluio ca be obai by oly oe ieraio sep due o he fac ha, o oliear eis so he Lagrage muliplier ca be eacly ideified., Assumig u () is he soluio of Lu. Havig deermied, he several approimaios ca be deermied. We ill rerie equaio (6) i he operaor form as follos: u () 1 ; here he operaor A akes he folloig form: u ( ) [ ], 1 A u X Theorem. Le (,. ) A[ u ( )] u ( ) (, ){ Lu ( ) Nu ( ) f ( )} d. be a Baach space ad A : X X is a oliear mappig ad suppose ha: A[ u ] A[ u ] u u, u, u X, for some cosa. The, A has a uique fied poi. Furhermore, he sequece (6) usig VIM ih a arbirary choice of, coverges o he fied poi of A ad u X m 1 jm1 j u u u u. Proof: See [16]. Cosequely, he eac soluio may be obaied by usig he Baach's fied poi Theorem [16]: u lim u. Accordig o he above heorem, a sufficie codiio for he covergece of he VIM is sricly coracio of A. Furhermore, sequece (6) coverges o he fied poi of A, hich is also he soluio of he equaio (5). Also, he rae of covergece depeds o. 3. Applicaio I his secio, he VIM is used for solvig he problem (1)-(4). I order o solve his problem by usig VIM, e require rasformig he problem ih oly oe uko fucio. This rasformaio is proposed by Cao, Li ad Xu [5]. Accordig o his procedure he erm i (1) is elimiaed by iroducig some rasformaio ad sysem (1)-(4) is rie i he caoical. This procedure is as follos: he erm i (1) elimiaed by iroducig he folloig rasformaio: a () r( ) ep{ a( s ) ds}, a () (7) (, ) u(, ) r( ). (8) Thus, e have: (, ) r( ) u (, ), a( ). r( ) r( ) (9) We reduce he origial iverse problem (1)-(4) o he folloig auiliary direc problem: r( ) f (, );, T, (1) (,) ( );, (11) (, ) g ( )r(); T, (1) Subjec o: JIC for coribuio: edior@jic.org.uk

3 Joural of Iformaio ad Compuig Sciece Vol. 11(16) No., p (, ) r( ) ; T. (13) E () This sysem ca be solved by VIM. No, e ca rie folloig correcio fucioal: (, ) 1(, ) (, ) (, ){ (, ) (, ) (, )} ;. f d E () Makig he above correcio fucioal saioary, oe ha (, ), e (, ) have: 1(, ) (, ) (, ){ (, ) (, ) (, )}. f d E () Therefore: 1(, ) (, ) (, ){ (, )}. d Thus, is saioary codiio ca be obaied as follo: (, ), 1 (, ). Therefore (, ) 1. Noe ha (,). No, he folloig ieraio formula ca be obaied as: (, ) 1(, ) (, ) { (, ) (, ) (, )} ;. f d (14) E () Accordig o Adomia's decomposiio mehod i -direcio hich is equivale o he VIM i - direcio [18], e choose is iiial approimae soluio as (, ) (,). Havig lim deermied [16], he he uko (u,a) ca be calculaed by usig he equaio (9). If he eac soluio of is o obaiable, i as foud ha a fe umber of approimaios ca be used for umerical purposes. 4. Numerical resuls I his secio e repor some resuls of our umerical calculaios usig he umerical procedures described i he previous secio. Eample 1: We cosider he folloig iverse problem: u u a( ) u ep( );, 1, u(,) ep();, u(, ) ep(); 1, u (, ) ep(); 1. The rue soluio is u(,) ep( ) hile a() 1. Le (, ) (,) ep(). By usig he equaio (14), e obai: Thus: 1(,) ep(), (,) (,). 1 lim ep(). Therefore, e obai a series hich is coverge o he eac soluio of he problem (1)-(13). Also from (13), e ca obai: r() ep( ). Thus, usig (9) e obai: u(, ) ep( ), JIC for subscripio: publishig@wau.org.uk

