A GENERALIZED CHEBYSHEV FINITE DIFFERENCE METHOD FOR HIGHER ORDER BOUNDARY VALUE PROBLEMS

Transcription

1 A GEERALIZED CHEBYSHEV FIITE DIFFERECE METHOD FOR HIGHER ORDER BOUDARY VALUE PROBLEMS Soer Aydilik Departmet of Mathematics, Faculty of Arts ad Scieces, Isik Uiversity, 34980, Istabul, Turkey Ahmet Kiris * Departmet of Mathematical Egieerig, Faculty of Arts ad Scieces, Istabul Techical Uiversity, 34469, Istabul, Turkey Abstract A geeral formula is preseted for ay order derivative of Chebyshev polyomials istead of the eistig recursive relatioship. Hece, the Chebyshev fiite differece method is made applicable ot oly to secod order problems but also to higher order boudary value problems. The geeralized method is applied to a variety of higher order boudary value problems ad it is see that the obtaied results are more accurate tha the other umerical methods i absolute error. Keywords: Chebyshev fiite differece method, umerical solutio, higher order boudary value problems.. Itroductio May egieerig problems icludig dyamical system problems such as harmoic oscillator, elasticity problems such as wave propagatio ad heat covectio, ad calculus of variatios problems have bee modelled with secod-order boudary value problems. However, whe mathematical costraits are slightly stretched i order to provide compatibility with physical realities or whe the factors that affect the problem are aalyzed cocomitatly, higher order boudary value problems arise. There eist some methods such as fiite differece ad the shootig method to solve secodorder boudary value problems, ulikely, more effective methods are required to solve higher order boudary value problems; Aouadi used the Chebyshev fiite differece method to solve the third-order boudary value problem arisig i the modellig of mass trasfer whe the material was cosidered as a micropolar material [, ] istead of a classical elastic material [3]. Fifth-order boudary value problems arise i the mathematical modellig of viscoelastic * Correspodig Author; kiris@itu.edu.tr

2 fluids [4, 5]. To solve these problems, Kha used the fiite differece method [6], Çağlar et al. used the sith-degree B-splie ad collocatio methods [7], ad Wazwaz used the Adomia method [8]. I order to solve si-degree problems occurrig i the modellig of may astrophysics problems, septic splie [9], Legedre-Galerki [0] ad Daftardar-Jafari methods [] were used. To solve eighth-order boudary value problems that arise i torsioal vibratio of uiform beams, the Adomia decompositio method [], the differetial quadrature method [3], the homotopy perturbatio method [4], ad the splie method [5] were used. Whe fluid layers are subect to rotatio, heat covectio is modelled with 0 th order differetial equatios, if magetic effects are also icluded [6, 7]. Ulla, Kha, ad Rahim used a ew iterative techique [8] to solve such problems. The Chebyshev fiite differece method is more advatageous i solutio of higher order problems tha the methods metioed above. Of these methods, for eample the Adomia method requires calculatio of Adomia polyomials ad the homotopy methods require may coditios i additio to fidig the appropriate parameters [9]. Because of their orthogoality property, the Chebyshev polyomials form a complete orthogoal set o a space of cotiuous fuctios ad sice recursive relatios ca be obtaied easily, especially derivatives ca be calculated recursively at ay order. Furthermore, ot oly values at specific poits of the solutio rage -as i may umerical methods-, but a approimatio polyomial valid throughout all the iterval is obtaied. The greatest advatage of Chebyshev polyomials compared to the other polyomial approimatios of the same order is that they are the most coveiet polyomials havig the lowest maimum error i the give rage [0]. Solutio of secod-order iitial or boudary value problems with the Chebyshev fiite differece method is widely used i the literature; El-Kady ad Elbarbary obtaied a geeral formula for the derivatives up to secod order of the Chebyshev approimatio polyomial istead of the recursive relatio ad solved secod-order boudary value problems with the help of this formula []. Saadatmadi ad Farsagi solved the secod-order oliear system ad Saadatmadi ad Degha solved some calculus of variatio problems with the Chebyshev fiite differece method; however, i these studies, oly secod order problems could be solved [, 3]. Aouadi obtaied aother formula cotaiig successive sums for the third order derivative to aalyze micropolar flow ad mass trasfer from a surface stretched with heat [3]. However, successive sums icrease compleity as the order icreases. This is the

3 mai reaso of why Chebyshev fiite differece method could ot be used for higher order differetial equatios. I this study, first, the derivatives of the Chebyshev polyomial of ay order are obtaied without ay recursive relatio ad the by the help of these derivatives, a geeral formula is preseted for the Chebyshev approimatio polyomial. With this formula, the Chebyshev fiite differece method became applicable ot oly to first ad secod order problems, but also to iitial or boudary value problems of ay order. The geeralized Chebyshev fiite differece method is applied to a variety of higher order boudary value problems give i the literature ad the obtaied results are more accurate tha the other umerical methods i absolute error.. Chebyshev Polyomials Chebyshev polyomials of the first kid are defied as [0]: T ( ) cos( ), cos, [0, ], 0, [,]. (.) Because of their trigoometric properties, the recursive relatio T ( ) T ( ) T ( ) (.) ca be easily observed. Chebyshev polyomials are orthogoal i the iterval [,] respect to the weight fuctio, w( ). Whe i the [,] they have roots at the poits ad at the poits k k cos, k,,..., k k cos, k 0,,,..., (.3) (.4) they have ( ) etrema. A importat iequality, which is kow as the ecoomizatio property of the Chebyshev polyomials is give by the theorem give below. Theorem: Let are the sets of all moic polyomials of the order T ( ) Chebyshev polyomials are defied by T ( ). The the iequality ad the moic

