1 Introduction We consider a class of singularly perturbed two point singular boundary value problems of the form: k x with boundary conditions

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1 Lakshm Sreesha Ch. Non Standard Fnte Dfference Method for Sngularly Perturbed Sngular wo Pont Boundary Value Problem usng Non Polynomal Splne LAKSHMI SIREESHA CH Department of Mathematcs Unversty College of Scence, Safabad Osmana Unversty, Hyderabad INDIA Abstract: - In ths paper, a non standard fnte dfference method, wth reference to the soluton of a class of sngularly perturbed sngular boundary value problems on a unform mesh, s dscussed. he non-polynomal splne forms the tool for the soluton of the problem. he dscretzed equaton of the problem s 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. he trdagonal scheme of the method s processed usng dscrete nvarant mbeddng algorthm. he convergence of the method s analyzed and mamum absolute errors n the soluton are tabulated. Root mean square errors n the soluton of the eamples are presented n comparson wth the methods chosen from the lterature, to establsh the proposed method. Key-Words: - Sngularly perturbed two pont sngular boundary value problem, Interor nodes, Sngular pont, Non-polynomal splne, Boundary layer Introducton We consder a class of sngularly perturbed two pont sngular boundary value problems of the form: k ε y ( ) = y ( ) + q( ) y( ) + r( ), 0, () wth boundary condtons y(0) = γ and y() = γ () 0< ε, q(), r() are bounded contnuous functons n (0, ), and γ, γ are fnte constants. k Let p ( ) =. If p ( ) M> 0throughout the doman [0, ], M s a postve constant, then the boundary layer est n the neghbourhood of = 0. If p ( ) N< 0throughout the nterval [0, ], N s a negatve constant, then the boundary layer wll be n the neghbourhood of =. he numercal treatment of these problems gves major computatonal dffcultes due to the presence of boundary and/or nteror layers. A wde varety of books and papers have been publshed, descrbng varous methods for solvng sngularly perturbed two-pont boundary value problems, among these, we menton [-, ]. Kadalbajoo and Aggarwal [] proposed a ftted mesh B-splne method for sngular sngularly perturbed boundary value problems. Mohanty et al. [8, 9, 0] establshed varous methods based on tenson splne and compresson splne methods both on a unform and non-unform mesh for sngularly perturbed two pont sngular boundary value problems. J. Rashdna [] used Cubc splne soluton of sngularly perturbed twopont boundary value problems on a unform mesh. he paper s organzed as follows: In secton, the non polynomal splne method s defned. In secton, descrpton of the numercal method s gven. In secton, truncaton error and classfcaton of varous orders of the proposed method are projected. In secton, convergence analyss of the method s dscussed. Fnally, Numercal results and comparson wth other methods are presented n secton 6. Non Polynomal Splne Method Decompose the doman of the ntegraton [a, b] nto N equal subntervals wth mesh sze h =, so that N = a + h, =0,,, N are the nodes wth a=, 0 b= N. Let y() be the eact soluton and y be an appromaton to y ( ) by the non polynomal cubc splne S () passng through the E-ISSN: - 0 Volume, 0

