Solving Singularly Perturbed Differential Difference Equations via Fitted Method

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1 Avalable at ttp://pvamu.edu/aam Appl. Appl. Mat. ISSN: Vol. 8, Issue 1 (June 013), pp Applcatons and Appled Matematcs: An Internatonal Journal (AAM) Solvng Sngularly Perturbed Dfferental Dfference Equatons va Ftted Metod Awoke Andarge Department of Matematcs Bar Dar Unversty Bar Dar, P.O. Box 53, Etopa awoke48@yaoo.com Y.N. Reddy Department of Matematcs Natonal Insttute of Tecnology Warangal, A.P., Inda ynreddy@ntw.ac.n Receved: July 30, 01; Accepted: June 3, 013 Abstract In ts paper, we presented a ftted approac to solve sngularly perturbed dfferental dfference equatons of second order wt boundary at one end (left or rgt) of te nterval. In ts approac, wt te elp of Taylor seres expanson, we approxmated te terms contanng negatve and postve sfts and modfed te sngularly perturbed dfferental dfference equaton to sngularly perturbed dfferental equaton. A fttng parameter n te coeffcent of te gest order dervatve of te new equaton s ntroduced and determned ts value from te teory of sngular perturbaton. Fnally, we obtaned a tree term recurrence relaton wc s solved usng Tomas Algortm. Te applcablty of te metod s tested wt sx lnear problems. Keywords: Ftted, Dfferental-Dfference, Sngular Perturbaton, Boundary Layer MSC 010 No.: 34K6, 65L10, 65L11 1. Introducton Te problems n wc te gest order dervatve term s multpled by a small parameter are known to be perturbed problems and te parameter s known as te perturbaton parameter. A sngularly perturbed dfferental-dfference equaton s an ordnary dfferental equaton n wc 318

2 AAM: Intern. J., Vol. 8, Issue 1 (June 013) 319 te gest dervatve s multpled by a small parameter and nvolvng at least one delay or advance term. Recently by constructng a specal type of mes, so tat te term contanng delay les on nodal ponts after dscretzaton, Rao and Cakravarty (011) presented a fourt order fnte dfference metod for solvng sngularly perturbed dfferental dfference equatons. Prasad and Reddy (01) consdered dfferental quadrature metod for fndng te numercal soluton of boundary-value problems for a sngularly perturbed dfferental-dfference equaton of mxed type. In te recent studes conducted by Kadalbajoo and Sarma (004, 008), Kadalbajoo and Rames (007) and Kadalbajoo and Kumar (008, 010), te terms negatve or left sft and postve or rgt sft ave been used for delay and advance respectvely. Te dfferental-dfference equaton plays an mportant role n te matematcal modelng of varous practcal penomena n te boscences and control teory. Any system nvolvng a feedback control wll almost always nvolve tme delays. Tese arse because a fnte tme s requred to sense nformaton and ten react to t. For a detaled dscusson on dfferentaldfference equaton, one may refer to Bellen and Zennaro (003), Drver (1977), Bellman and Cooke (1963) books and g level monograps. In El sgolt s (1964), smlar boundary value problems wt solutons tat exbt rapd oscllatons are studed. Kadalbajoo and Sarma (004, 008), Kadalbajoo and Rames (007) and Kadalbajoo and Kumar (008, 010) ntated an extensve numercal work for solvng sngularly perturbed delay dfferental equatons based on fnte dfference sceme, ftted mes and B-splne tecnque, pecewse unform mes. It s well known tat te classcal metods fal to provde relable numercal results for suc problems (n te sense tat te parameter and te mes sze cannot vary ndependently). Lange and Mura (1985, 1994) gave asymptotc approaces n te study of class of boundary value problems for lnear second order dfferental dfference equatons n wc te gest order dervatve s multpled by small parameter. Tey ave also dscussed te effect of small sfts on te oscllatory soluton of te problem. In ts paper, we presented a ftted approac to solve sngularly perturbed dfferental dfference equatons of second order wt boundary at end (left or rgt) of te nterval. In ts approac, te terms contanng negatve and postve sfts are approxmated usng Taylor seres expanson and te sngularly perturbed dfferental dfference equaton s modfed get sngularly perturbed dfferental equaton. A fttng parameter n te coeffcent of te gest order dervatve of te new equaton s ntroduced and determned ts value from te teory of sngular perturbaton. Fnally, we obtaned a tree term recurrence relaton wc s solved usng Tomas Algortm. Te applcablty of te metod s tested by consderng sx lnear problems (tree for left layer and tree for rgt layer).. Descrpton of te Metod To descrbe te metod, we frst consder a lnear sngularly perturbed dfferental dfference two-pont boundary value problem of te form: y( x) pxy ( ) ( x) qxyx ( ) ( ) rxyx ( ) ( ) sxyx ( ) ( ) f( x), 0 x 1. (1)

