Numerical Solution of Singular Perturbation Problems Via Deviating Argument and Exponential Fitting

Transcription

1 Amercan Journal of Computatonal and Appled Matematcs 0, (): DOI: 0.593/j.ajcam umercal Soluton of Sngular Perturbaton Problems Va Devatng Argument and Eponental Fttng GBSL. Soujanya, Y.. Reddy, K. Paneendra,* Department Matematcs, atonal Insttute of Tecnology, Warangal, , Inda Department Matematcs, Kakatya Insttute of Tecnology and Scence, Warangal, 50605, Inda Abstract In ts paper, an eponental ftted metod s presented for solvng sngularly perturbed two-pont boundary value problems wt te boundary layer at one end (left or rgt) pont va devatng argument. Te orgnal second order boundary value problem s transformed to frst order dfferental equaton wt a small devatng argument. Ts problem s solved effcently by usng eponental fttng and dscrete nvarant mbeddng metod. Mamum absolute errors of several standard eamples are presented to support te metod. Keywords Sngular Perturbaton Problems, Boundary Layer, Devatng Argument, Lnear Appromaton, Mamum Absolute Errors. Introducton umercal soluton of boundary value problems n sngularly perturbed second-order, generally nonlnear, ordnary dfferental equatons, s an establsed researc area. Many numercal metods ave been proposed for te soluton of sngularly perturbed two pont boundary value problems. It s well known tat te standard centered dfference sceme as order O, but t gves a non-pyscal oscllaton n te numercal soluton wen appled on a coarse mes because of a boundary layer. In order to remove ts non-pyscal oscllaton penomena, t s necessary to use suffcently small step sze compared to, teoretcally. But t s not practcal to use fner mes tan n real applcaton wen s very small. Adaptve metods present one possble approac. Te adaptve metods strve to automatcally concentrate computatonal grds wtn boundary layers, nteror layers, or oter knds of local soluton structures typcal of te sngularly perturbed ODE s. For te analyss of ts type of problems te readers can refer to te books by Bender and Orszag[], O Malley[8], ayfe [7], Doolan[3] and Roos[4]. Bran J. McCartn[] presented eponental ftted scemes for te numercal appromaton of delayed recrutment/ renewal equatons. Fev Erdogan[4] presented an eponentally ftted dfference sceme for sngularly perturbed ntal value problem for lnear frst-order delay dfferental equaton. Moan K. Kadalbajoo, Vkas Gupta[5] presented a bref survey on numercal metods for solvng sngularly * Correspondng autor: kollojupaneendra@yaoo.co.n (K. Paneendra) Publsed onlne at ttp://journal.sapub.org/ajcam Copyrgt 0 Scentfc & Academc Publsng. All Rgts Reserved perturbed problems. atesan and Bawa[6] constructed a ybrd numercal sceme on a pece-wse unform Sskn mes consstng of cubc splne sceme n te boundary layer regons and te classcal fnte dfference sceme n te regular regons, for solvng sngular perturbaton problems. J.I. Ramos[9] presented eponentally-ftted metods on layer-adapted meses based on te freezng of te coeffcents of te dfferental equaton and ntegraton of te resultng equatons subject to contnuty and smootness condtons at nodes. Rao and Kumar[0] presented an eponental B-splne collocaton metod for self-adjont sngularly perturbed boundary value problem and te metod s sown to ave second order unform convergence. Rasdna et al.[] used splne n compresson to develop te numercal metods for sngularly perturbed two-pont boundary-value problem. Tey dscussed te convergence analyss of te metod and sow tat proposed metods are second-order and fourt order accurate and applcable to problems bot n sngular and non-sngular cases. Reddy [] as dscussed te numercal soluton of sngular perturbaton problems va devatng arguments. Reddy and Cakravarty[3] constructed an eponentally ftted fnte dfference metod for solvng sngularly perturbed two-pont boundary value problem. A fttng factor s ntroduced n a trdagonal fnte dfference sceme and s obtaned from te teory of sngular perturbatons. In ts paper, we present an eponental ftted metod for solvng sngularly perturbed two-pont boundary value problems wt te boundary layer at one end (left or rgt) pont va devatng argument. Te orgnal second order boundary value problem s transformed to frst order dfferental equaton wt a small devatng argument. Ts problem s solved effcently by usng eponental fttng and dscrete nvarant mbeddng metod. umercal results and

