Solution of Singularly Perturbed Differential Difference Equations Using Higher Order Finite Differences

Transcription

1 Amercan Journal of Numercal Analyss, 05, Vol. 3, No., 8-7 Avalable onlne at Scence and Educaton Publshng DOI:0.69/ajna-3-- Soluton of Sngularly Perturbed Dfferental Dfference Equatons Usng Hgher Order Fnte Dfferences Lakshm Srsha, Y. N. Reddy * Department of Mathematcs, Natonal Insttute of Technology, Warangal, Inda *Correspondng author: ynreddy_ntw@yahoo.com Receved January, 05; Revsed February 0, 05; Accepted February 3, 05 Abstract In ths paper, we dscuss the soluton of sngularly perturbed dfferental-dfference equatons ehbtng dual layer usng the hgher order fnte dfferences. Frst, the second order sngularly perturbed dfferental-dfference equatons s replaced by an asymptotcally equvalent second order sngular perturbed ordnary dfferental equaton. Then, fourth order stable fnte dfference scheme s appled to get a three term recurrence relaton whch s easly solved by Thomas algorthm. Some numercal eamples have been solved to valdate the computatonal effcency of the proposed numercal scheme. To analyze the effect of the parameters on the soluton, the numercal soluton has also been plotted usng graphs. The error bound and convergence of the method have also been establshed. Keywords: dfferental-dfference equatons, delay parameter, advance parameter, dual layer Cte Ths Artcle: Lakshm Srsha, and Y. N. Reddy, Soluton of Sngularly Perturbed Dfferental Dfference Equatons Usng Hgher Order Fnte Dfferences. Amercan Journal of Numercal Analyss, vol. 3, no. (05): 8-7. do: 0.69/ajna Introducton A class of functonal dfferental equatons whch have the characterstcs of both classes.e., delay and/or advanced and sngularly perturbed behavour are known as Sngularly Perturbed Dfferental-Dfference Equatons (SPDDEs). These dfferental-dfference equaton models have rcher mathematcal framework for the analyss of bo system dynamcs, compared wth ordnary dfferental equatons. Also, they ehbt better consstency wth the nature of the underlyng process and predctve results. SPDDEs provde mathematcal models n optcs, physology, mechancs, epdemology, neural networks as well as many other applcatons Ref. [,,3,]. As a result, numercal treatment of SPDDEs has receved a great deal of attenton. Sten [5] gave stochastc effects due to neuronal ectaton wth help of a model that represents a dfferental-dfference equaton. Lange and Mura [6,7,8,9,0] publshed a seres of papers to study the numercal soluton of SPDDEs. They etended the matched asymptotc epansons, ntally developed for ordnary dfferental equatons, to obtan appromate solutons to dfferental dfference equatons. The same authors [7] etended the problems nvolvng boundary and nteror layer phenomenon, rapd oscllatons and resonance behavour n [6] and [0] to the problems havng solutons whch ehbt turnng pont behavour.e., transton regons between rapd oscllatons and eponental behavour. Kadalbajoo and Sharma [] devsed a fnte dfference scheme to obtan the soluton for sngularly perturbed dfferental dfference equatons of med type.e., whch contan negatve as well as postve shfts. The same authors [] descrbed a numercal approach based on fnte dfference method to solve a mathematcal model arsng from neuronal varablty. Sharma and Kaushk [3] dscussed two approaches, namely, ftted operator and ftted mesh method to solve SPDDEs wth small delay as well as advanced, wth layer behavour. Amralyev and Cmen [] used an eponentally ftted dfference scheme on unform mesh, accomplshed by the method of ntegral denttes, for lnear second order delay dfferental equatons. Prathma and Sharma [5] presented a numercal method to solve SPDDEs wth negatve shft and solated turnng pont at =0. Chakravarthy and Rao [6] presented a modfed fourth order Numerov method for solvng SPDDEs of med type. Wth ths motvaton, n ths paper, we dscuss the soluton of sngularly perturbed dfferental-dfference equatons ehbtng dual layer usng the hgher order fnte dfferences. Frst, the second order sngularly perturbed dfferental-dfference equatons s replaced by an asymptotcally equvalent second order sngular perturbed ordnary dfferental equaton. Then, fourth order stable fnte dfference scheme s appled to get a three term recurrence relaton whch s easly solved by Thomas algorthm. Some numercal eamples have been solved to valdate the computatonal effcency of the proposed numercal scheme. To analyze the effect of the parameters on the soluton, the numercal soluton has also been plotted usng graphs. The error bound and convergence of the method have also been establshed.. Descrpton of the Method