4 96 V. Eskadari e al.: VIM for Deermiig Uko Source Parameer i Parabolic Equaios a() 1, hich is equal o he eac soluio of his eample. From his eample, i ca be see ha he eac soluio is obai by usig oe ieraio sep oly. Eample : We cosider he folloig iverse problem: u u a( ) u;, 1, u(,) cos() si();, u(, ) ep( ); 1, u (, ) ep( ); 1. The rue soluio is u(, ) ep( )(cos() si()) hile. Le (, ) (,) cos() si(). By usig he equaio (14), e obai: 1(, ) (cos() si()) (cos() si()), 1 (, ) (cos() si()) (cos() si()) (cos() si()), (, ) (cos() si()) (cos() si()) (cos() si()) (cos() si()), 6 a() 1 1! 1 1 We ko ha (, ) (1 ( 1) ) is he! usig he fac ha: lim, (, ) (cos() si()) (cos() si()) (cos() si()) ( 1) (cos() si()). order Taylor series of ep( ). No ha leads o he eac soluio: (, ) ep( )(cos() si()). Therefore, e obai a series hich is coverge o he eac soluio of he problem (1)-(13). Also from (13), e ca obai: r() ep( ). Thus, usig (9) e obai: u(, ) ep( )(cos() si()), a(). This is equal o he eac soluio of his eample. Eample 3: Solve he folloig iverse problem: u u a( ) u ( )ep()cos( );, 1, u(,) cos( );, u(, ) ep(); 1, u (, ) ep(); 1. The rue soluio is u(,) ep()( cos( )) hile a() 1. JIC for coribuio: edior@jic.org.uk

5 Joural of Iformaio ad Compuig Sciece Vol. 11(16) No., p We ca selec (, ) (,) cos( ) ; by usig he give iiial value. Accordig o (14), oe ca obai he successive approimaios (, ) of (,) as follo: 1(, ) ( cos( )) ( cos( )), 4 (, ) ( cos( )) ( cos( )) ( cos( )), 4 6 3(, ) ( cos( )) ( cos( )) ( cos( )) ( cos( )), 6! 4 We ko ha (1 ) is he h order Taylor series of! No usig he fac ha: lim, 4 (, ) ( cos( )) ( cos( )) ( cos( )) ( cos( )). ep( ) ha leads o he eac soluio: (, ) ep( )( cos( )). Therefore, e obai a series hich is coverge o he eac soluio of he problem (1)-(13). Also from (13), e ca obai: r () ep( ). Thus, usig (9) e obai: u(, ) ep()( cos( )), a() 1 hich is equal o he eac soluio of his eample. Eample 4: We solve he problem (1)-(4) as: u u a( ) u 1si()( );, 1, u(,) ;, u(, ) ; 1, u (, ) 1; 1. for hich he eac soluio is u(,) ad a() si(). We ca selec (, ) ; by usig he give iiial value. Accordig o (14), oe ca obai he successive approimaios (, ) of (,) as follo: 1(, ) cos()( ) si(), (, ) cos ()() si() cos(). 4 4 Ad he res of he compoes of ieraio formula (14) are obaied usig he Maple 13Package. No form (13), e ca obai he successive approimaios r () of r () as: (, ) r (). E () Fially, usig (9), e ca obai he successive approimaios u (,) of u (, ) ad a () of a () as folloig: (, ) r () u(, ), a(). r () r (). JIC for subscripio: publishig@wau.org.uk

