Maejo International Journal of Science and Technology

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1 Maejo Int. J. Sc. Technol. () - Full Paper Maejo Internatonal Journal of Scence and Technology ISSN - Avalable onlne at Fourth-order method for sngularly perturbed sngular boundary value problems usng non-polynomal splne Kolloju Phaneendra * Emnen Sva Prasad and Ddd Kumara Swamy Department of Mathematcs Unversty College of Scence Osmana Unversty Hyderabad- Inda Department of Mathematcs Kavulguru Insttute of Technology and Scence Ramte Maharasthra 6 Inda Department of Mathematcs Chrstu Jyoth Insttute of Technology and Scence Warangal Inda * Correspondng author e-mal: ollojuphaneendra@yahoo.co.n Receved: January / Accepted: December / Publshed: January Abstract: Ths paper envsages a fourth-order fnte dfference method wth reference to the soluton of a class of sngularly perturbed sngular boundary value problems especally on a unform mesh. The non-polynomal splne forms the tool for the soluton of the problems. The dscretsaton equaton of the problems are developed usng the condton of contnuty for the frst-order dervatves of the non- polynomal splne at the nteror nodes and t s not vald at the sngularty. Hence at the sngularty the boundary value problem s modfed n order to get a three-term relaton. The trdagonal scheme of the method s processed usng dscrete nvarant mbeddng algorthm. The convergence of the method s analysed and maxmum absolute errors n the soluton are tabulated. Root mean square errors n the soluton of the examples are presented n comparson to the methods chosen from the lterature to establsh the proposed method. Keywords: sngularly perturbed two-pont sngular boundary value problem nteror nodes sngular pont non-polynomal splne boundary layer INTRODUCTION We consder a class of sngularly perturbed two-pont sngular boundary value problems of the form: y ( x) y( x) q( x) y( x) r( x) x () x wth boundary condtons y ( ) and y() ()

2 Maejo Int. J. Sc. Technol. () - where q(x) and r(x) are bounded contnuous functons n ( ) and are fnte constants. Let p( x). If p( x) M throughout the doman [ ] where M s a postve x constant then the boundary layer exsts n the neghbourhood of x =. If p( x) N throughout the nterval [ ] where N s a negatve constant then the boundary layer wll be n the neghbourhood of x =. Ths class of problems frequently occurs n many areas of appled mathematcs such as flud mechancs elastcty quantum mechancs optmal control chemcal-reactor theory aerodynamcs reacton dffuson process geophyscs and many other areas. Equatons of ths type exhbt solutons wth layers; that s the doman of soluton of the problem contans narrow regons where the soluton dervatves are extremely large. The numercal treatment of these problems gves major computatonal dffcultes due to the presence of boundary and/or nteror layers. A wde varety of boos have been publshed descrbng varous methods for solvng sngularly perturbed two-pont boundary value problems. Among these we menton Henre [] O Malley [] Bender and Orszag [] and Kress and Kress []. Bava [] nvestgated a fourth-order dfference scheme va cubc splne n compresson for the soluton of sngular perturbaton problems. Kadalbajoo and Aggarwal [6] proposed a ftted mesh B-splne method for sngular sngularly perturbed boundary value problems. Kadalbajoo and Patdar [] derved some dfference schemes for sngularly perturbed problems usng splne n compresson. Kadalbajoo and Reddy [] have dscussed a numercal method va devatng arguments to solve lnear sngular perturbaton problems. Mohanty et al. [ ] and Mohanty and Aurora [] have establshed varous methods based on tenson splne and compresson splne methods both on a unform and non-unform mesh for sngularly perturbed twopont sngular boundary value problems. Rashdna and Ghasem [] used cubc splne soluton of sngularly perturbed two-pont boundary value problems on a unform mesh. The approach presented n ths paper has the advantage over fnte dfference methods n that t provdes contnuous approxmatons not only for y(x) but also for y y and hgher dervatves at every pont of the range of ntegraton. Also the C - dfferentablty of the trgonometrc part of non-polynomal splnes compensates for the loss of smoothness nherted by polynomal splnes. Besdes a new parameter s ntroduced n ths method to acheve the desred fourth-order convergence for the problems represented by Eq. (). NON-POLYNOMIAL SPLINE METHOD The doman of the ntegraton [a b] s decomposed nto N equal subntervals wth mesh sze h so that x a h N are the nodes wth a x b xn. Let y(x) be the exact N soluton and y be an approxmaton to y ( x ) by the non-polynomal cubc splne S (x) passng through the ponts ( x y ) and ( x y ). Here S (x) satsfes nterpolatory condtons at x x ; also the contnuty of frst dervatve at the common nodes x y ) s fulflled. For each and th subnterval the cubc non-polynomal splne functon S (x) ( has the form: S ( x) a b ( x x ) c sn ( x x ) d cos ( x x ). N () where a b c and d are constants and s a free parameter.

