Approximate Solution of the Singular-Perturbation Problem on Chebyshev-Gauss Grid

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1 Amerca Joural of Computatoal Mathematcs,,, 9-8 do:.436/ajcm..44 Publshed Ole December ( Approxmate Soluto of the Sgular-Perturbato Problem o Chebyshev-Gauss Grd Abstract Mustafa Gülsu, Yalçı Öztürk Departmet of Mathematcs, Faculty of Scece, Mugla Uversty, Mugla, urkey E-mal: mgulsu@mu.edu.tr Receved Aprl 7, ; revsed May 4, ; accepted Jue 5, Matrx methods, ow-a-days, are playg a mportat role solvg the real lfe problems govered by ODEs ad/or by PDEs. May dfferetal models of sceces ad egeers for whch the exstg methodologes do ot gve relable results, these methods are solvg them compettvely. I ths work, a matrx methods s preseted for approxmate soluto of the secod-order sgularly-perturbed delay dfferetal equatos. he ma characterstc of ths techque s that t reduces these problems to those of solvg a system of algebrac equatos, thus greatly smplfyg the problem. he error aalyss ad covergece for the proposed method s troduced. Fally some expermets ad ther umercal solutos are gve. Keywords: Sgular Perturbato Problems, wo-pot Boudary Value Problems, he Shfted Chebyshev Polyomals, Approxmato Method, Matrx Method. Itroducto he boudary-value problems for sgularly perturbed delay-dfferetal equatos arse varous practcal problems bomechacs ad physcs such as varatoal problem cotrol theory. hese problems maly deped o a small postve parameter ad a delay parameter such a way that the soluto vares rapdly some parts of the doma ad vares slowly some other parts of the doma. Moreover, ths class of problems possess boudary layers,.e. regos of rapd chage the soluto ear oe of the boudary pots. here s a wde class of asymptotc expaso methods avalable for solvg the above type problems. But there ca be dffcultes applyg these asymptotc expaso methods, such as fdg the approprate asymptotc expasos the er ad outer regos, whch are ot route exercses but requre skll, sght ad expermetato. he umercal treatmet of sgularly perturbed problems preset some major computatoal dffcultes ad recet years a large umber of specalpurpose methods have bee proposed to provde accurate umercal solutos [-5] by Kadalbajoo. hs type of problem has bee tesvely studed aalytcally ad t s kow that ts soluto geerally has a multscale character;.e. t features regos called boudary layers the soluto vares rapdly. Ad these equa- tos as well as umercal methods have bee studed by several authors [6-]. he outer soluto correspods to the reduced problem,.e., that obtaed by settg the small perturbato parameter to zero. I recet years, the Chebyshev method has bee used to fd the approxmate solutos of dfferetal, dfferece, tegral ad tegro-dfferetal-dfferece equatos [,]. he ma characterstc of ths techque s that t reduces these problems to those of solvg a system of algebrac equatos, thus greatly smplfyg the problem. Cosder the of sgularly-perturbed delay dfferetal equatos form y( x) p( x) y( x ) q( x) y( x) g( x) () < x <, wth the boudary codtos y(), y() ad s a small postve parameter <, s also a small shftg parameter <, px ( ), qx ( ), rx ( ), s( x ) ad f ( x ) are suffcetly smooth fuctos. Our goal s to fd a approxmate soluto expressed polyomal of degree the form ( ) r r ( ) r y x a x () ar ukow coeffcets, r ( x) s the shfted Chebyshev polyomals of the frst kd ad s chose ay postve teger such that m. o obtaed a soluto () of the problem (), we ca use the zeroes of Copyrght ScRes.

