Research Article An Unconventional Finite Difference Scheme for Modified Korteweg-de Vries Equation

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1 Hindawi Advances in Mahemaical Physics Volume 27, Aricle ID 47967, 9 pages hps://doi.org/./27/47967 Research Aricle An Unconvenional Finie Difference Scheme for Modified Koreweg-de Vries Equaion Canan Koroglu and Ayhan Aydin 2 Mahemaics Deparmen, Science Faculy, Haceepe Universiy, Beyepe, 68 Ankara, Turkey 2 Mahemaics Deparmen, Ailim Universiy, Incek, 683 Ankara, Turkey Correspondence should be addressed o Canan Koroglu; ckoroglu@haceepe.edu.r Received 2 July 27; Acceped 9 Ocober 27; Published November 27 Academic Edior: Ming Mei Copyrigh 27 Canan Koroglu and Ayhan Aydin. 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. A numerical soluion of he modified Koreweg-de Vries (MKdV) equaion is presened by using a nonsandard finie difference (NSFD) scheme wih hea mehod which includes he implici Euler and a Crank-Nicolson ype discreizaion. Local runcaion error of he NSFD scheme and linear sabiliy analysis are discussed. To es he accuracy and efficiency of he mehod, some numerical eamples are given. The numerical resuls of NSFD scheme are compared wih he eac soluion and a sandard finie difference scheme. The numerical resuls illusrae ha he NSFD scheme is a robus numerical ool for he numerical inegraion of he MKdV equaion.. Inroducion This paper is concerned wih he nonsandard inegraion of modified Koreweg-de Vries (MKdV) equaion u +qu 2 u +ru =, (, ) [ L, R ] [, T] () wih iniial condiion and boundary condiions u (, ) =u (), [ L, R ] (2) u( L,)=f(), u( R,)=g(), [, T], q, r R. The analyical soluion of he MKdV equaion () can be epressed as [] u (, ) = 6r q (3) anh (+2r). (4) I plays an imporan role in he sudy of nonlinear physics such as fluid physics and quanum field heory. I is a model equaion for he weakly nonlinear long waves which occur in many differen physical sysems. I is an inegrable equaion and admis solion soluion obained by means of he inverse scaering mehod and Hiroa s direc mehod and by using Backlund ransformaions [2, 3]. I is well known ha () has a soliary wave soluion of he form u (, ) = 6r q k sech k( rk2 ). () Alhough he MKdV equaion has been eensively sudied by many auhors in solion heory, he soluion (4) is never considered before in he lieraure. For he purpose of nonsandard inegraion, he kink solion soluion (4) will be used hroughou he sudy. A nonsandard finie difference scheme can be consruced from he eac finie difference scheme [4]. An eac finie difference scheme can be consruced for any ordinary differenial equaion (ODE) or parial differenial equaion (PDE) from he analyical soluion of he differenial equaion [ 7]. Among he various numerical echniques such as classical finie difference, finie volume, adapive mesh, finie elemen, and specral mehod

2 2 Advances in Mahemaical Physics for solving ODEs and PDEs, NSFD schemes have been proved o be one of he mos efficien approaches in recen years. The auhors in [8] proposed a nonsandard finie volume mehod for he numerical soluion of a singularly perurbed Schrödinger equaion. They have shown ha he proposed nonsandard finie volume mehod is capable of reducing he compuaional cos associaed wih mos classical schemes. They have highlighed ha NSFD schemes have been efficien in ackling he deficiency of classical finie difference scheme for