A Mathematical model to Solve Reaction Diffusion Equation using Differential Transformation Method

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1 Inernaional Jornal of Mahemaics Trends and Technology- Volme Isse- A Mahemaical model o Solve Reacion Diffsion Eqaion sing Differenial Transformaion Mehod Rahl Bhadaria # A.K. Singh * D.P Singh # #Deparmen of Mahemaics RBS College Agra India #Deparmaen of Mahemaics FET RBS College Agra India Absrac The presen work is designed for differenial ransformaion mehod (DTM) o solve he linear and nonlinear reacion diffsion eqaions. To illsrae he capabiliy and reliabiliy of he mehods some cases have been defined. The approimae solion of his problem is calclaed in he form of a series wih easily compable erms and also he eac solions can be achieved by he known forms of he series solions. The mehod can easily be applied o many linear and non-linear problems and is capable of redcing he size of compaional work. The resls obained sing DTM are compared wih he resls of variaional ieraion mehod (VIM) and MATLAB solions. Keywords reacion diffsion eqaions differenial ransformaion mehod MATLAB I. INTRODUCTION In recen years here has been mch ineres in of sysems reacion diffsion eqaions which occr in many applicaions for eample mahemaical biology inclding models for mlispaces chemical reacions and predicor prey sysems. The reacion diffsion-eqaion and is varians have been he sbjec of sdy in varios branches of physical chemical and biological sciences. Besides is sdy in one dimension and higher dimensions his all depends on he nare of he phenomenon nder sdy as o wha form of his eqaion (wheher linear or nonlinear version) needs o be sed. While a variey of simplified versions of nonlinear reacion-diffsion eqaions are sdied [7- ] in he lierare a reaciondiffsion eqaion wih cbic nonlineariy has been fond recenly in several ineresing applicaions. Reacion-diffsion eqaion wih nonlocal bondary condiions has been given considerable aenion in recen years and varios mehods have been developed for he reamen of hese eqaions. The nmerical solions of he problem can be fond in many aricles. Mos of he discssions in he crren lierares are developed o he some ype nonlocal bondary condiions problem and mch less is given o he problem wih nonlocal Robin ype bondary condiions. The prpose of his aricle is o give a nmerical reamen o a class of reacion-diffsion eqaions wih nonlocal bondary condiions by differenial ransformaion mehod. The differenial ransformaion is a nmerical mehod for solving differenial eqaions. The concep of differenial ransform was inrodced by Zho (986) who was he firs one o se differenial ransform mehod (DTM) in engineering applicaions. He employed DTM in solion of iniial bondary vale problems in elecric circi analysis. In recen years concep of DTM has broadened o he problems involving parial differenial eqaions and sysems of differenial eqaions [-5]. Ayaz [-5] developed his mehod for PDEs and obained closed form series solions for linear and non-linear iniial vale problems. The differenial ransforms mehod an analyical solion in he form of a polynomial. I is differen from he radiional high order Taylor series mehod which reqires symbolic compaion of he necessary derivaives of he daa fncions. The Taylor series mehod is compaionally aken long ime for large orders. The presen mehod redces he size of compaional domain and applicable o many problems easily. A disincive pracical feare of he differenial ransformaion mehod DTM is haring abiliy o solve linear or non-linear differenial eqaions. Differen applicaions for he differenial ransformaion in he differenial eqaion were shown by Hassan [] where he was sed he differenial ransformaion echniqe which is applied o solve Eigen vale problems and parial differenial eqaions (P.D.E.) is proposed in his sdy. They were firs sing he one-dimensional differenial ransformaion o consrc he Eigen vales and he normalized Eigen fncions for he differenial eqaion of he second and he forh-order and he second sing he wodimensional differenial ransformaion o solve P.D.E. of he firs- and second-order wih consan coefficiens. In boh cases a se of difference eqaions is derived and he resls were compared wih he resls obained by oher analyical mehods. Kmar e al (9) discssed abo he eac and nmerical solions of non-linear reacion diffsion eqaion by sing he Cole Hopf ransformaion. In his work he main focs is o ge a consrcive mehod for obaining eac ISSN: -57 hp:// Page 6