3 Maejo Int. J. Sc. Technol. () - A non-polynomal functon ( ) S x of class C a b nterpolatng y(x) at the grd ponts x =.N depends on a parameter and reduces to ordnary cubc splne n [a b] as. To derve an expresson for the coeffcents of Eq. () n terms of y y M and M the followng are defned: S ( x ) y S ( x ) y '' S ( x ) M S '' ( x ) M Usng algebrac manpulaton the followng expressons are obtaned for the coeffcents: M y a y b M cos M c sn y h d M M where h for =. N-. Usng the contnuty of the frst dervatve at x y ) that s S ( x ) S( x ) we get the followng relatons for =.N- : where y y. M ( M M M () h y cos sn sn NUMERICAL METHOD '' M ( ) and. j y x j j h At the grd ponts x Eq. () may be dscretsed by Usng splne s second dervatves we have y p( x ) y q( x ) y r. M p( x ) y ( x) q( x ) y( x ) r( x ) for j. () j j j j j j Usng Eq.( ) n Eq.() and the followng approxmatons for the frst dervatve of y []: y y we get the trdagonal system: y y y h y y y h y y p p y h h q h[ p p ] E h q h[ p p ] y hr r h y F y G y H for. N (6) () where

4 Maejo Int. J. Sc. Technol. () - E p h ph p p p h q p h q h h p F p h p h p p p h q h G p h ph p p h pq p h q h h p H -h ph r r ph r p( x ) p q( x ) q r( x ) r for. N. For = the coeffcents y y and y n Eq. () are not defned; thus we need to develop a formula for ths case. Usng L-Hosptal rule and Eq. () we get the followng three-term formula for = : α h q β h q α h q h ε ε ε ε y y y α r β r α r. () Usng dscrete nvarant mbeddng algorthm [] the trdagonal system Eq.() together wth Eq.() for = N- s solved n order to get the approxmatons y y. yn of the soluton y(x) at x x. xn. TRUNCATION ERROR The local truncaton error assocated wth the scheme developed n Eq. () s T h h y x p x y x y x h O h 6 ( ) ( ) ( ) ( ) ( ) ( ) ( ). Thus for dfferent values of n the scheme of Eq. () the followng dfferent orders are ndcated: () For any choce of arbtrary and wth and for any value of the scheme of Eq. () gves the second-order method; () For and from Eq. () the fourth-order method s derved. CONVERGENCE ANALYSIS Incorporatng the boundary condtons (Eq. ) the system of Eq. () and () can be wrtten n the matrx form as: PY Q T( h) D ()

5 Maejo Int. J. Sc. Technol. () - where D and * * v w z v w z v w P z v w zn vn * h q * h q n whch v w α z α p h β ph ω p p ω p β h q p h α q h - hβ p v α p h β p h ω p p α p h β q h α w p h β ph ωp p ωh β pq α p h α q h hβ p for. N and h h q Q r r r q q. qn wn 6 wheren q h ph r r ph r for. N T ( h) ( h ) for and Y Y Y. YN T ( h) T T. TN O. are assocated vectors of Eq. ( ). T N Y Let y y y y satsfy the equaton Let e y Y. N Py Q T T T D. () be the dscretsaton error so that. Usng Eq.() from Eq.() we obtan the error equaton: D P E T(h) E e e e y Y. N. () Let p( x) C and q( x) C where C C are postve constants. If j P then h P C for P s the th T j element of P w h( ) C h C h C h CC. N