he approimaion of soluions of several differenial equaion models. A nonsandard symplecic Runge-Kua mehod is applied o Hamilonian sysems in [9]. In [9], i has been shown ha nonsandard schemes are beer han sandard finie difference schemes in long ime compuaions. Compared wih some oher mehods, NSFD mehod is more sable []. Up o he auhor s knowledge, a NSFD scheme for he numerical soluion of he MKdV equaion () is never sudied before. The aim of his paper is o designed a robus NSFD scheme for he numerical soluion of he MKdV equaion () ha is beer han he sandard scheme in he numerical precision for large spaial sep size which reduces he compuaional cos associaed wih mos classical schemes. This paper is organized as follows. In he ne secion we begin wih proposing he NSFD scheme for he MKdV equaion (). Sabiliy and local runcaion error of he NSFD scheme are eamined in Secion 3. In Secion 4 some numerical eperimens for he NSFD scheme are presened o show ha our proposed mehod is efficien and accurae. Finally, we summarize our observaion in Secion. 2. Nonsandard Discreizaion In his secion, we will propose he NSFD model for he numerical soluion of he MKdV equaion (). Firsly, we give hree basic definiions and properies of he NSFD discreizaion proposed by Mickens [, 2] o consruc a NSFD scheme. () The orders of he discree derivaives mus be eacly equal o he orders of he corresponding derivaives of he differenial equaions. (2) Denominaor funcions for he discree derivaives mus, in general, be epressed in erms of more complicaed funcions of he sep sizes han hose convenionally used. For eample, he discree derivaives u (, ) and u (, ) are generalized as u n u (, ) un+ Φ (Δ, λ), Φ(Δ, λ) =Δ+O (Δ2 ), u (, ) un + un (Δ,μ), u (, ) un un (Δ,μ), (Δ, μ) = Δ + O (Δ 2 ). (6) (3) Nonlinear erms mus, in general, be modeled nonlocally on he compuaional grid or laice; for eample, (u n )2 u n + un, (u n )2 ( un +un +un + )u n 3. (u n )3 (u n )2 ( un + +un ), 2 (u n )3 u n un un +. I is well known ha a NSFD mehod is consruced from he eac finie difference schemes. Bu Mickens [] discussed some difficulies of applying he nonsandard modeling rules in he acual consrucion of eac finie difference scheme for he MKdV equaion. Some pifalls in he procedures for consrucing an eac finie difference schemes in erms of basic rules of he NSFD mehods are invesigaed []. For he MKdV equaion () wo nonsandard finie difference models are proposed [], namely, he eplici scheme U n+ U n D 3 (Δ) +qu n+ (U n [ )2 (U n )2 ] D 2 (Δ) [ ] + Un +2 3Un + +3Un Un D (Δ) 2 D 2 (Δ) and he implici scheme U n+ U n D 3 (Δ) +qu n+ = (U n [ )2 (U n )2 ] D 2 (Δ) [ ] + Un+ +2 3Un+ + +3Un+ U n+ D (Δ) 2 =, D 2 (Δ) U n is he approimaion o he eac soluion u(, ) a he mesh poin (, n ) and D (Δ) = Δ, D 2 (Δ) = Δ, and D 3 (Δ) = Δ. The above consrucion processes do no give funcional relaion beween he space and ime sep sizes which is no known ye (see Mickens [], p: 228). The sep sizes for eac schemes mus saisfy some fied condiions. In order o release hese condiions for sep size, we follow hewayofzhangeal.[3]andconsruchefollowingnonsandard-hea scheme [3] U n+ Φ U n θ(u n+ +qu n Un+ U n+ )+( θ)(un Un ) +r θ(un+ +2 2Un+ + +2Un+ Un+ 2 ) +r ( θ) (Un +2 2Un + +2Un Un 2 ) = (7) (8) (9) ()