2 M. GÜLSU E AL. the shfted Chebyshev polyomals kd defed by ( ) x 3 π x cos,,,. Basc Idea of the frst Polyomals are the oly fuctos that a computer ca evaluate exactly, so we make approxmate fuctos ab, by polyomals. he uform orm (or Chebyshev orm, maxmum orm) s defed by f max x a, b f( x). (3) Defto.. For a gve cotuous fucto f Ca, b, a best approxmato polyomal of degree p f such that s a polyomal m : f p f f p p It s a good dea to approxmate fucto by polyomals, because the classcal Weerstrass heorem s a fudametal result the approxmato of cotuous fuctos by polyomals [3-5]. heorem.. Let f Ca, b. he for ay >, there exsts a polyomal p for whch f p. Proof: See Ref. [3-6]. heorem.3. For ay f Ca, b best approxmato polyomal uque. Proof: See Ref. [3-6]. Defto.4. Gve a teger ad the p f exsts ad s the a grd set of pots are x ab, such that a x < x < < x b. he pots ( x ) are called the odes of the grd. heorem.5. Gve a fucto f Ca, b ad a grd of odes, ( x ), there exsts a uque polyomal of degree, I f such that I f x f x, I f s called the terpolat (or the terpolatg polyomal) of f through the grd. Proof: See Ref. [3-6]. he terpolat I f ca be express the Lagrage form as x I f f x x s the -th Lagrace cardal polyomal assocated wth the grd : x x j ( x),. x x j, j j he Lagrage cardal polyomals are such that x,, j. j j he best approxmato polyomal p f s also a terpolat of f at odes the error s gve by formula: f I f f p f s the Lebesgue costat relatve to the grd, : max x x a, b he Lebesgue costat cotas all the formato o the effects of the choce of o f I f. heorem.6. For ay choce of the grd, there exst a costat C > such that l π C Proof: See Ref. [3-6]. Defto.7. he odal polyomal assocated wth the grd s the uque polyomal of degree ( ) ad leadg coeffcet whose zeroes are the odes of : w ( x) xx heorem.8. If f C a, b of odes, ad for ay x ab, to error at x s, the for ay grd, the terpola-! ( ) f f xi ( ) f x w x ( x) ab, ad w ( ) x odal polyomal assocated wth the grd. Proof: See Ref. [3-6]... he Shfted Chebyshev Polyomal of the Frst Kd Defto.9. he Chebyshev polyomal of the frst kd ( x ) s a polyomal x of degree, defed by the relato [7] ( x) cos whe x cos. If the rage of the varable x s the terval [,], the rage the correspodg varable ca be take [, π]. Sce the rage [,] s qute ofte more coveet to use tha the rage [,], we map the depedet varable x [,] to the varable s [,] by the trasformato (4) Copyrght ScRes.

3 M. GÜLSU E AL... Chebyshev-Gauss Grd s xor x s Defto.. he grd x such that the ad ths lead to a shfted Chebyshev polyomal of the x s are the zeroes of the Chebyshev polyomal frst kd ( x) of degree x o [,] gve of degree s called the Chebshev-Gauss (CG) by [7] grd. ( ) ( ). x s x heorem.. he polyomals of degree ad leadg coeffcet, the uque polyomal whch has hus we have the polyomals the smallest uform orm o [,] s the ( ) -th Chebyshev polyomal dvded by. ( x), ( x)x, ( x)8x 8x, Proof: See Ref. [3-6]. It s of course possble to defed ( x ), lke ( x), drectly by a trgoometrc relato. Ideed, we obtaed 3. Fudametal Matrx Relatos ( x) cos whe x cos x. Let us cosder the Equato () ad fd the matrx hs relato mght alteratvely be rewrtte, wth forms of each term of the equato. We frst cosder the replace by, the form soluto ( ) ( m y x ad ts dervatve y ) ( x ) defed by ( ) cos whe cos a trucated Chebyshev seres. he we ca put seres x x cos the matrx form () () () y( x) ( x) A, y ( x) ( x) A, y ( x) ote that the shfted Chebyshev polyomal ( (6) x ) s () ether eve or odd ad deed all powers of x from ( x) A x to x appear ( x). he leadg coeffcet of x ( x ) for > to be. hese polyomals have the followg propertes: ( x) ( x) ( x) ( x) ) ( ) x has exactly real zeroes o the terval [,]. he m-th zero x () () () () m, of ( ) ( ) ( x) ( x) ( x) ( x) x s located at () () () () ( ) ( x) ( x) ( x) ( x) 3 m π A x m, cos a a a By usg (5), we obtaed the correspodg matrx relato as follows: ) It s well kow that the relato betwee the powers x ad the shfted Chebyshev polyomals ( x) s x x k( ). k k (5) ad ( ) ( ) adso ( ) ( ) x D x x x D (7) ( x) x x D Copyrght ScRes.