3 30 Awoke Andarge and Y.N. Reddy Under te nterval and boundary condtons and yx () (), x on x 0, (a) yx ( ) ( x), on 1 x1, (b) were, px (),(),(),(),(), qx rx sx x () x and f() x are suffcently smoot functons, te sngular perturbaton parameter s a small postve parameter (0<<<1), and 0 o( ),0 o( ) are respectvely, te delay (negatve sft) and te advance (postve sft) parameters. Te soluton of (1) and () exbts, layer at te left end of te nterval f px ( ) qx ( ) sx ( ) 0 and layer at te rgt end of te nterval f px ( ) qx ( ) sx ( ) 0. If px ( ) 0, ten one may ave oscllatory soluton or two layers (one at eac end) dependng upon te cases weter qx ( ) rx ( ) sx ( ) s postve or negatve. Approxmatng yx ( ) and yx ( ) by te Taylor seres expanson, we ave yx ( ) yx ( ) y( x), (3a) yx ( ) yx ( ) y( x). (3b) Substtutng equaton (3) n to equaton (1), we get y( x) a( x) y( x) b( x) y( x) f( x), (4) ax ( ) px ( ) qx ( ) sx ( ), (5) bx ( ) qx ( ) rx ( ) sx ( ). (6) From te teory of sngular perturbatons t s known tat te soluton of (4) and () s of te form (O Malley (1974), pp. -6) x a( x) b( x) dx a(0) a( x) 0 ( ) 0( ) ( (0) 0(0)) ( ) yx y x y e O, (7) ax ( ) were y 0 ( x ) s te soluton of a( x) y 0( x) b( x) y0( x) f( x), y 0 (1) (1). By takng te Taylor s seres expanson for ax ( ) and bx ( ) about te pont 0 and restrctng to ter frst terms, (7) becomes, a(0) b(0) x a(0) a(0) yx ( ) y0( x) ( (0) y0(0)) e O( ). (8) ax ( )

4 AAM: Intern. J., Vol. 8, Issue 1 (June 013) 31 Now, we dvde te nterval [0, 1] nto N equal parts wt constant mes lengt. Let x, x,..., x 1be te mes ponts. Ten we ave x ; 0,1,..., N N From (8) we ave and a (0) b(0)) a(0) a(0) y ( ) y0( ) ( (0) y0(0)) e, (9) a ( ) a (0) b(0) a(0) lm y ( ) y(0) ( (0) y(0)) e,.1. Dervaton of te Specal Second Order Sceme for (4) Consder a typcal pvotal pont n te mes, at x x wrtten for y, y and y :. (10) g. Te followng expresson can be y g g y x g) E y( x ), (11a) ( y g Dy g y D g y g, (11b). (11c) Te sft operator D E e (11d) can be related to te central dfference operators, by usng te followng expressons D 5..., 6 30 (11e) D 6..., 1 90 (11f) D..., (11g) D , (11) 6 were, and denote te usual central dfference operators and E and D denote te sft (dsplacement) and te dfferental operators, respectvely. By substtutng (11d)-(11) nto (11a)-(11c), we get

5 3 Awoke Andarge and Y.N. Reddy g 1 3 g ( g 1) 4 yg [1 g g( g 1)...)] y, (1) (3g 1) 3 y g [ g g(g 1)...] y, (13) g 1 4 y g [ g g(g 3)...] y. (14) 1 6 By substtutng equatons (1-14) wt g 1/ nto equaton (4) we get: a 3y 6y - y ( - ) ( - ) ( ), 8 1,,..., N -1 1/ 1-1 y 1 y y-1 y 1 y b 1/ f 1/ (15) For furter dscusson on te specal second order sceme (one can see ()). Now, we ntroduce a fttng factor () n te above sceme (15) ( ) a 1/ 3y 1 6y - y-1 ( y 1- y y-1) ( y 1- y) b 1/( ) f 1/, 8. (16) 1,,..., N -1 Te fttng factor () s to be determned n suc a way tat te soluton of (16) converges unformly to te soluton of (1)-(). Multplyng (16) by and takng te lmt as, 0, we get lm y ( ) y ( ) y ( ) a ( /) y ( ) y ( ) 0 0. (17) By substtutng (10) n (17) and smplfyng, we get te constant fttng factor a (0) b(0) ( ) a(0) a(0) [1 e ] 4 a (0) b(0) [sn(( ) / )] a(0). (18) Te equvalent tree-term recurrence relaton of equaton (16) s gven by: Ey - Fy Gy H, 1,,..., N-1, (19) -1 1 were b 1/ a 1/ 6b 1/ a 1/ 3b 1/ E, F,, G H f 1/

