Research Article Invariant Solutions for Nonhomogeneous Discrete Diffusion Equation

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1 Discree Dynamics in Naure and Sociey Volume 2013, Aricle ID , 7 pages hp://d.doi.org/ /2013/ Research Aricle Invarian Soluions for Nonhomogeneous Discree Diffusion Equaion M. N. Qureshi, 1 A. Q. Khan, 1 M. Ayub, 2 and Q. Din 1 1 Deparmen of Mahemaics, Universiy of Azad Jammu & Kashmir, Muzaffarabad 13100, Pakisan 2 Deparmen of Mahemaics, COMSATS Insiue, Abboabad 22010, Pakisan Correspondence should be addressed o Q. Din; qamar.sms@gmail.com Received 29 April 2013; Acceped 18 Augus 2013 Academic Edior: R. Sahadevan Copyrigh 2013 M. N. Qureshi e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. One-dimensional opimal sysems for nonhomogeneous discree hea equaion wih differen source erms are calculaed. By uilizing hese opimal sysems invarian soluions are found. Also generaing soluions are calculaed, using he elemens of he symmery algebra. 1. Inroducion Mahemaics provides models, for real life phenomena, o precisely undersand he underlying laws governing hem. Dynamic changes in a physical phenomenon are usually modeled by differenial equaions. This means ha i is supposed ha he changes are coninuous and hese are aking place in a coninuous domain. Bu here are many real life siuaions where eiher he domain or he phenomenon iself or he boh are no coninuous. These siuaions occur bu are no limied o he fields of biology, physics (classical and quanum), geomery, mahemaical design, finance and so forh. Discree siuaions are beer modeled by difference equaions, in conras wih he differenial equaions. Therefore, difference equaions make heir appearance in almos every branch of mahemaics and heir imporance canno be overemphasized. Differences and heir calculus are as old as is he differenial calculus, bu apar from is imporance and usefulness he heory and mehods of solving difference equaions are no as developed as hose of differenial equaions. Since differenial equaion is a limiing case of difference equaion; herefore, i is naural o eend mehods available for solving differenial equaions o he respecive difference equaions, and he same is done mos of he imes [1 3]. Solving differenial equaions by eploiing symmery properies of heir soluion spaces is one of he sandard mehods, inroduced by he Norwegian mahemaician Sophus Lie ( ). In conras wih adhoc echniques his mehod provides an algorihmic and unified procedure o solve almos all ypes of differenial equaions, linear or nonlinear [4, 5]. A presen, he heory of difference equaions is a he same sage as was he heory of differenial equaions a he ime of S. Lie. A number of adhoc approaches are available o solve difference equaions, for eample, subsiuion, nonlinear funcional relaion, Schroder s generaion funcion, Maeda s mehod, or he heory of inegrable maps. Bu a number of auhors have aemped, following Lie s mehod for differenial equaions, o develop a unified inegraion procedurebasedoninvarianceproperiesofdifferenceequaion [6, 7]. I is worh noing ha he symmery properies are no only helpful o find soluion bu play an imporan role o undersand he physical phenomenon more deeply. In his direcion, a number of researchers have aemped