3 Advances in Mahemaical Physics 3 for he numerical inegraion of he MKdV equaion (). Here Φ is he ime sep funcion and and =2 3 are he space sep funcions; U n is an approimaion o he eac soluion u(, ) a he mesh poin (, n ).Asandardfiniedifference scheme for he MKdV equaion () can be proposed as U n+ Δ U n θ(u n+ +q(u n )2 U n+ )+( θ) (Un Un ) Δ +r θ(un+ +2 2Un+ + +2Un+ Un+ 2 ) 2Δ 3 +r ( θ) (Un +2 2Un + +2Un Un 2 ) 2Δ 3 =. According o (), we can wrie A=θ(U n+ () Φ= (Un Un+ ) qu nun+ A + rb, (2) U n+ )+( θ) (Un Un ) B=θ(U n+ +2 2Un+ + +2Un+ Un+ 2 ) + ( θ) (U n +2 2Un + +2Un Un 2 ). (3) When θ=,weselec(δ) = (h) = (e 2h )/2 and hence =2 3 =(e 2h ) 3 /4. Then subsiuing and ino (2), we can rewrie Φ as Φ= e4rδ 4r [4(+se 4rΔ )(+se 4rΔ 2h ) (+se 4rΔ+4h )(+se 4rΔ+2h ) (+se 4rΔ 4h )] [6(s ) (se 4rΔ )(+se 4rΔ+4h ) (+se 4rΔ+2h ) (+se 4rΔ 4h )2e 4rΔ 2h 4(e 2h +)(s+) ( + se 4rΔ ) 2 {s 2 e 2rΔ 4h se 8rΔ 6h (e 2h+ ) 2 +e 4rΔ 4h }], (4) s=s n =e2( +2r n ). If (Δ) = (h) = h + O(h 2 ) when h, Δ, afer edious calculaion we obain Φ (Δ) e4rδ 4r wih Φ (Δ) =Δ+O (Δ 2 ). () Similarly, for θ=and θ=/2,if is seleced o be (Δ) =(h) = e2h 2 (6) we obain he same denominaor funcion Φ Φ e4rδ. (7) 4r We noe ha Φ Δ, Δ,and 2Δ 3 as (Δ, Δ) (, ). 3. Sabiliy and Local Truncaion Error In his secion, sabiliy and local runcaion error of he nonsandard scheme () are eamined. For sabiliy analysis we use he von Neumann mehod. Since he mehod is applicable only for linear PDE, we consider he linearized MKdV equaion u +au +ru =, (, ) [ L, R ] [, T], (8) a=ma{qu 2 (, )} in he domain (, ) [ L, R ] [, T]. The applicaion of he nonsandard-hea scheme () o he linear equaion (8) yields U n+ Φ U n +a θ(un+ U n+ )+( θ)(un Un ) +r θ(un+ +2 2Un+ + +2Un+ Un+ 2 ) +r ( θ) (Un +2 2Un + +2Un Un 2 ) =. (9) We ake he difference beween he eac soluion u(, ) a he mesh poin (, n ) and he approimae soluion U n and define he error ε n = u(, n ) U n. Subsiuing Un = u(, n ) ε n ino he difference equaion (9), we see ha he error ε n saisfies he same discree equaion ε n+ Φ ε n +a θ(εn+ ε n+ )+( θ)(εn εn ) +r θ(εn+ +2 2εn+ + +2εn+ εn+ 2 )+ +r ( θ) (εn +2 2εn + +2εn εn 2 ) =. (2) The von Neumann sabiliy analysis uses he fac ha every linear consan coefficien difference equaion has a soluion of he form ε n =(eαδ ) n e iβδ = (ξ) n e iβδ, α,β R, i 2 =. (2) The funcion ξ is deermined from he difference equaion by subsiuing he Fourier mode (2) ino (2), and we obain ξ n+ e iβδ ( + A 2 +ib 2 )=( A ib )ξ n e iβδ, (22)

4 4 Advances in Mahemaical Physics A =2 Φa ( θ) sin2 ( βδ 2 ), A 2 =2 Φa θsin2 ( βδ 2 ), B = Φa ( θ) sin (βδ) 8 Φr ( θ) sin (βδ) sin2 ( βδ 2 ), B 2 = Φa θ sin (βδ) 8 Φr θ sin (βδ) sin2 ( βδ 2 ). Canceling he eponenial erm, we have ξ n+ =ρξ n, (23) ρ= A ib +A 2 +ib 2 (24) is he amplificaion facor of he mehod. If he mehod is o be sable, he sabiliy requiremen ρ should be saisfied. Now we consider he following cases. Crank-Nicolson Type Scheme. Whenθ=/2,heamplificaion facor is simplified o ρ= A ib +A+iB, (2) A= Φa sin2 ( βδ 2 ) (26) B= Φa sin (βδ) 4Φr 2 sin (βδ) sin2 ( βδ 2 ). From (2) we ge ρ 2 =ρρ ; hence he proposed mehod () is uncondiionally sable for θ = /2. The Fully Implici Scheme. Whenθ =,heamplificaion facor is simplified o ρ= +A+iB, (27) A=2 Φa sin2 ( βδ 2 ) B= Φa sin (βδ) 8Φr sin (βδ) sin2 ( βδ 2 ). (28) From (27) we ge ρ 2 =ρρ ; hence he proposed mehod () is uncondiionally sable for θ=. The Eplici Scheme. Whenθ=, he amplificaion facor is simplified o ρ = A ib, (29) A=2 Φa sin2 ( βδ 2 ) B= Φa sin (βδ) 8Φr sin (βδ) sin2 ( βδ 2 ). (3) I is well known ha he eplici mehods are