2 Lakshm Sreesha Ch. ponts (, y ) and ( +, y+ ). Here S ( ) satsfes nterpolatory condtons at and +, also the contnuty of frst dervatve at the common nodes (, y ) are fulflled. For each th subnterval, the cubc non-polynomal splne functon S ( ) has the form S ( ) = a + b( ) + c sn τ ( ) + d cos τ ( ), = 0,,..., N. () a, b, c and d are constants and τ s a free parameter. C A non-polynomal functon S ( ) of class a, b nterpolatng y() at the grd ponts, = 0,,..., N depends on a parameterτ, and reduce to ordnary cubc splne n [a, b] as τ 0. o derve an epresson for the coeffcents of Eq. () n term of y, y +, M and, defne M + S ( ) = y, S ( + ) = y+, '' '' S ( ) = M, S ( ) = M +. + Usng algebrac manpulaton, the followng epressons are obtaned for the coeffcents: M y y M M a y b τ h τθ Mcosθ M+ M c =, d = τ snθ τ + + = +,, = + θ = τh, for = 0,,..., N-. Usng the contnuty of the frst dervatve at (, y ), that s S ( ) ( ) = S, we get the followng relatons for =,,...,N-. y+ y + y αm+ + βm + α M = () h cosθ α = +, β =, θ θsnθ θ θsnθ = = + = '' M j y ( j), j,, and θ τh Numercal Scheme At the grd ponts, Eq. () may be dscretzed by ε y = p ( ) y + q ( ) y + r Usng splne s second dervatves, we have ε M = p ( ) y ( ) + q ( ) y ( ) + r ( ) j j j j j j for j =,, + () Usng Eq. () n Eq. () and wth the followng appromatons for the frst dervatve of y : y y + y+ y +, h y + y y+ y, h + ωhq+ + ωh[ p+ + p ] y y h ω p + p y + ω + hq h[ p+ + p ] [ r ] + ω hr + ω h + y (6) we get the trdagonal system E y + F y + G y = H () + + for =,,..., N E = ε α p h + β ph ω[ p+ + p ] α ω pβ hq + p+ h+ α q h hβ p F = ε + α p h β phω p + p + α p h + β qh + α G =ε p h + β phω p + p + ωh β pq + α p h+ α q h + hβ p ( α ωβ ph ) r + β r H = - h + ( α + ωβ ph ) r + [ ] E-ISSN: - Volume, 0

3 Lakshm Sreesha Ch. p ( ) = p, q ( ) = q, r ( ) = 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 Eqs. (), we get the followng three term formula for = : α h q 0 β h q α h q + y0 + + y + + y εk εk εk (8) h = [ α r0 + β r + α r] ε k We solve the trdagonal system Eq. () together wth the Eq. (8) for =,,, N- n order to get the appromatons y, y,..., yn of the soluton y() at,,..., N. runcaton error he local truncaton error assocated wth the scheme developed n Eq. (), s h hy ( ) = + ( α + β) ε ( ) α ωε + β p ( ) y ( ) 6 + h + Oh ( ) ε ( ) + ( + α ) y ( ) hus for dfferent values of α, βω, n the scheme Eq. (), ndcates dfferent orders: () for any choce of arbtrary α and β wth α + β = and for any value of ω, the scheme Eq.() gves second order method. () for α =, β = and ω =, from 0ε Eq. () fourth order method s derved. Convergence Analyss Incorporatng the boundary condtons Eq. (), the system of Eqs. () - (8) can be wrtten n the matr form as: ( D+ PY ) + Q+ h ( ) = 0 (9) D = [ ε, ε, ε] ε ε ε ε ε ε... 0 = ε ε and P= [ z, v, w ] * * v w z v w z v w z v = N N βhq αhq ε k ε k * * = +, w = + v zα = p h β ph+ ω p [ p+ + ] α ω p β h q + p h + α q h - hβ p + vα = p h β ph ω p p + + α p h + β q h + α w = β p h ωh + p p [ + + ] + ωh β pq + α p h+ α q h + hβ p for =,,..., N and E-ISSN: - Volume, 0