2 Amercan Journal of Numercal Analyss 9 Consder the sngularly perturbed dfferental dfference equaton of med type,.e., wth delay as well as advance parameters of the form: ε y ( ) + ay ( ) ( δ) + cy ( ) + by ( ) + = f( ) ( ) ( η ) () (0,) and subject to the nterval and boundary condtons y ( ) = ϕ( ), on -δ 0 () y( ) = γ( ), on + η (3) where a( ), b( ), c( ), f ( ), ϕ( ) and γ ( ) are bounded and contnuously dfferentable functons on (0,), 0< ε << s the sngular perturbaton parameter; and 0 < δ = o( ε) and 0 < η = o( ε) are the delay and the advance parameters respectvely. Usng Taylor seres epanson n the neghborhood of the pont, we have ( δ) ( ) δ ( ) y y y () ( η) ( ) η ( ) y + y+ y (5) Substtutng () and (5) nto Eq. (), we obtan an asymptotcally equvalent sngularly perturbed boundary value problem of the form: ε y ( ) + py ( ) ( ) + qy ( ) ( ) = f( ) (6) y(0) = ϕ( ) = ϕ0 (7) y() = γ( ) = γ (8) where p ( ) = ηb ( ) δa ( ) and q ( ) = a ( ) + b ( ) + c ( ). The transton from Eq.() to Eq. (6) s admtted, because of the condton that 0< δ << and 0< η << are suffcently small. Further detals on the valdty of ths transton can be found n Elsgolt s and Norkn [7]. If a ( ) + b ( ) + c ( ) 0 on the nterval [0, ], then the soluton of Eq. () ehbts boundary layers at both ends of the nterval [0,], whereas t ehbts oscllatory behavour for a ( ) + b ( ) + c ( ) > 0. Here, we have consdered the frst case where the solutons of the problem ehbt dual layers. Dvdng the nterval [0, ] nto N equal parts wth constant mesh length h, wth mesh ponts: 0 = 0,,,..., N =, we have = h, = 0,,,..., N.Assumng that y ( ) has contnuous fourth dervatves on [0,] and by makng use of Taylor s epanson, we have the fourth order central dfferences: y+ y + y h ( y ) = y ( ξ ) + T (9) h y+ y h y = y ( ς ) + T (0) h 6 ( 6 T h y ) ( ξ ) [ h h] where = 6! and ξς,, +. ( 5 ) ( ) = for T h y ς 5! Substtutng (9) and (0) nto Eq. (6) we can wrte the central dfference appromaton of Eq.(6) n the form that ncludes all the oh ( ) error terms as follows: ε p ε ε p y + q y + + y + h h h h h () h ( py ) + ε y + T= f where T = ε T + pt. Now, from Eq.(6) we have ε y = py qy + f () Dfferentatng both sdes of (), we get: ( ) ε y = py + py + qy + qy + f ε y ( ) + ( + ) + ( + ) + py p q y = + f p q y qy Usng (3) and () nto () we obtan ε h h h p ε y + q y ε p h p + + y p + q y h h ε h pp pq p + q y ε ε pq h p f q y T f f h = + + ε ε (3) () (5) Now, appromatng the converted error term, whch has a stablzng effect, n (5) by usng the central dfference formulas (9) and (0) for y and y we obtan ε p ε y + q y h h h ε p p + + y p + q y h h ε + p p + p + q y + + p + q y 6 ε ε (6) h pp pq p + q y+ ε ε h pp pq + + p + q y ε ε h pq h p f + + q y = f + + f + T ε ε Where q h ( ) qq T = + q + r y + + ε ε qr h + q + r y T ε 7