2 50 GBSL. Soujanya et al.: umercal Soluton of Sngular Perturbaton Problems Va Devatng Argument and Eponental Fttng mamum absolute errors of several standard eamples are presented to support te metod.. umercal Metod.. Left-end Boundary Layer Consder sngularly perturbed lnear two-pont boundary value problems of te form Ly y ay by = f,0 () wt boundary condtons y ( 0) = α (a) and y () = β (b) were 0 < <<, b(), f() are bounded contnuous functons n (0, ), f() > 0 and αβ, are fnte constants. Furter, we assume tat a M > 0 trougout te nterval [0, ], were M s postve constant. Ts assumpton merely mples tat te boundary layer wll be n te negbourood of = 0. By usng Taylor seres epanson about te devatng argument n te negbourood of te pont, we ave y( ) = y y y y = y y y (3) and consequently, equaton () s replaced by te followng frst order dfferental equaton wt a small devatng argument: y = p( ) y( ) q( ) y( ) r( ), for were (4) p = a b q = a f r = a Te transton from equaton () to equaton (4) s admtted, because of te condton tat s small. ow we dvde te nterval [0, ] nto equal subntervals of mes sze =/ so tat =, = 0,,,,. Here, for consoldatons our deas of te metod, we assume tat a(), b() and f() are constants. Hence, p(), q() and r() are constant. Rearrangng equaton (4) as y qy = py ( ) r q e We ten apply an ntegratng factor, producng (as n[]) d e q y e q = py ( ) r d et, ntegratng from to, we obtan q q q q e y e y = e p y( ) d e r d By makng use of lnear appromaton on [, ], wc we nsert nto te above equaton we ave p ( y ) ( ) q q q e y = e y e p ( ) y q e r r d p ( y ) ( ) q q ( I ) y e y e = d p ( ) y q ( I ) e r r d p q ( ) y = e y y( ) e ( ) d p q ( I ) y( ) e ( ) d q ( I ) e r r d After evaluatng te ntegrals n equaton (5), we get e e r p y = e y y q q q r p ( ) e q q q y e y y y q q q p e e = p e q q y q r e e r e q q q q q q y e e e Let A =, B = q q q q q q Ten, we ave p Ap Bp y = e y A y y y Bp ra r y B Te equaton (6) can be wrtten as a trdagonal system of equatons gven by E y F y G y = H, for =,,, n- (7) were Ap E = Ap Bp F = e Bp G = (5) (6)

3 Amercan Journal of Computatonal and Appled Matematcs 0, (): H = ra r B.. Rgt-end Boundary Layer [ ] We assume tat a M< 0trougout te nterval [0, ], were M s negatve constant. Ts assumpton merely mples tat te boundary layer wll be n te negbourood of =. By usng Taylor seres epanson about te devatng argument n te negbourood of te pont, we ave ( ) y y y y y = y y y (8) and consequently, equaton () s replaced by te followng frst order dfferental equaton wt a small devaton argument: for 0 were y = py ( ) qy r, (9) p = a b q = a f r = a Te transton from equaton () to equaton (9) s admtted, because of te condton tat s small. Ts replacement s sgnfcant from te computatonal pont of vew. ow we dvde te nterval [0, ] nto equal subntervals of mes sze =/ so tat =, = 0,,,,. Here, for consoldatons our deas of te metod we assume tat a(), b() and f() are constants. Hence, p(), q() and r() are constant. We ten apply an ntegratng factor n[]) Integratng from q e d e q y e q = py ( ) r d to, we obtan, producng (as q q q q = e y e y e p y( ) d e r d q q q e y e y e p y ( ) = p y ( ) q e r r d p q ( ) p y ( I ) ( q ) e ( ) d q ( ) e r r d y = e y y( ) e ( ) d (0) After evaluatng te ntegrals n equaton (0), we get e e r p y = e y y ( ) q q q r p( ) e q q q p e e y = e y y y q q q p e y y q q q r e e r e q q q q q q e e e Let A =, B = q q q q q q Ten, we ave p Ap y = e y A y y () Bp Bp r A r y y B Te equaton () can be wrtten as a trdagonal system of equatons gven by E y F y G y = H, for =,,, n- () Ap E =e Ap Bp F = e Bp G = were H = [ r A rb ] Te trdagonal system of equatons (7) and () olds for =,,,-, we ave - lnear equatons n te - unknowns y, y,..., y. We assume te matr of ts set of lnear equatons as A. Lemma: for all > 0 and all =/ te matr A s an rreducble and dagonally domnant matr. Proof. Clearly, A s a trdagonal matr. Hence, A s rreducble f ts codagonals contan non-zero elements only. It s easly seen tat te codagonals E, G do not vans for all > 0, > 0 and a R. Hence A s rreducble. Snce E, G do not vans for all > 0, > 0 and a R tese epressons are of constant sgn. Ten obv-