4 Lakshm Sreesha Ch. h ( αr0 + βr + αr) ε k Q = αhq γ0, q, q,..., qn + wn γ ε k ( α ωβ ) q = h, =,,..., N ( ) + β r + α + ωβ ph r + and Y = Y, Y,..., Y, ( h) =,,...,, N N ph r 6 h ( ) = 0( h) for α =, β =, ω = 0ε O = 0,0,...,0 are assocated vectors of Eq. (8). Let [,,..., ] y = y y yn Y satsfes the equaton ( D+ P) y+ Q= 0 (0) Let e = y Y, =,,..., N be the dscretzaton error so that E = e, e,..., e = y Y. N Usng Eq. (9) from Eq. (0), we get the error equaton ( D+ PE ) = h ( ) () Let p ( ) C and q ( ) C C, C are th postve constants. If P, j be the (, j) element of P, then P, + = w h α P, + = + C 0 for = ε k ( hα β C hαc βωh C h βωcc ) ( + ) + + +, =,,..., N P z, = ( hα β C hαc βωh C h βωcc ) ( + ) + + +, =,,..., N hus for suffcently small h P, + < ε, =,,..., N and P, < ε, =,,..., N () Hence, (D + P) s rreducble (see Ref. []). Let S be the sum of the elements of the th row of the matr (D + P), then we have h S = + ( βq + αq) for = ε k ( α β α ) β ω( ) S = h q + q + q + h p q q + + for =,,..., N αh S = ε + p p+ hβp + h αq + βq h βω p p + p h βω pq ( ) ( ) ( ) + for = N * Let C * = mn p ( ) and C = ma p ( ), N N * * N N C = mn q ( ) and C = ma q ( ). Snce 0 < ε t s possble or easy to verfy that for a gven, D+ P s monotone [,]. h ( ) Hence ( D P) ( D P) usng Eq.(0), we have + ests and + 0. hus ( ) E D+ P () th Let ( D+ P) k ( D+ P) and we defne be the (, ) element of k, N N k, N k = ( ) ( ) D+ P = ma D+ P, h ( ) = ma h ( ), N k, k, k = D+ P 0 and D+ P Sk =. Snce ( ) ( ) for =,,..., N Hence ε k ( D+ P) < k, S h α + β C, = ( ) * (a) (b) E-ISSN: - Volume, 0

5 Lakshm Sreesha Ch. + < S h C C ( D P) ( α + β ) * βω * k, for = N Furthermore, N ( D+ P) < k = mn S h C k, N ( ( α + β) * ). (c) (d) Eample. Consder the boundary value problem ε y + y = f( ), 0 < <. he eact soluton of ths problem s y ( ) = snh. he mamum absolute errors are presented n able for dfferent valuesε and h. Comparson of root mean square errors the estng methods are presented n able. Usng Eqs. (a) - (d), from Eq. (), we get E Oh ( ). () N k, k, k = D+ P 0 and D+ P Sk =. Snce ( ) ( ) for =,,..., N Hence the method Eq. (6) s fourth order convergent for α =, β =, ω =. 0ε Numercal eamples o demonstrate the proposed method computatonally, we consder three problems of the type Eq. (). hese problems have been chosen because they have been wdely dscussed n the lterature. Eample. Consder the sngularly perturbed sngular boundary value problem ε y + ( / y ) + ( + ) y= f( ), 0 < <. he eact soluton s y ( ) = ep( ). he mamum absolute errors are tabulated n able for dfferent valuesε and h. Comparson of root mean square errors wth the estng methods are presented n able. Dscussons and Concluson In ths paper, non-polynomal splne method s dscussed for a class of sngularly perturbed sngular two-pont boundary value problems. he dscretzed equaton s developed for the problem usng the condton of contnuty especally for the frst order dervatves of the non polynomal splne at the nteror nodes. It s not vald at the sngularty zero. A three term relaton s obtaned by modfyng the boundary value problem at the sngularty zero. Usng ths, the dscretzed equaton of the problem s solved usng dscrete nvarant mbeddng algorthm. Convergence of the method s eplaned. he mamum absolute errors are tabulated for the estng standard eamples chosen from the lterature wth a vew to demonstrate the method. Root mean square errors n the soluton of the sad eamples are presented wth comparson n order to justfy the method. he proposed method s also applcable to non-sngular problems and sngularly perturbed delay dfferental equatons. Based on the numercal results, t s observed that the method gves good results for smaller values of ε also. ABLE. Mamum errors n the soluton of Eamples 6 ε /N Eample.0(-).8(-).0(-).0(-).(-8).60(-).96(-).(-).0(-6).8(-8).0(-).80(-).(-).6(-6).(-).0(-) 6.(-).(-).(-6).(-) E-ISSN: - Volume, 0