6 98 V. Eskadari e al.: VIM for Deermiig Uko Source Parameer i Parabolic Equaios fucios The obaied umerical resuls are summarized i Tables 1 ad. I addiio, he graphs of he error u u 8 ad a a 8 are ploed i Figure 1. u u Table1. Absolue errors of u u 4 u a u u 6.5 for Eample 4. u u 8 u u E E-7.66E E E E E-7.66E E E E E-7.66E E E E E-7.66E E E E E-7.66E E E E E-7.66E E E E E-7.66E E E-18 a a Table : Absolue errors of a a 4 a a 6 a for Eample 4. a a 8 a a E E E-1 1.3E E E E E E E E-.67E-5.651E E E E- 3.45E-4 1.5E E E-1 a) u(,) u8(,) o he [1,1] ad [,1]. b) a() a8 () o he [,1]. Figure 1. Graph of absolue error by usig VIM by for Eample Coclusio I he prese ork, e have demosraed he applicabiliy of he VIM for solvig a class of parabolic iverse problem. The illusraive eamples sho he efficiecy of he mehod. This mehod provides he soluios of he problems i closed form, Moreover, by usig oly oe ieraio sep; e may ge he eac soluio. I ca be cocluded ha he VIM is a very poerful ad relaively easy ool for solvig iverse hea problem. 6. Refereces [1] Abdou, M. A., Solima, A. A., Variaioal ieraio mehod for solvig Burger's ad coupled Burger's equaios, J. Compu. Appl. Mah., 181 () (5), [] Akmaz, H. K., Variaioal ieraio mehod for elasodyamic Gree's fucio, No-liear Aalysis, 71 (9), [3] Bara, E. C., Numerical procedures for deermiig of a uko parameer i parabolic equaio, Appl. Mah. JIC for coribuio: edior@jic.org.uk

7 Joural of Iformaio ad Compuig Sciece Vol. 11(16) No., p Compu., 16 (5), [4] Cao, J. R., The Oe-Dimesioal Hea Equaio, Addiso Wesley, Readig, MA, [5] Cao, J. R., Li, Y., Xu, S., Numerical procedures for deermiaio of a uko coefficie i semi-liear parabolic differeial equaio, Iverse Problems, 1 (1994), [6] Colo, D., Eig, R., Rudell, W., Iverse Problems i Parial Differeial Equaio, SIAM, Philadelphia, PA, 199. [7] He, J. H., Approimae aalyical soluio for seepage flo ih fracioal derivaives i porous media, Compu. Mehods Appl. Mech. Eg., 167 (1998), [8] He, J. H., Approimae soluio of o-liear differeial equaios ih covoluio produc o-lieariies, Compu. Mehods Appl. Mech. Eg., 167 (1998), [9] He, J. H., Wu, G. C., Ausi, F., The variaioal ieraio mehod hich should be folloed, No-liear Sci. Le., 1 (1), 1-3. [1] He, J. H., Wu, X. H., Variaioal ieraio mehod: Ne developme ad applicaios, Compu. Mah. Appl., 54 (7), [11] Iokui, M., Sekie, H., Mura, T., Geeral use of he Lagrage muliplier i oliear mahemaical physics, Variaioal Mehod i he Mechaics of Solids, Pergamo Press, NeYork, 1978, pp [1] MacBai, J. A., Bedar, J. B., Eisece ad uiqueess properies for oe-dimesioal mageoellurics iversio problem, J. Mah. Phys., 7 (1986), [13] Shidfar, A., Tavakoli, K., A iverse hea coducio problem, Souheas Asia Bullei of Mahemaics, 6 (3) (3), [14] Shou, D. H., He, J. H., Beyod Adomia mehod: The variaioal ieraio mehod for solvig hea-like ad avelike equaios ih variable coefficies,phys. Le., 37 (8), [15] Solima, A. A., A umerical simulaio ad eplici soluios of KdV-Burgers ad La's seveh-order KdV equaios, Chaos Solios Fracals, 9 () (6), [16] Taari, M., Dehgha, M., O he covergece of He's variaioal ieraio mehod, J. Compu. Appl. Mah., 7 (7), [17] Waga, W., Haa, B., Yamamoob, M., Iverse hea problem of deermiig ime-depede source parameer i reproducig kerel space,oliear Aalysis: real orld applicaios, 14 (13), [18] Wazaz, A. M., A compariso beee he variaioal ieraio mehod ad Adomia decomposiio mehod, Joural of Compuaioal ad Applied Mahemaics,7 (1) (7), [19] Xu, L., The variaioal ieraio mehod for fourh order boudary value problems, Chaos Solios ad Fracals, 39 (9), JIC for subscripio: publishig@wau.org.uk

8 1 V. Eskadari e al.: VIM for Deermiig Uko Source Parameer i Parabolic Equaios JIC for coribuio: edior@jic.org.uk