6 AAM: Intern. J., Vol. 8, Issue 1 (June 013) 33 Ts gves us te tr dagonal system wc can be solved easly by Tomas Algortm... Tomas Algortm A bref dscusson on solvng te tree term recurrence relaton usng Tomas Algortm, wc also called Dscrete Invarant Imbeddng (Bellman and Cooke (1963)), s presented as follows: Consder te sceme gven n (19).e., Ey -1 - Fy Gy 1 H, 1,,..., N 1 subject to te boundary condtons We set y 0 y(0), y y(1). (0a) N y Wy 1 T, N1, N,...,,1, (0b) were, W = W(x ) and T = T(x ) wc are to be determned. From (0b), we ave y1. (0c) W 1y T 1 By substtutng (0c) n (19), we get y G ET -1 - H y1. (0d) F -EW -1 F -EW -1 By comparng (0d) and (0b), we get te recurrence relatons G W, (0e) F - EW -1 and ET 1 H T. (0f) F EW 1 To solve tese recurrence relatons for = 0, 1,, N 1, we need te ntal condtons for W 0 and T 0. For ts we take y0 W0 y1 T0. We coose W 0 = 0 so tat te value of T 0. Wt tese ntal values, we compute W and T for = 0, 1,, N 1, from (0e) and (0f) n forward process, and ten obtan y n te backward process from (0b) and (0a). Te condtons for te dscrete nvarant embeddng algortm to be stable are (Andarge and Reddy (007) and Bellman and Cooke (1963)):

7 34 Awoke Andarge and Y.N. Reddy E 0, F 0, F E G and E G. (0g) In ts metod, f te assumptons px ( ) qx ( ) sx ( ) 0, qx ( ) rx ( ) sx ( ) 0 and 0 old, one can easly sow tat te condtons gven n (16g) old and, tus, te nvarant mbeddng algortm are stable. 3. Rgt Layer Problems For problems wt layer at te rgt end of te nterval, from te teory of sngular perturbatons t s known tat te soluton of (4) and () s of te form (O Malley (1974)) x a( x) b( x) dx a(1) a( x) 1 ( ) 0( ) ( (1) 0(1)) ( ) yx y x y e O, (1) ax ( ) were y ( x) s te soluton of a( x) y 0 0( x) b( x) y0( x) f ( x), y 0 (0) (0). By applyng Taylor s seres expanson on a(x) and b (x) about te pont 1 and restrctng to ter frst terms, (1) becomes, 0 0 a(1) b(1) - (1- x) a(1) yx ( ) y( x) ( (1)- y(1)) e O( ). () Now, we dvde te nterval [0, 1] nto N equal parts wt constant mes lengt. Let x, x,..., x 1be te mes ponts. Ten, we ave x ; 0,1,..., N. From () we ave N 0 0 a(1) b(1) - (1- ) a(1) y ( ) y( ) ( (1)- y(1)) e O( ), (3) and a (1) b(1) 1 a(1) lm y ( ) y(0) ( (1) y(1)) e,. (4) By substtutng equatons (1)-(14) wt g = -1/ nto equaton (4) and ntroducng te fttng factor (), we get: a-1/ -y 1 6y 3y-1 ( y 1- y y-1) ( y - y-1) b-1/( ) f-1/, 8. (5) 1,,..., N -1 Multplyng (5) by and takng te lmt as 0, we get