6 Maejo Int. J. Sc. Technol. () - P z h( ) C h C h C h CC. N. Thus for suffcently small h we have 6 Hence (D + P) s rreducble []. P N and P N. () Let S be the sum of the elements of the th row of the matrx (D + P); then we have h S q q for S h q q q h p q q for. N h S p p h p h q q h p p p h p q for N. * * Let C * mn p( x) C max p( x) and C * mn q( x) C max q( x). Snce N N N N and O( h) t s possble or easy to verfy that for small P exsts and D P. D Thus usng Eq. () we have Let th D P be the ( ) element of D P. h D P s monotone [ ]. Hence E D P T () and we defne N D P max D P and T ( h) max T( h). (a) N Snce hence Furthermore N N D P and D P S for. N D P D P Usng Eq.(a) - (d) from Eq.() we get * S h C (b) N. (c) S h C C * D P (d) mn S h N N E * C. * O( h ). () Hence the method (Eq. ) s fourth-order convergent for.

7 Maejo Int. J. Sc. Technol. () - NUMERICAL EXAMPLES To demonstrate the proposed method computatonally we consder three problems of the type n Eq.(). These problems are chosen because they have been wdely dscussed n the lterature and exact solutons are also avalable for comparson. Example. Consder a sngularly perturbed sngular boundary value problem: y (/ x) y ( x ) y f ( x) x. The exact soluton s y( x) exp( x ). The maxmum absolute errors are tabulated n Table for dfferent values of and h. A comparson of the root mean square errors wth another method s presented n Table. Example. Consder a boundary value problem: y y f ( x) x. x The exact soluton of ths problem s y( x) x snh x. The maxmum absolute errors are presented n Table for dfferent values of and h. A comparson of the root mean square errors s presented n Table. Table. Maxmum absolute errors n solutons of Examples - /h 6 Example.(-) 6.(-).(-).(-6).6(-).6(-).(-).(-6).(-).66(-).(-).(-).(-).(-).6(-).66(-) Example.6(-).6(-).(-).(-6).(-).(-).(-).(-6).(-).(-).(-).(-6).(-).(-).66(-).(-6) Example.(-). (-).6(-) 6.(-).(-). (-). (-).6 (-).6(-).(-).(-).6(-).(-).(-).(-).(-)

8 Maejo Int. J. Sc. Technol. () - Table. Comparson of root mean square errors n soluton of Example 6 /N Mohanty-Arora method [].(-).6(-).(-) 6.(-).(-).(-).(-).6(-) 6.(-).(-).6(-).(-) 6.(-).6(-).(-).(-).(-).(-).(-).(-).(-).(-).(-).(-) -.(-).(-).(-).6(-).(-).(-).(-) -.6(-).(-).6(-).(-).(-).(-).(-) -.(-).(-).(-) 6.(-).(-).6(-).(-) -.(-).(-).(-).(-) 6.6(-).(-).(-) Proposed method.(-).(-).(-6) 6.(-).(-).(-).(-).(-).(-).66(-).(-).(-) 6.(-).6(-).(-).(-).(-).(-).(-).(-6).(-).6(-).(-).(-).(-).(-).(-).(-6).(-).(-).66(-).(-).(-) 6.(-) 6.(-) 6.(-6).(-).(-).(-).(-).(-).(-).(-).(-).6(-) 6.(-).(-).(-).(-).(-).(-).6(-).(-6).(-).(-).6(-) Table. Comparson of root mean square errors n soluton of Example 6 /N Mohanty-Arora method [].(-).(-).(-).(-).6(-).(-).(-).(-).(-).(-).(-).6(-).(-).(-).(-) 6.6(-).(-).(-).(-).(-).(-).(-).(-).(-) 6.6(-).6(-).(-).(-).(-).(-).(-).6(-).(-).(-).(-).(-).(-).(-).(-).(-).(-).(-).(-) 6.(-).(-).(-).(-).6(-).(-).(-).(-) 6.6(-).6(-).(-).(-).6(-).(-).(-).66(-) 6.(-).(-).6(-).(-).6(-) Proposed method 6.(-).(-).(-6).(-).6(-).(-).(-).(-).(-) 6.(-).(-6).6(-).6(-).(-).6(-).(-).(-).(-).(-6).6(-).(-).(-).6(-).(-)