o apply symmeries o analyze differen discree physical phenomena [6 19]. The discree diffusion equaion is widely used in many cones [20, 21]. For insance, i has been applied o he area of populaion growh where one wishes o model geographic spread in addiion o growh in number. In he area of physics i is used o model ionic diffusion on a laice. I has also been used in digial filering in he form of diffusion filering [10]. Due o is wide uiliy, a number of researchers have analyzed differen aspecs of he discree diffusion process. Recenly, in [6], Levi e al. have aemped o generalize he

2 2 Discree Dynamics in Naure and Sociey Lie infiniesimal formulism for calculaion of symmeries of difference equaions and calculaed generalized symmeries of diffusion ype difference equaions. The symmery group, which hey have found for he discree equaion, is a minimal eension of he Lie poin symmeries of he corresponding differenial equaion. They also found ha in he case of linear discree equaion he symmery algebra is isomorphic o he coninuous limi. Symmeries are used no only o inegrae bu also o analyze he soluion space for more insigh ino he physical sysem a hand. So, once symmeries are given i is naural o ask for furher analysis and find soluions of he equaion. Thereislimiedlieraureonheanalysisofsoluionsof diffusion ype difference equaions via symmeries. In his paper one-dimensional opimal sysems are obained for he symmery algebra of diffusion difference equaions wih differen source erms found in [6]. We have also calculaed invarian soluions corresponding o each represenaive of he one-dimensional opimal sysem, wherever hey eis. Using elemens of he symmery algebra generaing soluions are also obained. Moreover, all hose cases in which no group invarian soluion eis are menioned. The res of he paper is arranged as follows. In he ne wosecionswegivesomepreliminariesandsummarizehe resuls of [6], for compleeness. In Secion 3 one-dimensional opimal sysems are obained for all cases and corresponding group invarian soluions are given. Secion 4 is devoed o obain generaing soluions corresponding o each symmery generaor. A brief conclusion is given in Secion Preliminaries Le u() be a coninuous scalar funcion of p independen variables = ( 1, 2,..., i,..., p ) evaluaed a finiely many poins on a uniformly disribued laice wih spacing σ i >0. Then parial finie differences are defined by Δ i u () = 1 σ i { u( 1, 2,..., i 1, i +σ i, i+1,..., p ) u ( 1, 2,..., i 1, i, i+1,..., p ) }. (1) I is someimes convenien o epress finie difference operaor in erms of shif operaor defined as follows: T i u () =u( 1, 2,..., i 1, i +σ i, i+1,..., p ). (2) Consequenly, Δ i u () = 1 σ i (T i 1)u(). (3) For eample, for he facorial funcion (n) = ( h)( 2h)( 3h) [ (n 1)h](2)gives Also he acion of T 1 i T i (n) =σ i n (n 1) + (n). (4) on (n) is given by T 1 i (n) = σ i n (n 1) + (n). (5) Consider a parial difference equaion in he following form: E(,T α u (),T β i Δ i u (),T γ ij Δ i Δ j u (),...)