3 0 Amercan Journal of Numercal Analyss s the truncaton error and T ε T qt O( h ) = + =. Rearrangng Eq. (6) we obtan the three term recurrence relaton of the form for = () N-. where Ey Fy + Gy + = H, (7) ε p p E = + + p + q h h ε h pp pq + + p + q ε ε ε p F = q + + p 6 + q h ε h pq + q ε ε p p G = p + q + h h ε h pp pq + + p + q ε ε h pf = + + H f f ε Ths gves us the tr-dagonal system whch can easly be solved by Thomas Algorthm. 3. Thomas Algorthm We gve a bref descrpton of Thomas algorthm for solvng the tr-dagonal system. Consder the system: Ey Fy + Gy + = H (8) =,,..., N subject to the boundary condtons We set y 0= y(0) = ϕ0 (9) y N = y() = γ (0) y = Wy + + T () for = N, N,...,,, where W = W( ) and T = T( ) whch are to be determned. From (), we have: y = Wy + T () By substtutng () n (8) and comparng wth () we get the recurrence relatons: G W = F EW ET H T = F EW (3) () To solve these recurrence relatons for =,,..., N, we need the ntal condtons for W 0 and T 0. For ths we take y 0 = ϕ 0 = W 0 y + T 0. We choose W 0 = 0 so that the value of T 0 = ϕ 0. Wth these ntal values, we compute W and T for =,,... N from (3) and () n forward process, and then obtan y n the backward process from (0) and ().. Error Analyss In ths secton we dscuss the error analyss of the method. Wrtng the tr-dagonal system (8) n matrvector form, we get AY where, A ( m ),, j N- = C (6) = j s a tr dagonal matr of order N-, wth 3 h h p h m+ = ε + p + p + q + ε pp pq + + p + q ε ε h p m = ε hq+ + p 6 + q ε h pq + q ε 3 h h p h m = ε p + p + q ε pp pq + + p + q ε ε ( d ) and C = s a column vector wth h pf d = h f + f, where,,..., - ε = N wth local truncaton error where We also have T 6 7 ( h) = h K+ oh ( ) (7) p () K = - + p + q y ε pp pq + + p 7 + q y ε ε ` AY Th ( ) = C (8) T where Y = y0, y, y,, y N denotes the actual soluton and

4 Amercan Journal of Numercal Analyss T h = T T h T h TN h s the local truncaton error. From (6) and (8), we get ( ) ( ( ), ( ),, ( )) A Y Y = Th ( ) Thus, we obtan the error equaton (9) AE = T ( h) (30) T where E = Y Y = ( e0, e, e,, e N ). Let S be the sum of elements of the th row of A, then we have 3 h h p pq h S = ε + p + + p 6 + q + ε ε pp h pq 5 + p + q q + O( h ) for = ε ε h pq 3 S = hq q + Oh ( ) for =,3,..N- ε 3 h h h S = ε p + p + p hq 6 ε ( ) pp + pq for N- + p q + Oh = ε ε ( d ) and C = s a column vector wth h pf d = h f + f, where,... - ε = N wth local truncaton error 6 p () T ( h ) = h - + p + q y ε Snce 0 ε pp pq p ( ) 7 + q y + oh ε ε < << and δ o( ε) =, for suffcently small h the matr A s rreducble and monotone Ref. [8]. Then t follows that A ests and ts elements are non negatve. Hence, from Eq. (30), we get and E= A Th ( ) (3) ( ). (3) E A T h th Let m k, be the ( k, ) element of A mk, 0, from the theory of matrces we have Therefore,. Snce N mk, S =, k =,,... N- (33) = We N - _ mk, = (3) = mn S h q N defne ( ) ma ( ) T h = T h. N From (3), (7), (8) and (30), we get whch mples where N A = ma mk, and k N = N ej = mk. T( h), j =,,3.. N- = 6 h K h K ej =, j =,... N- (35) h q B 0 p () K = + p + q y ε. pp pq + + p 7 + q y ε ε Therefore, E = o( h ) Hence, the method gves a fourth order convergence for unform mesh as gven n Ref. [9]. 5. Numercal Eamples To valdate the computatonal effcency of the scheme, we have appled t to two cases of the problem ()-(,3). These cases have been chosen because they have been wdely dscussed n lterature and because appromate solutons are avalable for comparson. ( η ) ( ) ε y ( ) + ay ( ) ( δ) + cy ( ) + by ( ) + = f( ) (0,) and subject to the condtons y ( ) = ϕ( ), on -δ 0 y ( ) = γ ( ), on + η The eact soluton of such boundary value problems havng constant coeffcents (.e. a ( ) = a, b ( ) = b, c ( ) = c, f( ) = f, ϕ( ) = ϕ and γ( ) = γ are constants) s gven by: y ( ) = ([( abc) ep( m) ] ep ( m) [ ( abc) ep( m) ] ep ( m)) / [( abc) ep( m) ] (36) ep ( m) [( a+ b+ c) ( ep( m) ep( m) )] + /( a+ b+ c) where ( ) ( ) aδ bη + bηaδ ε ( a+ b+ c) m = ε