2 Inernaional Jornal of Mahemaics Trends and Technology- Volme Isse- solions of cerain classes of non-linear eqaions arising in mahemaical biology. The mehod is based on he Cole Hopf ransformaion of non-linear parial differenial eqaion. Wih he help of his mehod new eac solions were obained for non-linear reacion-diffsion eqaions of varios forms which are he generalizaions of he Fisher and Brgers eqaions. Finally he governing parial differenial eqaions are hen solved sing MATLAB. Again Kmar e al () worked on he solion of reacion diffsion eqaions by sing homoopy perrbaion mehod. Applicaion of He's homoopy perrbaion mehod for solving he Cachy reacion diffsion problem have also discssed by Yildirim (9) where he presened he solion of Cachy reacion diffsion problem by he homoopy perrbaion mehod. Or work in his paper relies mainly on he mos recenly mehods DTM. The DTM mehod which accraely compe he solions in a series form or in an eac form are of grea ineres o applied sciences. The effeciveness and he seflness of mehod is demonsraed by finding eac solions o he below problems - ha will be invesigaed. However he mehod has is own characerisics and significance ha will be eamined. II. MATHEMATICAL FORMULATION In his paper consider he following one-dimensional ime dependen reacion-diffsion eqaion for his model- = D + r ( ) Ω R () ( ) = g () R () ( ) = f () = f () R () () where = ( ) is he concenraion r = r( ) is he reacion parameer and D > is he diffsion coefficien sbjec o he iniial or bondary condiions The eqaions () and () is called he characerisic Cachy reacion-diffsion eqaion in he domain R R while he eqaions () and () is called he non-characerisic Cachy reacion-diffsion eqaion in he domain R R. If r = hen he eqaion () is called Kolmogorov Perovsky Pisknov eqaions. The DTM inrodces a promising approach for many applicaions in varios domains of science. However DTM has some drawbacks. By sing he DTM we obain a series solion or we can say a rncaed series solion. This series solion does no ehibi he real behaviors of he problem b gives a good approimaion o he re solion in a very small region. While homoopy perrbaion mehod can be sed sccessflly finding he solion of linear and non-linear bondary vale problems and he sysem of differenial eqaions. I may be conclded ha his echniqe is very powerfl and efficien in finding he analyical solions for a large class of inegral and differenial eqaions. This echniqe provides more realisic series solions as compared wih he Adomain decomposiion and variaional ieraion mehod and ohers echniqes. III. BASIC IDEA OF DIFFERENTIAL TRANSFORMATION METHOD The basic definiions of he differenial ransformaion are inrodced as follows: (i) One-dimensional differenial ransformaion: The differenial ransformaion of he k derivaive of a fncion () is defined as follows: U(k) =! () (4) () is he original fncion while U(k) is he ransformed fncion. As discssed by [] and [] he differenial inverse ransformaion of U(k) may defined as follows: () = U(k)( ) In fac from (4) and (5) we may obain: () = k! ( ) d () d (6) Eqaion (6) implies ha he concep of differenial ransformaion is derived from he Taylor series epansion a =. (ii) Two-dimensional differenial ransformaion: Similarly consider a fncion of wo variables ( y) analyic in he domain R and le ( y) = ( y ) in his domain. The fncion ( y) is hen represened by one power series whose cener is locaed a ( y ). Hence he differenial ransformaion of fncion ( y) is having he following form: U(k h) = ()!! ( ) where ( y) is he original fncion and U(k h) is he ransformed fncion. The ransformaion is called T-fncion while he lower case and pper case leers represen he original and ransformed fncions respecively. The differenial inverse ransform of U(k h) is defined as: ( y) = U(k h) ( ) (y y ) (8) and from eqaion (7) and (8) i may be conclded ha ( y) = () k! h! y ( ) (y ( ) y ) (9) When we apply eqaion (8) a ( y ) = ( ) hen (8) can be wrien as: (7) ISSN: -57 hp:// Page 7