4 M. GÜLSU E AL. Moreover t s clearly see that the relato betwee ( k the matrx ( x ) ad ts dervatve ) ( x ), () () ( x) ( x) B, ( x) ( x) B (8) B. he dervatve of the matrx ( x) defed (7), by usg the relato (8), ca expressed as ( x) x D ( x) B ( D ), k,,(9) ( k) ( k) k () ( x) ( x), () ( x) ( x ), B B. By substtutg (9) to (6), we obta ( k ) k y ( x) ( x) B D A, k,, y () ( x) y( x ). Moreover, we kow that; () x x B () B ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Usg relato (8) ad (), we ca wrte () ( x ) ( x) B B I a smlarly way as (), we obta () y x x A x BB D A. (3) () () Matrx Represetato of the Codtos Usg the relato (), the matrx form of the codtos gve by () ca be wrtte as y D A y D A (4) 4. Method of Soluto. We are ready to costruct the fudametal matrx equato correspodg to Equato (). For ths propose, frstly substtutg the matrx relato () ad (3) to () we obtaed x B D p x x BB D q x x D A g x (5) For computg the Chebyshev coeffcet matrx A umercally, the zeroes of the shfted Chebyshev polyomals of the frst kd defed by (3) are puttg above relato (5) ad orgazed. We obtaed, x B D p x x BB D q x x D A gx So, the fudametal matrx equato s gaed EB D PBB D Q D A G (6) Copyrght ScRes.

5 M. GÜLSU E AL. 3 px ( ) qx ( ) px ( ) qx ( ) P px ( ) Q qx ( ) px ( ) ( ) qx x x x gx ( ) gx ( ) x x x x G g( x x x ) E x gx ( ) x x he fudametal matrx equato (6) for Equato() correspods to a system of ( ) algebrac equato for the ( ) ukow coeffcets a, a,, a. Brefly we ca wrte Equato(5) as so that, for W w WA G or W; G (7) pq,,,, pq EB D PBB D Q D Brefly, the matrx form for codtos () are CA or C;,, (8) C ( ) D c c c o obta the soluto of Equato () uder the codtos (), by replacg the rows matrces (8) by the last m rows of the matrx (7), we have the requred augmeted matrx w w w ; g x w w w ; g x W ; G w w w g x c c c ; c c c ; or correspodg matrx equato ; W A G (9) If rak W rak W ; G, the we ca wrte A W G. are uquely de- hus the coeffcets a,,, termed by Equato (). () 4.. Covergece ad Error Aalyss Sce, we coclude that f we choose the grd odes x to be zero the ( ) zeroes of the shfted Chebyshev polyomal, we have w ad ths s the smallest possble value. I partcular, for ay f C, we have [4] y y ( )! y If y s uformly bouded, the covergece of the terpolato y towards y( x) whe s the extrem ely fast. Also the Le besgue costat assocated wth the Chebyshev-Gauss grd s small l as. π hs s much better tha uform grds ad close to the optmal value. 4.. Checkg of Soluto We ca easly check the accuracy of the obtaed solu- the shfted Cheby- tos as follows: Sce the obtaed shev polyomal of the frst kd expaso s a approxmate soluto of Equato (), whe the fucto y ( ) x ad ts dervatves are substtuted Equato (), the resultg equato must be satsfed approxmately, that s for x, E, px y qx yx gx y x x 5. Illustratve Example o demostrate the effect of delay o the layer behavor Copyrght ScRes.