=0, (6) where he operaor T α operaes on a funcion u() as follows: T α u () =T α 1 1 T α 2 2 T α p p u (), (7) wih m i α i n i, i = 1,2,...,p,andm i, n i aresomefied inegers. The shif operaors T β i, T γ ij are defined likewise. Adiscreevecorfieldinisevoluionaryform,X Q = Q(/ wih characerisic Q given by Q= ξ i (, T α T b Δ i u Φ(,T c (8) i is an infiniesimal generalized symmery generaor of he difference equaion (6) if i saisfies he infiniesimal symmery crierion: X Q (n) E(,T α u (),T β i Δ i u (),T γ ij Δ i Δ j u (),...) E(,T α u(),t β i Δ i u(),t γ ij Δ i Δ j u(),...)=0 =0. (9) Here, X Q (n) is he discree analog of he nh prolongaion of he vecor field X Q. Prolonged vecor field acs on he nh eended laice and herefore has he following form: Here, he coefficien funcions Q i, Q i j,...are he discree oal variaions of Q given by Q i =Δ T i Q, Q i j =Δ T i Δ T j Q, (11) X (n) Q = T α Q α T α u+ T β i Q i β i Δ T β i u i + T γ ij Q i j γ ij T γ Δ ij i Δ j u+. (10) where he oal variaion Δ T acs on funcions of, u, Δ u,... as follows: Δ T i f(, u (),Δ u (),...) = 1 σ i {f(( 1, 2,..., i +σ i,..., p ),

3 Discree Dynamics in Naure and Sociey 3 u( 1, 2,..., i +σ i,..., p ), (Δ ( 1, 2,..., i +σ i,..., p ),...) f(( 1, 2,..., p ), u( 1, 2,..., p ), (Δ ( 1, 2,..., p ),...)}. (12) 3. Symmery Algebra of Nonhomogeneous Discree Hea Equaion In hissecionwe summarizehe resuls of [6]. The difference equaion Δ u Δ u+g(,,t,t )u=0 (13) is a discree hea or diffusion equaion. Here u is a funcion of space variable, ime and parial shifs T and T wih respec o and, respecively. Le he following be he generalized symmery generaor in he evoluionary form: X Q =(ξδ u+τδ u+f u =Q u. (14) If (14) isadmiedby(13) hen he infiniesimal symmery crierion gives X (2) Q (Δ u Δ u+g(,,t,t ) Δ u= Δ u+g(,,t,t )u where =0, X (2) Q =Q u +Q +Q +Q u u +Q +Q u u u (15) (16) ishesecondprolongaionofx Q.Using(16)in(15)oneges he following se of deermining equaions: Δ (τ) =0, (17) Δ (τ) T +2Δ (ξ) T + [τ, g] =0, (18) Δ (ξ) T +Δ (ξ) T 2 +2Δ (f) T + [ξ, g] = 0, Δ (f) T +Δ (f) T 2 +2Δ (ξ) T g+ξδ (g) T (19) +τδ (g) T +[f,g]=0. (20) To obain he symmery generaor (14) one needs o solve he se of equaions (17) o(20) for unknown funcions ξ, τ, and f.in[6] hese funcions have been found for hree special source erms g(,, T,T ). Here, we give he infiniesimal symmery generaors obained in [6] andreferohepaper for deails Free Hea Equaion. In his case g(,, T,T )=0and (13)akesheform Δ u Δ u=0. (21) The se of equaions (17)o(20) are hen solved o obain he following symmery algebra: X 6e X 1e =Δ u X 2e =Δ u X 4e = (2Δ ut 1 X 5e = (2Δ ut 1 =( 2 T 2 Δ u σ T 2 =u T 2 u 1 4 σ T 2 +T 1 Δ +T 1 Δ u+t 1 T 1 Δ u u+1 2 T 1 u. (22) 3.2. Nonzero Poenial. In his case he poenial g is aken o be a nonzero funcion g = g(,t ).Plugginginhis assumpion on g in he se of equaions (17)o(20)oneges ξ= 1 2 (1) Δ (τ) T T 1 +α(,t ), (23) where α(, T ) is an arbirary funcion o be known. And f= 1 8 (2) Δ (τ) T 2 T (1) Δ αt T Δ 1 [(1) Δ (τ) T T 1,g]T 1 +β(,t ), (24) where β(, T ) is an arbirary funcion o be known. To findheunknownfuncionsf and ξ one pus a condiion on he poenial funcion g such ha he commuaor [ (1) Δ (τ)t T 1,g]vanishes and (24)becomes f= 1 8 (2) Δ (τ) T 2 T (1) Δ αt T 1 Using (25)in(20) and simplifying one ges 1 8 (2) Δ (τ) T 3 T (1) Δ αt 2 T 1 +β(,t ). (25) Δ βt Δ (τ) T 2 +Δ (τ) T g+ 1 2 (1) Δ (τ) T T 1 Δ (g) T +α(,t )Δ (g) T =0. (26) The auhors hen consider he following wo paricular physically imporan models ha obey he assumpions imposedonhefunciong.