4 5 GBSL. Soujanya et al.: umercal Soluton of Sngular Perturbaton Problems Va Devatng Argument and Eponental Fttng ously, E > 0, G > 0. ow n eac row of A, te sum of te two off-dagonal elements less tan or equal to te modulus of te dagonal element. Ts proves te dagonal domnant of A. Under tese condtons te dscrete nvarant mbeddng algortm s stable[0]. 3. umercal Eamples To demonstrate te applcablty of te metod, we ave appled t to tree lnear sngular perturbaton problems wt left-end boundary layer and two lnear problems wt rgt-end boundary layer. Tese eamples ave been cosen because tey ave been wdely dscussed n lterature and because eact solutons are avalable for comparson. We ave also present te mamum absolute errors for te problems. Eample. Consder te followng omogeneous sngular perturbaton problem y y y = 0; [0, ] wt y(0) = and y() =. Te eact soluton s gven by m m m m ] m m y = [( e ) e ( e ) e ] / [ e e ] were m = ( 4 ) / and m = ( 4 ) / Te numercal results are gven n tables and for dfferent values of. Table. umercal results of Eample wt 4 = 0, = 0 X umercal soluton Eact soluton Mamum absolute error = 6.68e-003 Table. umercal results of Eample wt 5 = 0, = 0 numercal soluton Eact soluton Mamum absolute error =.0064e-003 Eample. ow consder te followng non- omogeneous sngular perturbaton problem y ( ) y y = 0; [0,] wt y(0) = 0 and y() =. Clearly ts problem as a boundary layer at = 0. / ( e e ) Te eact soluton s gven by y() = / e e Te numercal results are presented n tables 3 and 4 for dfferent values of. Table 3. umercal results of Eample wt 4 = 0, = 0 numercal soluton Eact soluton Mamum absolute error = e-00

5 Amercan Journal of Computatonal and Appled Matematcs 0, (): Table 4. umercal results of Eample wt 5 = 0, = 0 numercal soluton Eact soluton Mamum absolute error = e-00 Eample 3. Consder te followng sngular perturbaton problem y y = ; [0,] wt y(0) = 0 and y() =.Te eact soluton s gven by y = e ( ) e ( ). Te numercal results are presented n tables 5 and 6 for dfferent values of. Eample 4. Consder te followng sngular perturbaton problem y y = 0; [0, ] wt y (0) = and y () =0. Clearly, ts problem as a boundary layer at =.e., at te rgt end of te underlyng nterval. Table 5. umercal results of Eample 3 wt 4 = 0, = 0 numercal soluton Eact soluton e e Mamum absolute error = e-003 Table 6. umercal results of Eample 3 wt 5 = 0, = 0 umercal soluton Eact soluton e e Mamum absolute error = e-003 / e y = / e Te eact soluton s gven by Te numercal results are presented n tables 7 and 8 for dfferent values of. Table 7. umercal results of Eample 4 wt 4 = 0, = 0 umercal soluton Eact soluton Mamum absolute error = e-003 Table 8. umercal results of Eample 4 wt 5 = 0, = 0 umercal soluton Eact soluton Mamum absolute error = e-003