6 4 M. GÜLSU E AL. of the soluto ad the effcecy of the method, we cosder the examples gve below ad solve them usg the preset method ad all of them were performed o the computer usg a program wrtte Maple 9 software the solvg process. We have plotted the graphs of the soluto of the problem for dfferet wth dfferet values of to show the effect of delay o the boudary layer soluto. he maxmum errors deoted by E, at all the grd pots are evaluated usg the formula E max y x y x, Example 5.. Let us cosder the secod-order sgu- larly-perturbed delay dfferetal equato [3] y y.5 y x uder the codtos y (), y (). he exact sowe compare umercal results Fgures (a) ad (b) luto for the cosdered examples s ot avalable but gve by Kadalbajoo ([3], Example ). I Fgure (a), we show the umercal result usg preset method ad for the method usg by Kadalbajoo s show Fgure (b). he graphs of the soluto of the cosdered examples for dfferet values of delay are plotted Fgures (a)-(d) to exame the questos o the effect of delay o the boudary layer behavor of the soluto. (a) (b) (c) (d) Fgure. (a) umercal results of Example 5. for varous., ; (b) umercal results of Example 5. for varous [3].; (c) umercal results of Example 5. for varous., ; (d) umercal results of Example 5. for varous [3].. Copyrght ScRes.

7 M. GÜLSU E AL. 5 Example 5.. Let us cosder the secod-order sgu- larly-perturbed delay dfferetal equato [3] uder the codtos to s gve by y x y y y x y (), y(). Its exact solu- e e e e m m x m m x e e m m 4 m 4 m. We solved ths problem usg the method preseted here ad compared the result wth the exact soluto of the problems able. Also for, we have plotted the graphs of the computed soluto of the problem for, for dfferet values Fgure (a) ad we ca easly check the accuracy of the obtaed solutos Fgure (b). Example 5.3. Let us cosder the secod-order sgularly-perturbed delay dfferetal Equato [6] y y y x uder the codtos y (), y(). Its exact so- luto s gve by e e e e yx m m e e m mx m mx 4 m 4 m. We calculated umercal results for.5 ad dsplay results for some varous. Moreover, we compare the results wth o-stadard fte dfferece methods (SFDMs) able ad we dsplay results for, Fgure 3. able. E values of Example 5.., 3 E, () 8.35E 4.5E 4.46E.38E 4.53E 5.39E.34E 4.935E 7.384E 4.33E 4.594E 8.38E delta 8 delta delta delta (a) Copyrght ScRes.

8 6 M. GÜLSU E AL. 5 x 3 4 E, 8() E, () E, () E, 4() (b) Fgure. (a) umercal results of Example 5. for varous ( ); (b) For E (), umercal results of Example 5. for, varous. able. For.5, E, values of Example FSDs ( 3) FSDs ( 64) 6.3E.E.58E.E.34E.7E 7.3E.538E.347E.65E.34E.7E 8.63E.4E.336E.8E.34E.7E 9.94E.34E.E.598E.34E.7E.36E.656E.36E.3E.34E.7E 6. Coclusos I recet years, the studes of the sgularly-perturbed delay dfferetal equatos have attracted the atteto of may sceces ad egeers. he Chebyshev expaso methods are used to solve the sgularly-perturbed delay dfferetal equatos umercally. A cosderable advatage of the method s that the Chebyshev polyomal coeffcets of the soluto are foud very easly by usg computer programs Maple 9. Shorter computato tme ad lower operato cout results reducto of cumulatve trucato errors ad mprovemet of overall accuracy. Illustratve examples are cluded to demostrate the valdty ad applcablty of the techque. o get the best approxmatg soluto of the equato, we take more forms from the Chebyshev expaso of fuctos, that s, the trucato lmt must be chose large eough. Suggested approxmatos make ths method very attractve ad cotrbuted to the good agreemet betwee approxmate ad exact values the umercal example. As a result, the power of the employed method s cofrmed. We assured the correctess of the obtaed solutos by puttg them back to the orgal equato wth Copyrght ScRes.