8 AAM: Intern. J., Vol. 8, Issue 1 (June 013) 35 ( ) lm y( )- y( ) y( - ) a( - /) y( )- y(( -1) ) 0 0. (6) Usng (4) n (6) and smplfyng, we get a(0) 4 a (1) b(1) ( ) a(1) [ e 1] a(1) b(1) [sn(( ) )] a(1). (7) An equvalent tree-term recurrence relaton for equaton (5) s: Ey - Fy Gy H, 1,,..., N 1, (8) -1 1 were a 1/ 3b 1/ a 1/ 6b 1/ b 1/ E -, F - -, -, G H f 1/ Numercal Examples To demonstrate te applcablty of te metod, we consdered sx lnear problems (tree wt left layer and tree wt rgt layer). We presented te tables of te numercal results along wt te exact soluton. Now let us consder te sngularly perturbed dfferental dfference equaton y( x) p( x) y( x) q x y( x) r x y x s x y x f( x), x(0,1), subject to te nterval and boundary condtons yx ( ) x, -x0; yx ( ) x,1x1. Te exact soluton of suc boundary value problems avng constant coeffcents s gven by: y f 1 exp 1 exp, c x c m x c m x

9 36 Awoke Andarge and Y.N. Reddy c qrs, pqs pqs 4c m1, pqs pqs 4c m, f c m f c c1, exp exp m1 expmc exp 1 m expm c f c m f c c, y(0), y(1) exp 1 Example 1: Consder te sngularly perturbed dfferental dfference equaton wt layer at te left end y( x) y( x) y( x ) 3 y( x) 0, and yx ( ) 1, x0, yx ( ) 1, 1 x1. Example : Consder te sngularly perturbed dfferental dfference equaton wt layer at te left end y( x) y( x) 3 yx ( ) yx ( ) 0, and yx ( ) 1, x0, yx ( ) 1, 1 x1. Example 3: Consder te sngularly perturbed dfferental dfference equaton wt layer at te left end y( x) y( x) yx ( ) 5 yx ( ) yx ( ) 0, and yx ( ) 1, x0, yx ( ) 1, 1 x1. Example 4: Consder te sngularly perturbed dfferental dfference equaton wt rgt layer y( x) y( x) yx ( ) yx ( ) 0, and yx ( ) 1, x0, yx ( ) 1, 1 x1. Example 5: Consder te sngularly perturbed dfferental dfference equaton wt rgt layer y( x) y( x) yx ( ) yx ( ) 0, and yx ( ) 1, x0, yx ( ) 1, 1 x1.

10 AAM: Intern. J., Vol. 8, Issue 1 (June 013) 37 Example 6: Consder te sngularly perturbed dfferental dfference equaton wt rgt layer y( x) y( x) yx ( ) yx ( ) yx ( ) 0, and yx ( ) 1, x0, yx ( ) 1, 1 x1. 5. Dscusson and Conclusons We presented a ftted approac to solve sngularly perturbed dfferental equatons of second order wt boundary at one (left or rgt) end of te nterval. In ts approac, te sngularly perturbed dfferental dfference equaton s modfed to a sngularly perturbed dfferental equaton by approxmatng te term contanng small sfts (negatve and postve) usng Taylor seres expanson. In te new sngularly perturbed problem, we ntroduced a fttng parameter and determned ts value usng te teory of sngular Perturbaton. Fnally, we get a tree term recurrence relaton wc s solved usng Tomas algortm. Many metods ave been presented producng good approxmaton n te outer regon of te sngularly perturbed problems. Here, to test te applcablty of ts metod, we consdered sx lnear problems (tree for left layer and tree for rgt layer). We presented tables of values and te pont wse error for meses n te nner regon of te problems by takng dfferent values of te perturbaton parameter, te delay parameter and te advance parameter. Te results demonstrate tat te present metod produced good approxmaton to te exact soluton. Table 1. (a) Absolute Maxmum Error for Example 1 wt n=100, =0.01, =0.5* δ =0.00 δ =0.005 δ =0.009 x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00

11 38 Awoke Andarge and Y.N. Reddy Table 1. (b) Absolute Maxmum Error for Example 1 wt n=100, =0.005, 0.1* δ =0.000 δ = δ = x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table. (a) Absolute Maxmum Error for Example wt n=100, =0.01, δ=0.5*ε =0.00 =0.005 =0.009 x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table. (b) Absolute Maxmum Error for Example wt n=100, =0.005, δ=0.1*ε =0.000 = = x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00

12 AAM: Intern. J., Vol. 8, Issue 1 (June 013) 39 Table 3. (a) Absolute Maxmum Error for Example 3 wt n=100, =0.01 δ =0.000, =0.009 δ =0.005, =0.005 δ =0.009, =0.000 x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-16 Table 3. (b) Absolute Maxmum Error for Example 3 wt n=100, =0.005 δ =0.000, = δ =0.0005, = δ =0.0009, =0.000 x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 4. (a) Absolute Maxmum Error for Example 4 wt n=100, =0.01,δ=0.1*ε,=0.5* δ =0.00 δ =0.005 δ =0.009 x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00