condiionally sable. Numerical eperimens show ha he amplificaion facor ρ when Φ/ <.3 (or Δ/Δ <.) forhe parameers q =.3 and r =.. Now, we will discuss he local runcaion error of he nonsandard scheme (). In heory, we can approimae he original problem as accuraely as we wish by making he ime sep Δ and Δ small enough. I is said in his case ha he approimaion is consisen. The local runcaion error and he sabiliy play imporan roles in he convergence of he numerical mehod. In a convergen mehod, he order of he error is deermined by he order of he local runcaion error. The local runcaion error T n of he nonsandard scheme () is defined by T n =( u n u (, n )) +q(u n un+ n u n u(, n ) 2 u (, n )) +r( u n u (, n )), u n = un+ Φ u n u n = θ(un+, u n+ )+( θ) (un un ) u n = θ(un+ +2 2un+ + +2un+ un+ 2 ) + ( θ) (un +2 2un + +2un un 2 ), (3) (32) and u n is he approimae soluion for he MKDV equaion () obained from he nonsandard scheme () and u(, n ) is he eac soluion a he mesh poin (Δ, nδ). For θ=, he principal par of he local runcaion error is T n =(Δ Φ )u + Δ2 2Φ u + Δ3 6Φ u +( Δ )qu2 u Δ2 2 qu2 u + ΔΔ qu2 u + Δ3 6 qu2 u

5 Advances in Mahemaical Physics Δ2 Δ 2 + ΔΔ + ΔΔ2 Δ2 Δ 2 2 qu2 u + ΔΔ2 2 qu2 u quu Δ2 Δ 2 quu quu u + Δ3 Δ 6 quu u quu u + ΔΔ3 2 quu u + ΔΔ2 2 quu u Δ2 Δ 2 quu 4 u + ΔΔ3 2 quu u + Δ3 Δ 2 2 quu u Δ2 Δ 3 quu 4 u +r( 2Δ3 )u. (33) I is clear ha (Φ,,) (Δ,Δ,2Δ 3 ) as (Δ, Δ) (, ). Thus, we ge he local runcaion error T n = O(Δ+Δ). Similarly, one can show ha he local runcaion error for θ=is firs-order in ime and space. The principal par of he local runcaion error for θ=/2is T n =(Δ Φ )u + Δ2 2Φ u + Δ3 6Φ u +( Δ )qu2 u Δ2 2 qu2 u + ΔΔ 2 qu2 u + Δ3 6 qu2 u Δ2 Δ 4 qu2 u + ΔΔ2 4 qu2 u + ΔΔ2 2 quu u Δ2 Δ 2 quu 4 u Δ2 Δ 2 4 quu u + ΔΔ3 4 quu u + ΔΔ4 8 quu u +( 2Δ3 )ru + Δ3 Δ 2 2 ru + Δ 2 ru. (34) I is clear ha (Φ,,) (Δ,Δ,2Δ 3 ) as (Δ, Δ) (, ). Thus, we ge he local runcaion error T n = O(Δ2 + Δ). We deduce ha he nonsandard scheme () is consisen since he local runcaion T n endsozeroasδ and Δ end o zero. The cenerpiece for he heory of convergence of linear difference approimaions of ime-dependen parial differenial equaions is he La Equivalence Theorem [4]. Since he proposed scheme () is consisen and sable, i is convergen according o he La heorem. 4. Numerical Resuls In his secion we presen some numerical eperimens o es he accuracy and efficiency of he proposed NSFD scheme () for he numerical soluion of he MKdV equaion () over he soluion domains L R and T.The soluion domain is divided ino equal inervals wih lengh Δ=( R L )/M in he direcion of he spaial variable and Δ = T/M in he direcion of ime such ha = L +Δ, =,,...,J, n = nδ,andn =,,...,M. The iniial condiion u () and boundary condiions are aken from he eac soluion (4) u () = 6r q anh (), [ L, R ] (3) u ( L,) = 6r q anh ( L +2r), (36) u ( R,) = 6r q anh ( R +2r),, (37) respecively, r = β/2, β, q R. We use he following error norms: L = ma J u e () u a (), M L 2 = = [u e () u a ()] 2, (38) o assess he performance of he NSFD scheme. Here u e is he eac soluion obained from (4) and u a is he approimae soluion obained from he NSFD scheme (). Table represens L and L 2 errors of he NSFD scheme () and he sandard finie difference scheme () a differen imes for he MKdV equaion () wih θ=, β =.,and q=in he spaial domain wih Δ=2. and Δ =.. From he able we see ha he NSFD scheme () is more accurae han sandard finie difference scheme () in all cases. We obained similar resuls for θ = and θ = /2. The absolue error is defined by u(, n ) U n (39) for he sandard scheme () and he nonsandard scheme () are provided in Table 2 a various and values. From he eperimens i is readily seen ha our mehod is more accurae han he sandard mehod. Table 3 also measures he accuracy and he versailiy of he NSFD scheme () wih θ = by using he absolue error a he mesh poin (, n ).Fromheableweseeha he nonsandard scheme is very accurae and efficien. In addiion, we noice ha increasing he value of he nonlinear erm q does no affec he accuracy for large spaial sep sizes. Similar resuls are obained for θ=and θ = /2. I is well known ha numerical insabiliies occur in many discree models unless cerain numerical condiions on spaial and emporal sep sizes are saisfied. For eamples,

6 6 Advances in Mahemaical Physics Table : L and L 2 errors for MKDV equaion () wih θ=, q=,andβ =., on wih Δ =. and Δ = 2.. N: nonsandard; S: sandard. T L (N) L (S) L 2 (N) L 2 (S) E 6.466E 8.7E 6.6E 4..2E 2.92E.6E 3.4E E 4.339E 2.42E 4.63E E.747E 3.22E 6.4E. 3.E 7.37E 4.2E 7.632E Table 2: Absolue errors for MKDV equaion () wih θ=, q =.3, andβ =. on wih Δ =. and Δ = =.. N: nonsandard; S: sandard. N S.2.23E 8.24E E 2.2E E 3.949E E 2.448E E E E 2.833E E 6.636E E.969E E E 9 Table 3: Nonlinear effec: absolue errors for θ=and β =. on wih Δ =. and T=wih Δ =.. q =.3 q = q =.2.23E 2.83E 6.88E E.8E.87E E 7.734E 2.446E.2.983E E 3.86E E E E E 2.37E E E.73E.87E E.6E.7E.8.668E E 9.36E forward and backward Euler and cenral difference for decay equaions produce numerical insabiliy for large sep sizes [].Figurecompareshenumericalsoluionofnonsandard scheme () and he eac wave soluion (4). This picure represens he resul of an inegraion wih q =.3, β =., and θ = /2,over he spaial domain and emporal inerval wih spaial sep size Δ =. and emporal sep size Δ =.. From he figure we see ha nonsandard scheme well simulaes he eac soluion wihou showing any numerical insabiliies. Figures 2 and 3 represen he L and L 2 errors and Figures 4 and represen he absolue errors for various spaial sep sizes of he NSFD scheme () and he sandard finie difference scheme () for he MKdV equaion () wih he same se of parameers of Figure. From he figures i is obvious ha he NSFD scheme is beer han he sandard scheme in he numerical precision forlargespaialsepsize,buiisinferiorforsmallspaialsep size. We obained similar resuls for θ=and θ=. A final issue o consider is he effec of he hird-order dispersion coefficien r = β/2 in (). We see ha for various values of β, as shown in Table 4, dispersion-dominaed soluion demonsraed ha he error is increased a he place heshockwaveoccurs.bohtables3and4showha he errors are concenraed in he spaial region here are seep soluions.. Conclusion I is well known ha eplici finie difference models for soluion of differenial equaion require resricion on sep size o preven he numerical insabiliies. For his reason, small sep sizes are used o ensure he numerical sabiliy. This causes

7 Advances in Mahemaical Physics 7 u (, ) u (, ) (a) (b) Figure : Surface for θ=/2. Nonsandard (a). Eac (b)..7.2 L error L 2 error Sandard Nonsandard Sandard Nonsandard Figure 2: θ=/2, L and L 2 errors for he NSFD scheme (), and sandard scheme (): Δ = L error L 2 error Sandard Nonsandard Sandard Nonsandard Figure 3: θ=/2, L and L 2 errors for he NSFD scheme (), and sandard scheme (): Δ =..