4 4 Discree Dynamics in Naure and Sociey Discree Harmonic Oscillaor. The discree harmonic oscillaor follows he discree hea equaion wih he following poenial funcion: g (, T ) =k 2 (2) T 2, k R>0. (27) Solving he se of deermining equaions (17)o(20)for above value of g one ges he following symmery algebra: X 5e =E 4k () X 1e =Δ u X 2e =u =E 2k () (Δ u+ (1) kt 1 X 4e =E 2k () (Δ u (1) kt 1 (28) (Δ u+2 (2) k 2 T 2 u+2(1) kt 1 Δ u+k X 6e =E 4k () (Δ u+2 (2) k 2 T 2 u 2(1) kt 1 Δ u k u. (29) Discree Cenrifugal Barrier. Likewise, he discree cenrifugal barrier wih poenial funcion hashefollowingsymmeryalgebra: g(,t )=T ( 2) T (30) X 1e =Δ u X 2e =u =( (1) T 1 Δ u+ 1 2 (1) T 1 Δ X 4e =( (1) (1) T 1 T 1 Δ u+ (2) T 2 Δ u+ 1 4 (2) T 2 u T 1 (1) τ 2 u. (31) Table 1: Opimal sysems for free hea equaion and respecive invarian soluions. Opimal sysems Invarian soluions X 1e u (, ) =c 1 + (1) c 2 X 2e u (, ) =c No invarian soluion X 2e X 1e u (, ) =c 1 +c w(y+1)+ yw (y) = 0 (reduced form) 2 X 4e 0 (((2) (1) σ ) ( 1) 1) u=c (2 (1) σ ln (1) ) ( 1) e 2 d X 4e +X 1e u (, ) =c 1 +c 2 ( (2) + (1) σ ) Table 2: The opimal sysems for discree harmonic oscillaor and respecive invarian soluions. Opimal sysems X 1e X 2e X 4e X 5e Invarian soluions u (, ) =c 1 + (1) c 2 k 2 T 2 ((2) +6 (1) +6) No invarian soluion u (, ) =c(k+1) 2 k2 T 2 k+1 ((2) + 2(1) K (k + 1) ) 2 u (, ) =c+k 2 T 2 ((2) +2 (1) 1) u (, ) =c 1 + (1) c 2 +4 (1) +(2+k 2 T 2 )(2) +(8+4k 2 T 2 )(1) +(2σ 1+6k 2 T 2 ) Table 3: The opimal sysems for discree cenrifugal barrier and respecive invarian soluions. Opimal sysems Invarian soluions X 1e u (, ) =c 1 + (1) c 2 +T 2 (4( 1) ( 2) 6) X 2e No invarian soluion Δ y w+ 1 2 yw(y) = T ( 2) T (reduced form) 0 u (, ) =c e (((2) / (2) ) 1) d + 2(1) T 2 (2) ( 2) Δ y V H(, ) V = T 2 ( 2) X 2e + Where H (, ) = 1 (2) 2 σ (2) + 1 (2) +1 2 (1) We consider he following case where he poenial funcion also depends on he variables, T and ye saisfies he required condiion. hen he values of coefficien funcions of infiniesimal generaor (14)are given by: Time-Dependen Discree Harmonic Oscillaor. Now, if he poenial funcion considered in (27) depends on all he four variables,, T,andT andisgivenbyhefollowing g(,,t,t )=k 2 (2) T 2 +k2 (2) T 2, (32) f= (1) ke 2k () α 1 T 1 τ=0, ξ=e 2k () α 1 +E 2k () α 2, (1) ke 2k () α 2 T 1 +β 0. (33)

5 Discree Dynamics in Naure and Sociey 5 Table 4: Generaing soluions for free hea equaion. Generaors Infiniesimal ransformaions Generaing soluion X 1e G 1 ( (1), (1) +ε, u(, ) =f( (1), (1) ε) X 2e G 2 ( (1) +ε, (1), u(, ) =f( (1) ε, (1) ) G 3 ( (1), (1),ue ε ) u(, ) =e ε f( (1), (1) ) X 4e G 4 ( (1) e ε, (1) e 2ε, u(, ) =f( (1) e ε, (1) e 2ε ) X 5e G 5 ( (1) +2 (1) ε, (1),uep ( ( (1) ε+ε 2 (1) ))) u (, ) = ep (ε 2 (1) (1) ε) f( (1) 2 (1) e ε, (1) ) X 6e G 6 ( (1) 1 4ε, (1) (1) 1 4ε,u 1 4ε (1) ep ( (2) ε+ (1) σ ε 1 ( (1) σ )ε )) u (, ) = ep ( ) (1) 1 4ε (1) 4π (1) 4 (1) And one ges 3-dimensional subalgebra of he discree harmonic oscillaor obained in ((28)and(29)). This is generaed by he following infiniesimal generaors: X 1e =u X 2e =(E 2k () Δ u+ (1) ke 2k () T 1 =(E 2k () Δ u (1) ke 2k () T 1 u. 4. One-Dimensional Opimal Sysem and Invarian Soluion (34) In his secion one-dimensional opimal sysem for each of he above cases of discree diffusion equaion is calculaed. I is