6 Lakshm Sreesha Ch. 0.90(-).0(-).0(-).06(-).8(-6) Eample.90(-).69(-).0(-6).8(-).9(-8).00(-).98(-).9(-6).(-).0(-8).0(-).0(-) 9.(-6) 6.9(-).0(-8).90(-).6(-).0(-).0(-6).(-) 0.90(-).0(-) 8.0(-) 9.(-6) 8.(-) ABLE. Comparson of Root mean square errors n the soluton of Eample 6 ε /N R.K. Mohanty, Urvash Arora method [0].9(-).69(-) 9.8(-) 6.(-).(-) 6.(-).(-).6(-) 9.(-) 6.9(-) 9.(-).9(-).0(-).(-) 8.0(-) -.(-).0(-).9(-).06(-) 0-8.8(-) 8.0(-).(-).(-) Proposed method.80(-).(-) 8.9(-6) 6.0(-).8(-8).0(-).66(-).(-) 9.(-) 6.(-8).80(-).(-).8(-).0(-6).0(-).00(-).80(-).(-).(-6).9(-) 0.0(-) 8.9(-).(-).60(-).(-6) ABLE. Comparson of Root mean square errors n soluton of Eample 6 ε /N R.K. Mohanty, Urvash Arora method [0].(-).8(-).(-).09(-).68(-).(-).0(-).(-).60(-).8(-).(-) 8.8(-).(-).0(-).8(-).9(-).(-) 8.(-).8(-).(-) 0.(-).(-) 9.66(-) 6.(-).(-) Proposed method 6.8(-).(-).9(-6).9(-).6(-8) 8.(-) 6.(-).8(-6).6(-).6(-8).0(-) 9.9(-).(-6).6(-).8(-8).0(-).88(-).9(-).(-6).0(-) 0.0(-).88(-).(-) 6.0(-6) 6.00(-) References: [] R.K.Bawa, Splne based computatonal technque for lnear sngularly perturbed boundary value problems, Appl. Math. Comput., 6, 00, pp. -6. [] C.M. Bender, S.A. Orszag, Advanced mathematcal methods for scentsts and engneers, Mc. Graw-Hll, New York, 98. [] P.Henre, Dscrete Varable Methods n Ordnary dfferental equatons, Wley, New York, 96. [] M.K.Kadalbajoo, V.K.Aggarwal, Ftted mesh B- splne method for solvng a class of sngular sngularly perturbed boundary value problems, Internatonal Journal of Computer Mathematcs., 8 (), 00, pp [] M.K.Kadalbajoo, K.C.Patdar, Numercal soluton of sngularly perturbed two pont boundary value problems by splne n compresson, Internatonal Journal of Computer Mathematcs,, 00, pp [6] M.K.Kadalbajoo, Y.N.Reddy, Numercal soluton of sngularly perturbaton problems va devatng arguments, Appled Mathematcs and Computaton, (), 98, pp.. [] B.Kress, H.O.Kress Numercal methods for sngular perturbaton problems, SIAM J. Numer. Anal., 6, 98, pp [8] R.K.Mohanty, Navnt Jha, D.J Evans, Splne n compresson method for the numercal soluton of sngularly perturbed two pont sngular boundary value problems, Int. J. Comput. Math., 8 (), 00, pp [9] R.K.Mohanty, D.J.Evans, U.Aurora, Convergent splne n tenson methods for sngularly perturbed two pont sngular boundary value problems, Internatonal Journal of Computer Mathematcs., 8, 00, pp. 66. [0] R.K.Mohanty, Urvash Aurora, A famly of non-unform mesh tenson splne methods for sngularly perturbed two pont sngular boundary value problems wth sgnfcant frst E-ISSN: - Volume, 0

7 Lakshm Sreesha Ch. dervatves, Appled Mathematcs and Computaton.,, 006, pp. -. [] R.E. O Malley, Introducton to Sngular Perturbatons, Academc Press, New York, 9. [] J. Rashdna, M. Ghasem, Cubc splne soluton of sngularly perturbed boundary value problems wth sgnfcant frst dervatves, Appled Mathematcs and Computaton, 90, 00, pp [] R.S.Varga, Matr Iteratve Analyss, Prentce- Hall, Englewood Clffs, New Jersey,96. [] D.M. Young, Iteratve Solutons of Large Lnear Systems, Academc press, New York, 9. E-ISSN: - 6 Volume, 0