9 Maejo Int. J. Sc. Technol. () - Table. (Contnued) 6 /N.(-).(-).(-).(-6).(-).(-) 6.(-).(-).(-).6(-).(-).(-6).(-).6(-).(-).(-).(-).(-).(-).(-6).(-).6(-).(-).6(-).(-).(-).(-) 6.(-6) 6.(-).(-).(-).(-) Example. Consder a boundary value problem: y y y x x wth boundary condtons y() y() exp whose exact soluton s not nown. The maxmum absolute errors for ths example are calculated by usng the double mesh prncple E N max y N N y N and tabulated n Table for dfferent values of and h. CONCLUSIONS In ths paper the non-polynomal splne method s dscussed for a class of sngularly perturbed sngular two-pont boundary value problems. Convergence of the numercal method s analysed. The maxmum absolute errors n the soluton are tabulated for the exstng standard examples chosen from the lterature wth a vew to demonstratng the method. Root mean square errors n the soluton of the examples are presented wth comparson n order to justfy the method. Based on the numercal results t s observed that the method also affords good results for smaller values of. The proposed method s also extendable to non-sngular problems and sngularly perturbed delay dfferental equatons. REFERENCES. P. Henre Dscrete Varable Methods n Ordnary Dfferental Equatons Wley New Yor 6.. R. E. O Malley Introducton to Sngular Perturbatons Academc Press New Yor.. C. M. Bender and S. A. Orszag Advanced Mathematcal Methods for Scentsts and Engneers McGraw-Hll New Yor.. B. Kress and H.-O. Kress Numercal methods for sngular perturbaton problems SIAM J. Numer. Anal R. K. Bawa Splne based computatonal technque for lnear sngularly perturbed boundary value problems Appl. Math. Comput M. K. Kadalbajoo and V. K. Aggarwal Ftted mesh B-splne method for solvng a class of sngular sngularly perturbed boundary value problems Int. J. Comput. Math. 6-6.

10 Maejo Int. J. Sc. Technol. () -. M. K. Kadalbajoo and K. C. Patdar Numercal soluton of sngularly perturbed two pont boundary value problems by splne n compresson Int. J. Comput. Math M. K. Kadalbajoo Y. N. Reddy Numercal soluton of sngular perturbaton problems va devatng arguments Appl. Math. Comput. -.. R. K. Mohanty N. Jha and D. J. Evans Splne n compresson method for the numercal soluton of sngularly perturbed two-pont sngular boundary-value problems Int. J. Comput. Math R. K. Mohanty D. J. Evans and U. Aurora Convergent splne n tenson methods for sngularly perturbed two-pont sngular boundary value problems Int. J. Comput. Math R. K. Mohanty and U. Aurora A famly of non-unform mesh tenson splne methods for sngularly perturbed two-pont sngular boundary value problems wth sgnfcant frst dervatves Appl. Math. Comput J. Rashdna R. Mohammad M. Ghasem Cubc splne soluton of sngularly perturbed boundary value problems wth sgnfcant frst dervatves Appl. Math. Comput R. S. Varga Matrx Iteratve Analyss Prentce-Hall Englewood Clffs 6.. D. M. Young Iteratve Solutons of Large Lnear Systems Academc Press New Yor. by Maejo Unversty San Sa Chang Ma Thaland. Reproducton s permtted for noncommercal purposes.