6 54 GBSL. Soujanya et al.: umercal Soluton of Sngular Perturbaton Problems Va Devatng Argument and Eponental Fttng Eample 5. : ow we consder te followng sngular perturbaton problem y y ( ) y = 0; [0, ] wt y(0) = ep(-()/); and y() = /e. Table 9. umercal results of Eample 5 wt 4 = 0, = 0 umercal soluton Eact soluton Mamum absolute error = e-003 Table 0. umercal results of Eample 5 wt 5 = 0, = 0 umercal soluton Eact soluton Mamum absolute error = e-003 Clearly ts problem as a boundary layer at =. Te eact soluton s gven by ( )( / ) y = e e Te numercal results are presented n tables 9 and 0 for dfferent values of. 4. Dscussons and Conclusons We ave presented a numercal metod for solvng sngularly perturbed two-pont boundary value problems wt te boundary layer at one end (left or rgt) pont va devatng argument and an eponental ftted metod. Te orgnal second order boundary value problem s transformed to frst order dfferental equaton wt a small devatng argument. We obtaned te trdagonal system by usng eponental fttng and lnear appromaton and dscrete nvarant mbeddng metod s used to solve te system. Te metod s very easy to mplement. umercal results and mamum absolute errors of standard eamples cosen from te lterature are presented to support te metod. REFERECES [] Bender, C.M. and Orszag, S.A., Advanced Matematcal Metods for Scentsts and Engneers, McGraw-Hll, ew York, 978. [] Bran J. McCartn, "Eponental fttng of te delayed re-crutment/renewal equaton", Journal of Computatonal and Appled Matematcs, vol.36, , 00. [3] Doolan, E.P., Mller, J.J.H. and Sclders, W.H.A., Unform umercal Metods for problems wt Intal and Boundary Layers, Boole Press, Dubln, 980. [4] Fevz Erdogan, An Eponentally-ftted metod for sngularly perturbed delay dfferental equatons, Advances n Dfference Equatons, volume 009, Artcle ID 78579, 9 pages, 009. [5] Kadalbajoo, M. K., Vkas Gupta., "A bref survey on numercal metods for solvng sngularly perturbed problems", Appled Matematcs and Computaton, vol.7, , 00. [6] atesan, S. and Bawa, R.K., "Second-order numercal sceme for sngularly perturbed reacton dffuson robn problems", JAIAM J. umer. Anal. Ind. Appl. Mat. vol., 77 9, 007. [7] ayfe, A.H., Problems n Perturbaton, Wley, ew York,985. [8] O Malley, R.E., Introducton to Sngular Perturbatons,Academc Press, ew York, 974. [9] Ramos, J.I., Eponentally-ftted metods on layer-adapted meses, Appled Matematcs and Computaton, vol. 67,3-330, 005. [0] Rao, S.C.S. and Kumar, M., "Eponental b-splne collocaton metod for self-adjont sngularly perturbed boundary value problems", Appl. umer. Mat. vol. 58, 57 58, 008. [] Rasdna, J, Gasem, M, and Mamood, Z, "Splne approac to te soluton of a sngularly-perturbed boun-dary-value problems", Appl. Mat. Comput. vol. 89, 7 78, 007. [] Reddy, Y.., umercal Treatment of Sngularly Perturbed two pont boundary value problems, P.D. Tess, IIT, Kanpur, Inda, 986. [3] Reddy, Y.. and Pramod Cakravarty, P., "An eponen-tally ftted fnte dfference metod for sngular perturbaton problems", Appl. Mat. Comput. vol. 54, 83 0, 004. [4] Roos, G., Stynes, M and Tobska, L., umercal metods for sngularly perturbed dfferental equatons, Sprnger Verlag, Berln, 996.