5 Amercan Journal of Numercal Analyss ( ) ( ) aδ bη bηaδ ε ( a+ b+ c) m = Eample : Consder the model problem gven by equatons ()-(3) wth ε a( ) =, b( ) =, c( ) =, f( ) =, ϕ( ) =, γ( ) = 0. The eact soluton of the problem s gven by Eq. (36). The numercal results are gven n Table -Table for ε =0., 0.0 and dfferent values of the delay and advance parameters. Table. Numercal soluton of Eample for ε = 0. ; δ = 0.07 and N=00 η = 0.00 η = 0.03 η = Table. Numercal soluton of Eample for ε = 0. ; η = 0.05 and N=00 δ = 0.00 δ = 0.03 δ = Table 3. Numercal soluton of Eample for ε = 0.0 ; η = and N=00 δ = δ = δ =

6 Amercan Journal of Numercal Analyss 3 Table. Numercal soluton of Eample for ε = 0.0 ; δ = and N=00 η = η = η = Eample : Consder the model problem gven by equatons ()-(3) wth a( ) = 0.5, b( ) = 0.5, c( ) =, f( ) =, ϕ( ) =, γ( ) = 0. The eact soluton of the problem s gven by Eq. (36). The numercal results are gven n Table 5- Table 8 for ε =0., 0.0 and dfferent values of the delay and advance parameters. δ = and N=00 Table 5. Numercal soluton of Eample for ε = 0. ; 0.07 η = 0.00 η = 0.03 η = Table 6. Numercal soluton of Eample for ε = 0. ; η = 0.05 and N=00 δ = 0.00 δ = 0.03 δ =

7 Amercan Journal of Numercal Analyss Table 7. Numercal soluton of Eample for ε = 0.0 ; η = and N=00 δ = δ = δ = Table 8. Numercal soluton of Eample for ε = 0.0 ; δ = and N=00 η = η = η = Fgure. Numercal soluton of Eample for ε = 0. and δ = 0.07

8 Amercan Journal of Numercal Analyss 5 6. Dscusson and Concluson In ths paper, we have dscussed the soluton of sngularly perturbed dfferental-dfference equatons ehbtng dual layer usng the hgher order fnte dfferences. Frst, the second order sngularly perturbed dfferental-dfference equatons s replaced by an asymptotcally equvalent second order sngular perturbed ordnary dfferental equaton. Then, fourth order stable fnte dfference scheme s appled to get a three term recurrence relaton whch s easly solved by Thomas algorthm. The error bound and convergence of the method have also been establshed. Some model eamples (.e., eamples n whch both the negatve and postve shfts are non zero) have been solved to demonstrate the applcablty of the proposed approach, by takng varous values of the delay, advanced and perturbaton parameters. These cases have been chosen because they have been wdely dscussed n lterature Ref. [0,,] and also because the eact solutons are avalable for comparson. The numercal solutons are presented n tables and compared wth the eact soluton. To analyze the effect of the parameters on the soluton, the numercal soluton has also been plotted usng graphs. From the graphs, t s observed that by varyng ether δ or η the thckness of the left boundary layer decreases and that of the rght boundary layer ncreases. The effect of negatve shft s more domnant than the postve shft. Fgure. Numercal soluton of Eample for ε = 0. and η = 0.05 Fgure 3. Numercal soluton of Eample for ε = 0.0 and δ = 0.007

9 6 Amercan Journal of Numercal Analyss Fgure. Numercal soluton of Eample for ε = 0. and δ = 0.07 Fgure 5. Numercal soluton of Eample for ε = 0. and η = 0.05 Fgure 6. Numercal soluton of Eample for ε = 0.0 and η = 0.007