9 M. GÜLSU E AL epslo 6 epslo 8 epslo Fgure 3. umercal results of Example 5.3 for varous. the ad of Maple, t provdes a extra measure of cofdece the results. 7. Refereces [] M. K. Kadalbajoo ad D. Kumar, Ftted Mesh B-Sple Collocato Method for Sgularly Perturbed Dfferetalatos wth Small Delay, Appled Mathe- Dfferece Equ matcs ad Computato, Vol. 4, o., 8, pp do:.6/j.amc [] M. K. Kadalbajoo ad K. K. Sharma, A umercal Method Based o Fte Dfferece for Boudary Value Problems for Sgularly Perturbed Delay Dfferetal Equatos, Appled Mathematcs ad Computato, Vol. 97, o., 8, pp do:.6/j.amc [3] M. K. Kadalbajoo ad V. P. Ramesh, umercal Methods o Shshk Mesh for Sgularly Perturbed Delay Dfferetal Equatos wth a Grd Adaptato Strategy, Appled Mathematcs ad Computato, Vol. 88, o., 7, pp do:.6/j.amc [4] M. K. Kadalbajoo ad V. P. Ramesh, Hybrd Method for umercal Soluto of Sgularly Perturbed Delay Dfferetal Equatosy, Appled Mathematcs ad Computato, Vol. 87, o., 7, pp do:.6/j.amc [5] M. K. Kadalbajoo ad K. K. Sharma, umercal Aalyss of Sgularly Perturbed Delay Dfferetal Equatos wth Layer Behavor, Appled Mathematcs ad Computato, Vol. 57, o., 4, pp. -8. do:.6/j.amc.3.6. [6] K. C. Patdar ad K. K. Sharma, ε-uformly Cover- get o-stadard Fte Dfferece Methods for Sgularly Perturbed Dfferetal Dfferece Equatos wth Small Delay, Appl ed Mathematcs ad Computato, Vol. 75, o., 6, pp do:.6/j.amc [7] M. H. Adhkar, E. A. Coutsas ad J. K. Mclver, Per- Delay Dffere- odc Solutos of a Sgularly Perturbed tal Equato, Physca D, Vol. 37, o. 4, 8, pp do:.6/j.physd [8] I. G. Amrslyeva, F. Erdoga ad G. M. Amralyev, A Uform umercal Method for Dealg wth a Sgularly Perturbed Delay Ital Value Problem, Appled Mathematcs Letters, Vol. 3, o.,, pp. -5. do:.6/j.aml..6. [9] S.. Chow ad J. M. Paret, Sgularly Perturbed Delay- Dfferetal Equatos, Cuopled olear Oscllators, orth-hollad Publshg Compay, Amsterdam, 983. [] R. E. O Malley Jr., Itroducto to Sgular Perturbato, Academc Press, ew York, 979. [] M. Glsu, Y. Öztürk ad M. Sezer, A ewcollocato Method for Solutoof the Mxed Lear Itegro-Dfferetal-Dfferece Equatos, Appled Mathematcs ad Computato, Vol. 6, o. 7,, pp do:.6/j.amc [] M. Sezer ad M. Gulsu, Polyomal Soluto of the Most Geeral Lear Fredholm Itegro-Dfferetal-Dfferece Equato by Meas of aylor Matrx Method, Iteratoal Joural of Complex Varables, Vol. 5, o. 5, 5, pp [3]. J. Rvl, Itroducto to the Approxmato of Fuctos, Courer Dover Publcatos, Lodo, 969. Copyrght ScRes.

10 8 M. GÜLSU E AL. [4] P. J. Davs, Iterpolato ad Ap proxmato, Dover Pub- [6] J. P. Body, Chebyshev ad Fourer Spectral Methods, lcatos, ew York, 963. Uversty of Mchga, ew York,. [5] K. Atkso ad W. Ha, heoretcal umercal Aalyss, 3rd Edto, Sprger, 9. mals, Chapma ad Hall/CRC, ew York, [7] J. C. Maso ad D. C. Hadscomb, Chebyshev Polyo- 3. Copyrght ScRes.