8 8 Advances in Mahemaical Physics Error (a) Error (b) Figure 4: θ=/2, absolue errors for he NSFD scheme () (a) and sandard scheme () (b). Δ =., q =.3, β =.,andδ = 2. Error (a) Error (b) Figure : θ=/2, absolue errors for he NSFD scheme () (a) and sandard scheme () (b). Δ =., q =.3, β =.,andδ =.. Table 4: Dispersion effec: absolue errors for θ=and q =.3 on wih Δ =. and T=wih Δ =.. β =. β =. β =.2.232E 3 3.6E 9.69E E 3.364E E E 3 6.7E 9.3E E E E E E E E 3 2.E 7.2E E 2.23E 6.236E many grid poins and hence round-off errors in numerical compuaions. Moreover he use of small sep size is compuaionally epensive. In his sudy a nonsandard finie difference (NSFD) model is proposed for he numerical soluion of he modified Koreweg-de Vries (MKdV) equaion based on kink solion soluion. Local runcaion error is discussed. Linear sabiliy analysis is performed for he linearized equaion. The numerical resuls obained by he NSFD scheme is compared o he eac soluion and a sandard finie difference scheme. Proposed scheme indicaes ha NSFD scheme shows beer performance for large sep sizes. The effec of nonlinear erm and he dispersive erm is also

9 Advances in Mahemaical Physics 9 sudied numerically. The numerical resuls ha have been obained in his paper ehibi ha he coefficien of he nonlinear erm does no change he numerical resuls much. Tables and figures illusrae ha he NSFD scheme can be a robus ool for numerical soluion of various nonlinear problems such as 2D-KdV. Conflics of Ineres The auhors declare ha here are no conflics of ineres regarding he publicaion of his paper. References [] A. Bekir, On raveling wave soluions o combined KDVmKDV Equaion and modified Burgers-KDV Equaion, Communicaions in Nonlinear Science and Numerical Simulaion,vol. 4, no. 4, pp , 29. [2] G.L.LambJr.,Elemens of Solion Theory, JohnWiley&Sons, Inc, New York, NY, USA, 98. [3] V.E.Zakharov,S.V.Manakov,S.P.Novikov,andL.P.Piaevski Consulans Bureau, New York, NY, USA, 98. [4] P. M. Manning and G. F. Margrave, Inroducion o non-sandard finie-difference modelling, CREWES Research Repor 8, 26. [] R. P. Agarwal, Difference Equaions and Inequaliies: Theory, Mehods, and Applicaions, in Chapman & Hall/CRC, Pure and Applied Mahemaics, vol. 228, CRC Press, New York, NY, USA, 2nd ediion, 2. [6] C. Liao and X. Ding, Nonsandard finie difference variaional inegraors for mulisymplecic PDEs, Applied Mahemaics,vol.22,AricleID779,22. [7] L. W. Roeger, Eac finie-difference schemes for wo-dimensional linear sysems wih consan coefficiens, Compuaional and Applied Mahemaics,vol.29,no.,pp.2 9, 28. [8] A. A. Aderogba, M. Chapwanya, J. Doko Kamdem, and J. M. Lubuma, Coupling finie volume and nonsandard finie difference schemes for a singularly perurbed Schrödinger equaion, Inernaional Compuer Mahemaics, vol.93,no., pp ,26. [9] B. Karasözen, Runge-Kua mehods for Hamilonian sysems in non-sandard symplecic wo-form, Inernaional Compuer Mahemaics, vol. 66, no. -2, pp. 3 22, 998. [] L. Zhang, L. Wang, and X. Ding, Eac finie-difference scheme and nonsandard finie-difference scheme for coupled Burgers equaion, Advances in Difference Equaions,24:22,24pages, 24. [] R. E. Mickens, Nonsandard finie difference models of differenial equaions, World Scienific, 994. [2] R. E. Mickens, Advances in he Applicaions of Nonsandard Finie Difference Schemes, R.E.Mickens,Ed.,Wiley-Inerscience, Singapore, 2. [3] L. Zhang, L. Wang, and X. Ding, Eac finie difference scheme and nonsandard finie difference scheme for Burgers and Burgers-Fisher equaions, Applied Mahemaics, Aricle ID 97926, Ar. ID 97926, 2 pages, 24. [4]P.D.LaandR.D.Richmyer, Surveyofhesabiliyof linear finie difference equaions, Communicaions on Pure and Applied Mahemaics, vol. 9, pp , 96.

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