10 Amercan Journal of Numercal Analyss 7 References [] Bellman and R. K. L. Cooke, Dfferental-Dfference Equatons, Academc Press, New York, 963. [] Derstne, M. W., Gbbs, H. M., Hopf, F. A. and Kalplan, D. L.: Bfurcaton gap n hybrd optcal system, Phys. Rev. A6(98) [3] Kolmanovsk, V. and Myshks, A.:Appled Theory of Functonal Dfferental Equatons, Volume 85 of Mathematcs and ts Applcatons(Sovet Seres), Kluwer Academc Publshers Group, Dordrecht, 99. [] H. C. Tuckwell and W. Rchter, Neuronal nter-spke tme dstrbutons and the estmaton of neuro-physologcal and neuroanatomcal parameters, J. Theor. Bol., 7 (978) [5] Sten R.B., A theoretcal analyss of neuronal varablty, Bophys. J., 5 (965) [6] C.G. Lange R.M. Mura, Sngular perturbaton analyss of boundary-value problems for dfferental-dfference equatons. II. Rapd oscllatons and resonances, SIAM J. Appl. Math., 5 (985) [7] C.G. Lange R.M. Mura, Sngular perturbaton analyss of boundary- value problems for dfferental-dfference equatons. III. Turnng pont problems, SIAM J. Appl. Math., 5 (985) [8] C.G. Lange R.M. Mura, Sngular perturbaton analyss of boundary-value problems for dfferental-dfference equatons. V. Small shfts wth layer behavour, SIAM J. Appl. Math., 5 (99) 9-7. [9] C.G. Lange R.M. Mura, Sngular perturbaton analyss of boundary-value problems for dfferental-dfference equatons. VI. Small shfts wth rapd oscllatons, SIAM J. Appl. Math., 5 (99) [0] C.G. Lange R.M. Mura, Sngular perturbaton analyss of boundary-value problems for dfferental dfference equatons, SIAM J. Appl. Math., (98) [] M. K. Kadalbajoo and K. K. Sharma, Numercal analyss of boundary-value problems for sngularly perturbed dfferentaldfference equatons wth small shfts of med type, J. Optm. Theory Appl., 5 () (00) [] M. K. Kadalbajoo and K. K. Sharma, Numercal treatment of a mathematcal model arsng from a model of neuronal varablty, J. Math. Anal. Appl., 307 (005) [3] Sharma, K. K. and Kaushk, A.: A soluton of the dscrepancy occurs due to usng the ftted mesh approach rather than to the ftted operator for solvng sngularly perturbed dfferental equatons, Appl. Math. Comput., 8(006) [] Amralyeva, I. G., and Cmen, E: Numercal method for a sngularly perturbed convecton-dffuson problem wth delay, Appled Mathematcs and Computaton, 6(00) [5] Pratma, R. and Sharma, K. K.: Numercal method for sngularly perturbed dfferental-dfference equatons wth turnng pont, Internatonal Journal of Pure and Appled Mathematcs, 73() (0) [6] Chakarvarthy, P. P. and Rao R. N.: A modfed Numerov method for solvng sngularly perturbed dfferental-dfference equatons arsng n scence and engneerng, Results n Physcs, (0) [7] L. E. El sgolts and S. B. Norkn, Introducton to the Theory and Applcatons of Dfferental Equatons wth Devatng Arguments, Academc Press, New York, 973. [8] R. K. Mohanty and N. Jha, A class of varable mesh splne n compresson methods for sngularly perturbed two pont sngular boundary value problems, Appled Mathematcs and Computaton, 68 (005) [9] J. Y. Choo and D. H.Schultz: Stable hgher order methods for dfferental equatons wth small coeffcents for the second order terms, Journal of Computers and Math. wth Applcatons, 5 (993), [0] Pratma, R. and Sharma, K. K.: Numercal study of sngularly perturbed dfferental dfference equaton arsng n the modellng of neuronal varablty, Computers and Mathematcs wth Applcatons, 63 (0) 8-3. [] Rao, R. N. and Chakravarthy, P. P.: An ntal value technque for sngularly perturbed dfferental-dfference equatons wth a small negatve shft, J. Appl. Math. & Informatcs, 3 (03) 3-5. [] Gemechs FD and Reddy Y.N.: A Non-Asymptotc Method for Sngularly Perturbed Delay Dfferental Equatons, Journal of Appled Mathematcs and Informatcs, 3:(0)