3 Inernaional Jornal of Mahemaics Trends and Technology- Volme Isse- ( y) = U(k h) y k! h! ( y) () = () k! h! y y () () The fndamenal mahemaical operaions performed by wodimensional differenial ransform mehod and ha can be readily obained hrogh Table. Table : Some operaions of he wo dimensional differenial ransformaion. Original fncion Transformed fncion ( y) = v( y) ± w( y) U(k h) = V(k h) ± W(k h) ( y) = α v( y) U(k h) = α V(k h) (α =Consan) v( y) ( y) = v( y) ( y) = y ( y) = v( y) y ( y) = v( y)w( y) ( y) = y ( y) = e () ( y) = sin(a + b) ( y) = e U(k h) = (k + )V(k + h) U(k h) = (h + )V(k h + ) (k + r)! (h + s)! U(k h) = V(k + r h + s) k! h! U(k h) = V(r h s) W(k r s) U(k h) = δ(k m h n) k = m and h = n = oherwise U(k h) = λ() k! h! U(k h) = a π δ(k m)sin h h! + b U(k h) = a δ(k m) h! The applicaion of he differenial ransformaion mehod (DTM) will be discssed for solving problems ()-(). According o he (DTM) and he operaions mahemaics of he mehod consider eqaion () afer aking he differenial ransformaion of boh sides in he following form: D(k + )(k + )U(k + h) U(k h) = (h + ) () + P(r h s)u(k r s) and by applying he differenial ransformaion o iniial and bondary condiions eqaion () and () are obained as follows: U(k ) = G(k) U( h) = F (h) and U( h) = F (h) Problem : Consider he following he Kolmogorov- Perovsky-Pisknov (KPP) eqaion: = ( ) Ω R (5) wih iniial and bondary condiions ( ) = e + = g() R (6) ( ) = = f () (7) ( ) = e = f () R (8) According o he differenial ransformaion mehod (DTM) When aking he differenial ransformaion of (4) can be obain U(k h + ) = [(k + )(k + )U(k + h) U(k h)] (9) The relaed iniial condiions (5) shold be also ransformed as follows U(k ) = +!! + 4! + + δ(k h) () which gives if k = U(k ) = if k = () () if k = 4! And he bondary condiion (6) shold be ransformed as follows: if k = h = U( h) = δ(k h) = if oherwise () and U( h) = +!! + 4! 5! + + δ(k h) () which gives if k = U(k ) = if k = (4) ()! if h = 4 For each k h sbsiing eqaions () () and () ino eqaion (9) and by recrsive mehod all oher of U(k h) are eqal o zero when sbsie all vales U(k h) ino () and obain he series for ( ). Then rearrange he solion and obain he following closed form solion: ( ) = U(k h) ( ) = ( ) ISSN: -57 hp:// Page 8

4 Inernaional Jornal of Mahemaics Trends and Technology- Volme Isse- ( ) = e + e (5) From eqaions (4) he approimae solion of he given problem () by sing differenial ransformaion mehod is having he same resls as ha obained by he homoopyperrbaion mehod [] and i clearly appears ha he approimae solion remains closed form o eac solion Figre (a) Figre (b) Figre (a) represens he HPM or DTM solion for he problem () of eqaion (4) from = o = and = o = while Figre (b) represens he by MATLAB solion of problem () from = o = and = o =. The accracy of he DTM for he combined KPP eqaion is conrollable and absole errors are very small wih he presen choice of and. These resls also obain by MATLAB and boh he resls are compare hrogh he figre (a) & (b) and here are no visible differences in he wo solions. Eample: Consider he following he reacion-diffsion eqaion: = ( ) Ω R (6) wih iniial and bondary condiions ( ) = e = g() R (7) ( ) = e = f () ( ) = e = f () R (8) According o he DTM eqaion (5) may be wrien as: (k U(k h + ) = + )(k + )U(k + h) h + (9) δ(r h s ) U(k r s) From he iniial condiion (6) can be wrien as if k = U(k ) = if k =.. And he bondary condiion (7) can calclaed from he following Table. Table. U( h) = U( h) h = U() = U() = U() = U() = U() = U() = U() = U() = U(4) = U(4) = U(5) = U(5) = U(6) = U(6) = U() = U() = Using he vales of able ino (8) and by recrsive mehod we have U() = U() = U(4) = U(5) = U(6) =. U() = U() = U(4) = U(4) = U(4) = U(5) = U(5) = U(45) = Sbsiing all vales U(k h) ino () and obain series for ( ). Then rearrange he solion and ge he following closed form solion: ( y) = U(k h) = e () From eqaions () he approimae solion of he given problem () by sing differenial ransformaion mehod is he same resls as ha obained by he homoopy-perrbaion mehod and i clearly appears ha he approimae solion remains closed form o eac solion () ISSN: -57 hp:// Page 9

5 Inernaional Jornal of Mahemaics Trends and Technology- Volme Isse- Figre (a) if h = 5 U( h) = if h = 4 6. (8)! and U( h) = for all h. (9) Using (6)-(8) ino (5) by recrsive mehod we have he following vales in Table when h we have U(k h) = Figre (b) Figre (a) represens he HPM solion for he problem () of eqaion () from = o = and = o = while he figre (b) represens he MATLAB solion of Problem () from = o = and = o = The accracy of he DTM for he combined KPP eqaion is conrollable and absole errors are very small wih he presen choice of and. These resls also obain by MATLAB and boh he resls are comparing hrogh he figres (a) and (b). There are no visible differences in