done in analogy wih he procedure o calculae opimal sysems for differenial equaion, laid down in [4]. These opimalsysemsarehenusedocalculaeinvariansoluions Free Hea Equaion. Acalculaiongiveseighequivalence classes of one-dimensional opimal sysems represened by he following generaors: X 6e, X 5e, X 1e +X 4e, X 4e,, X 2e X 1e, X 2e,and X 1e Invarian Soluion. Employinghemehodgivenin[4] for calculaing invarian soluion for differenial equaions, he invarian soluion corresponding o he represenaives of he opimal sysems given above are obained below. To obain invarian soluion corresponding o he subalgebra X 2e X 1e we find he group invarians u=h( (1) + (1) ) and define new variables in erms of hese invarians by y= (1) + (1), u = h(y).thischangeofvariablesreduceshe parial difference equaion (21) o he ordinary 2nd order difference equaion Δ yy h Δ y h=0. This equaion is hen solved o ge he invarian soluion u = c 1 +c 2 2 ((1) + (1)). Similarly, corresponding o X 4e, u = h(( (2) (1) σ ) ( 1) ) is he group invarian. Subsiuing y=( (2) (1) σ () ) ( 1), u= h(y) (21) reduces o he ordinary difference equaion Δ yy h+ (1/2)yΔ y h=0.afurherchangeofvariableδ y h=w(y) reduces he equaion o he following firs-order difference equaion wih variable coefficien: w(y+1)+ 1 yw (y) = 0. (35) 2 To solve (35) weuselaplacemehodgivenin[3]. Le w(y) = b a y 1 u()d, yw(y) = y u() b a b a y Du()d, and w(y + 1) = b a y u()d. Onsubsiuinghesevalues(35) becomes 1 2 y u () b b a + y (u () 1 a 2 Du ())d=0. (36) Now here are wo possibiliies eiher u() (1/2)Du() = 0 and his gives u() = ce 2 or c y e 2 =0; herefore, 0 or.so,w(y) = c 0 y 1 e 2 d, andonegeshe following invarian soluion: 0 u=c (2 (1) σ ln ) ( 1) e 2 d. (37) Similar calculaions give invarian soluions corresponding o oher represenaives of opimal classes. The resuls are summarizedintable 1. (((2) (1) σ ) ( 1) 1) 4.2. Discree Harmonic Oscillaors and Cenrifugal Barrier. For he discree harmonic oscillaor and he discree cenrifugal barrier he represenaives of heir one-dimensional opimal sysems and he corresponding invarian soluions are summarized in Tables 2 and 3. Remark 1. The soluion corresponding o is obained by applying Laplace mehod in reduced form. Since, in case of ime-dependen harmonic oscillaor, he symmery algebra is a subalgebra of he ime-independen case herefore one does no ge new invarian soluions. 5. Generaing Soluions Since a symmery of a given equaion maps is soluion space ono iself, so, one can find new soluions by applying symmery ransformaions o a known soluion of he

6 6 Discree Dynamics in Naure and Sociey Table 5: Generaing soluions for discree harmonic oscillaor. Generaors Infiniesimal ransformaions Generaing soluion X 1e G 1 ( (1), (1) +ε, u (, ) =f( (1), (1) ε) X 2e G 2 ( (1), (1),ue ε ) u (, ) =e ε f ( (1), (1) ) G 3 ( (1) +ε, (1),uep ( (1) kε + kε2 2 )) u (, ) = ep ((1) kε kε2 2 )f((1) ε, (1) ) X 4e G 4 ( (1) +ε, (1),uep ( (1) kε kε2 2 )) u (, ) = ep ( (1) kε + kε2 2 )f((1) ε, (1) ) G 5 ( (1) e ε, (1) +ε,uep (ep ( 1 X 5e 2 ((2) (1) σ )e 4ε + ε))), k=1 G 5 ( (1) e 2ε, (1) +ε,uep (ep ( 1 X 6e 2 ((2) (1) σ )e 4ε ε))), k=1 u (, ) = ep (ep ( 1 2 ((2) (1) σ )+ε))f( (1) e 2ε, (1) ε) u (, ) = ep (ep ( 1 2 ((2) (1) σ )e 8ε ε))f( (1) e 2ε, (1) ε) Table 6: Generaing soluions for discree cenrifugal barrier. Generaors Infiniesimal