13 330 Awoke Andarge and Y.N. Reddy Table 4. (b) Absolute Maxmum Error for Example 4 wt n=100, =0.005,δ=0.1*ε,=0.1* δ =0.000 δ = δ = x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-16 Table 5. (a) Absolute Maxmum Error for Example 5 wt n=100, =0.01,δ=0.1*ε,δ=0.5* =0.00 =0.005 =0.009 x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 5. (b) Absolute Maxmum Error for Example 4 wt n=100, =0.005,δ=0.1*ε, δ =0.1* =0.000 = = x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00

14 AAM: Intern. J., Vol. 8, Issue 1 (June 013) 331 Table 6. (a) Absolute Maxmum Error for Example 5 wt n=100, =0.01 δ =0.000, =0.009 δ =0.005, =0.005 δ =0.009, =0.000 x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 6. (b) Absolute Maxmum Error for Example 4 wt n=100, =0.005,δ=0.1*ε,=0.1* δ =0.000, = δ =0.0005, = δ =0.0009, =0.000 x() y() Exact Error y() Exact Error y() Exact Error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 REFERENCES Andarge, A., Reddy, Y. N. (007). An exponentally ftted specal second-order fnte dfference metod for solvng sngular perturbaton problems, Appled Matematcs and Computaton Vol. 190, pp Bellen, A., Zennaro, M. (003). Numercal Metods for Delay Dfferental Equatons, Oxford Unversty Press, Oxford. Bellman, R. E., Cooke, K. L. (1963). Dfferental-Dfference Equatons, Academy Press, New York. Drver, R. D. (1977). Ordnary and Delay Dfferental Equatons, Sprnger-Verlag, New York. El sgol ts, L. E. (1964). Qualtatve Metods n Matematcal Ana-lyses, Translatons of Matematcal Monograps 1, Amercan matematcal socety, Provdence, RI. Kadalbajoo, M. K., Sarma, K. K. (004). Numercal analyss of sngularly perturbed delay dfferental equatons wt layer beavor, Appled Matematcs and Computaton Vol. 157, pp Kadalbajoo, M. K., Rames, V.P. (007). Hybrd metod for numercal soluton of sngularly perturbed delay dfferental equatons, Appled Matematcs and Computaton Vol. 187, pp

15 33 Awoke Andarge and Y.N. Reddy Kadalbajoo, M. K., Sarma, K. K. (008). A numercal metod based on fnte dfference for boundary value problems for sngularly perturbed delay dfferental equatons, Appled Matematcs and Computaton Vol. 197, pp Kadalbajoo, M. K., Kumar, D. (008). Ftted mes B-splne collocaton metod for sngularly perturbed dfferental dfference equatons wt small delay, Appled Matematcs and Computaton, Vol. 04, pp Kadalbajoo, M. K., Kumar, D. (010). A computatonal metod for sngularly perturbed nonlnear dfferental-dfference equatons wt small sft, Appled Matematcal Modelng Vol. 34, pp Lange, C. G., Mura, R. M. (1994). Sngular perturbaton analyss of boundary value problems for dfferental dfference equatons, V. Small sfts wt layer beavor, SIAM J. Appl. Mat., Vol. 54, pp Lange, C.G., Mura, R.M. (1994). Sngular Perturbaton Analyss of Boundary Value Problems for dfferental dfference equatons, (VI), Small sfts wt Rapd Oscllatons, SIAM J. Appl. Mat., Vol. 54, pp Lange, C.G., Mura, R.M. (1985). Sngular perturbaton analyss of boundary value problems for dfferental dfference equatons, SIAM Journal of Appl. Mat., Vol.45, pp O Malley, R. E. (1974). Introducton to Sngular Perturbatons, Academc Press, New York. Prasad, H. S., Reddy, Y. N. (01). Numercal Soluton of Sngularly Perturbed Dfferental- Dfference Equatons wt Small Sfts of Mxed Type by Dfferental Quadrature Metod, Amercan Journal of Computatonal and Appled Matematcs 01, Vol., No.1, pp Rao, R. N., Cakravarty, P. P. (011). A Fourt Order Fnte Dfference Metod for Sngularly Perturbed Dfferental-Dfference Equatons, Amercan Journal of Computatonal and Appled Matematcs, Vol. 1, No. 1, pp