he wo solions. Eample: Consider he following he reacion-diffsion: = (4 + ) ( ) Ω R wih iniial and bondary condiions ( ) = e = g() R ( ) = e = f () ( ) = = f () R () () (4) (5) According o he DTM Taking he differenial ransformaion of () U(k h + ) = h +. (k + )(k + )U(k + h) + δ(r h s ) U(k r s) 4 δ(r h s) U(k rs) U(k h) From he iniial condiion () of he form we may have: if k = 5 U(k ) = if k = 4 6. (7)! (6) Table U( ) = U(4 ) = U(6 ) = U(8 ) = U( ) = U( 4) = U(4 4) = U(6 4) = U(8 4) = U( 4) = U( 6) = U(4 6) = U(6 6) = U(8 6) = U( 6) = U( 8) = U(4 8) = U(6 8) = U(8 8) = U( ) = U(8 ) = U( 8) = U(4 ) = U(6 ) = U( ) =... Sbsiing all vales U(k h) ino () and obain series for ( ). Then rearrange he solion and ge he following closed form solion: ( y) = U(k h) ( y) = e ( ) (4) From eqaions (9) he approimae solion of he given problem () by sing differenial ransformaion mehod is he same resls as ha obained by he homoopy-perrbaion mehod and i clearly appears ha he approimae solion remains closed form o eac solion Figre (a) Figre (a) represens he DTM solion for he problem () of he eqaions (9) from = o = and = o = ISSN: -57 hp:// Page

6 Inernaional Jornal of Mahemaics Trends and Technology- Volme Isse- while he figre (b) represens he MATLAB solion of he problem () from = o = and = o = Figre (b) The accracy of he DTM for he combined KPP eqaion is conrollable and absole errors are very small wih he presen choice of and. These resls also obain by MATLAB and boh he resls are comparing hrogh he figres (a) and (b). There are no visible differences in he wo solions [] A. Arikogl and I. Ozkol (6); Solion of differenial-difference eqaions by sing differenial ransform mehod Appl. Mah. Comp. 8 pp.5-6. [] Ayaz F. (); On he wo-dimensional differenial ransform mehod Appl. Mah. Comp. 4 pp [4] Ayaz F. (4); Solions of he sysem of differenial eqaions by differenial ransform mehod Appl. Mah. Comp. 47 pp [5] Ayaz F. (4); Applicaions of differenial ransform mehod o differenial-algebraic eqaions Appl. Mah. Comp. 5 pp [6] Baaineh A.S. Noorani M.S.M. and Hashim I. (8); The homoopy analysis mehod for Cachy reacion-diffsion problems Physics Leers A 7 pp [7] Kmar S. and Singh R. (9): Eac and nmerical solions of nonlinear reacion diffsion eqaion by sing he Cole Hopf ransformaion Inernaional Transacions in Mahemaical Sciences & Comper Vol. () pp 4-5. [8] Kmar S. and Singh R.(): Solion of Reacion Diffsion Eqaions by sing Homoopy Perrbaion Mehod Inernaional Jornal of Sabiliy and Flid Mechanics Vol () 5-4. [9] Kmar S. and Singh R. (): A Mahemaical Model o Solve Reacion Diffsion Eqaion Using Homoopy Perrbaion Mehod and Adomain Decomposiion Mehod VSRD-TNTJ Vol. (4) 86-. [] Lesnic D. (7); The decomposiion mehod for Cachy reaciondiffsion problems Applied Mahemaics Leers pp [] Mrray J.D. (989); Mahemaical Biology Springer-Verlag Berlin. [] Yildirim A. (9); Applicaion of He's homoopy perrbaion mehod for solving he Cachy reacion diffsion problem Compers and Mahemaics wih Applicaions 57 pp [] Zho J. K. (986); Differenial ransformaion and is applicaion for elecrical circis Harjng Universiy Press Whahn China. IV. CONCLUSIONS Reacion diffsion eqaions have special imporance in engineering and sciences and consie a good model for many sysems in varios fields. In his work we sdied differenial ransformaion mehod for solving reacion diffsion eqaions. This mehod is applied o solve hree bondary vale problems. All problems we obained closed form eac series solions. The differenial ransformaion mehod is so powerfl and efficien ha hey boh give approimaions of higher accracy and closed form solions if eising. I is observed ha he mehod is an effecive and reliable ool for he solion of sch problems. Frher he resls of or eample ell s he sccessflly mehod can be alernaive way for he solion of he linear and non-linear higher-order iniial and bondary vale problems. In fac DTM is very efficien mehods o find he nmerical and analyic solions of differenial-difference eqaions delay differenial eqaions as well as inegral eqaions. The resls show ha DTM is a powerfl mahemaical ool for solving linear and nonlinear parial differenial eqaions and herefore can be widely applied in engineering problems. In work MATLAB is sed o calclae he series obained from he DTM. REFERENCES [] I. H. Abdel-Halim Hassan Differen applicaions for he differenial ransformaion in he differenial eqaions Appl. Mah. Comp. 9 pp 8- ();. ISSN: -57 hp:// Page