ransformaions Generaing soluion X 1e G 1 ( (1), (1) +ε, u (, ) =f( (1), (1) ε) X 2e G 2 ( (1), (1), ue ε ) u (, ) =e ε f( (1), (1) ) G 3 ( (1) e ε, (1) e 2ε, u(, ) =f( (1) e ε, (1) e 2ε ) X 4e (1) 1 4ε, (1) (1) 1 4ε, (1) u (, ) = G 6 ( u ( (1) ) σ )ε 1+4ε (1) ep ( ((2) (1) σ )ε )f( ep ( ) 1+4ε (1) 1+4ε, (1) 1+4ε ) (1) 1 4ε (1) 1 4ε (1) equaion.thesoluionhusobainediscalledageneraing soluion. Here, hese generaing soluions are obained by firs obaining he one parameer ransformaion corresponding o each infiniesimal generaors obained in Secion 2. These soluions are obained in analogy wih he mehod given in [4] for differenial equaions. The resuls are summarized in he form of Tables 4, 5,and6. 6. Conclusion In his paper, in analogy wih differenial equaions, he symmery analysis of a nonhomogeneous discree hea equaion (13) has been carried ou. The one-dimensional opimal sysems are obained for he equaion wih differen source erms by using symmery algebra given in [6]. These opimal sysems are hen used o obain invarian soluions if hey eis. Moreover, one parameer group of ransformaions corresponding o all infiniesimal generaors of he symmery algebra has been calculaed o obain generaing soluions of he equaion. Main resuls are summarized in Tables 1, 2, 3, 4, 5, and6. Working on he same lines one can calculae conservaion laws which we will repor soon. Acknowledgmen The auhors hank he main edior and anonymous referees for heir valuable commens and suggesions leading o improvemen of his paper. This work was suppored by he Higher Educaion Commission of Pakisan. References [1] S. Goldberg, Inroducion o Difference Equaions, wih Illusraive Eamples from Economics, Psychology, and Sociology, John Wiley & Sons, New York, NY, USA, [2] C. Jordan, Calculus of Finie Differences, Chelsea Publishing CompanyBron, New York, NY, USA, 3rd ediion, [3] R.BellmanandK.L.Cooke,Differenial-Difference Equaions, Academic Press, New York, NY, USA, [4] P. J. Olver, Applicaions of Lie Groups o Differenial Equaions, Springer,NewYork,NY,USA,1986. [5] G. W. Bluman and S. Kumei, Symmeries and Differenial Equaions, Springer, New York, NY, USA, [6] D. Levi, L. Vine, and P. Winerniz, Lie group formalism for difference equaions, Physics A,vol.30,no.2,pp , [7] S. Maeda, The similariy mehod for difference equaions, IMA Applied Mahemaics,vol.38,no.2,pp , [8] G.R.W.QuispelandR.Sahadevan, Liesymmeriesandhe inegraion of difference equaions, Physics Leers A, vol.184, no. 1, pp , [9] Z. Jiang, Lie symmeries and heir local deerminacy for a class of differenial-difference equaions, Physics Leers A,vol.240, no.3,pp ,1998. [10] J. Campbell, The SMM Model as a boundary value problem using he discree hea equaion, NASA Langley Research Cener Hampon Virginia,vol.72,no.4,pp ,2007.

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8 Advances in Operaions Research Advances in Decision Sciences Applied Mahemaics Algebra Probabiliy and Saisics The Scienific World Journal Inernaional Differenial Equaions Submi your manuscrips a Inernaional Advances in Combinaorics Mahemaical Physics Comple Analysis Inernaional Mahemaics and Mahemaical Sciences Mahemaical Problems in Engineering Mahemaics Discree Mahemaics Discree Dynamics in Naure and Sociey Funcion Spaces Absrac and Applied Analysis Inernaional Sochasic Analysis Opimizaion