4 T ( ) ( ) P (.5) holds for P( ), here the equality is satisfied oly whe P ( ) T ( ). The proof ca be fid i [0]. The ecoomizatio property states that whe approachig a fuctio f ( ) polyomials, i order to miimize the maimum error i the give rage, the Gauss-Lobatto with poits, i.e. the roots of the polyomial iterpolatio. This is epressed with T ( ) have to be take as the ode poits i the ( )! ( ) ma f ( ) P ( ) ma f ( ).,, (.6) 3. Chebyshev Fiite Differece Method Cleshaw ad Curtis defied the solutio for a give iitial or boudary value problem as a series of Chebyshev polyomials [4] where, the superscript () have to be take ad y( ) '' a T ( ) (3.) 0 i the sum symbol meas that half of the first ad the last terms shows the order of the approimatio polyomial. Usig the orthogoality property of Chebyshev polyomials ad the Lagrage iterpolatio, the ukow a coefficiets are foud as a y( ) T ( ). (3.) The value of the th m order derivative of y i (3.) at the poits k is give by ( m) ( m) k k, 0 y ( ) d y( ) (3.3) where, d '' T ( ) T ( ) (3.4) ( m) ( m) k, k 0 ad oly the derivatives of Chebyshev polyomials, ( m T ) ( ) are ukows. Elbarbary ad El- ( m) Kady showed that the first two of the coefficiets, d are give as [] k,

5 4 d T ( ) T ( ), k, 0,,..., () k, l k 0 l0 cl ( l) odd, (3.5) () ( l ) k, l k 0 l0 cl ( l) eve (3.6) d T ( ) T ( ), k, 0,,..., where 0 /, c0, ci,,...,, i. (3.7) O the other had, Aouadi gives the coefficiets for the third order as [3] 4 l d ( l) ) T ( ) T ( ), ( l) eve, ( i l) odd L l (3) k, l k L 0 l0 i0 cl ci. (3.8) As it ca be see i (3.8), a ew sum symbol comes for third order. Emergig a ew sum symbol for every order i (3.8) ad ot kowig ( m T ) ( ) i (3.4) make the calculatio of higher order coefficiets impossible i a similar way. I the preseted study, i order to make the coefficiets ( m) d k, computable for each order, the relatio betwee Chebyshev polyomials T ( ), 0 T ( ) T ( ), (3.9) ( ) T ( ) T ( ), ( ) ( ) are used, ad the the recursive relatio is elimiated by takig successive derivatives of (3.9). Cosequetly, the derivatives of the Chebyshev polyomials are foud as T ( ) ( ( l ) )( ( l ) ) T ( ), 3 (3) l0 4cl ( l)odd T ( ) ( ( l ) )( l )( ( l ) ) T ( ), 4 (4) l0 4cl ( l)eve T ( ) ( ( l 3) )( ( l ) )( ( l ) )( ( l 3) ) T 5 (5) l0 9cl ( l)odd l l l ( ) (3.0) ad followig a similar process as above, fially the derivative of ay order is give as [5]

6 T ( ) ( l i) T ( ), m ( m)! m m ( m) l. (3.) ( m) * l0 im cl ( l m) eve m From this geeral formula, the value of the th m order derivative at the poits k, T m ( ) k is calculated ad substitutig this ito (3.4) gives the coefficiets ( m) d k, as d l i T T m k m m ( ) ( ) ( ),,, 0,,..., (3.) m ( m) * k, ( 3) l k 0 l0 im cl ( m )! ( l m) eve The symbol (*) i the multiplicatio shows that the multiplicatio ide icreases two by two i both (3.) ad (3.). Thus, the may be easily calculated i (3.3) by the help of (3.). th m order derivative of the approimatio polyomial Fially, writig the give iitial or boudary value problem i terms of approimatio polyomial ad its derivatives, the problem is trasformed to a liear or oliear algebraic equatio. Evaluatig this equatio ad the iitial or boudary coditios at the Gauss-Lobatto poits, i.e.,, 0,,...,, a system cosistig of equatios is obtaied. This system ca be solved usig ay appropriate method ad the approimatio polyomial s (+) ukows, which costitute the solutio to the origial differetial equatio, are determied. The Chebyshev fiite differece method is made applicable ot oly for the first two order but also to higher order iitial or boudary value problems by the geeralized formulas (3.) ad (3.). I the followig sectio, the method is applied to some higher order boudary value problems ad compared to the methods used commoly i the literature. It is show that the error of the preseted method is much lower tha those of other methods. 4. umerical Eamples I this sectio, five oliear problems are solved by geeralized Chebyshev fiite differece method metioed above. Eample : Cosider the oliear boudary value problem (4) y ( ) si si ( y ( )) 0 < <, (4.) with the give boudary coditios

7 y(0) 0, y(0), y() si, y() cos. (4.) The eact solutio for the above problem is y( ) si. I order to solve this problem with usig the Chebyshev fiite differece method, the iterval of, the iterval of coditios are by t 0, should be trasferred to. The trasferred differetial equatio ad the boudary ( t) ( t) 4 (4) 4 y ( t) si si y (t), t, y( ) 0, y( ), cos y() si, y(). (4.3) (4.4) (4.3) ad (4.4) are coverted to the followig form with the help of (3.3) 4 4 tk tk 4 dk, y( t ) si si dk, y( t ), k,,..., 0 0 (4.5) y( t ) 0, d y( t ),, 0 cos y( t ) si, d y( t ). 0 0, 0 (4.6) Equatios (4.5) ad (4.6) gives oliear algebraic equatios cotaiig ukow coefficiets, y( t), y( t),..., y( t ) ad ca be solved by ay appropriate umerical root fidig method. Here, the obtaied values of yt () from the system (4.5) ad (4.6) costitutes the solutio of the (4.3) i virtue of (3.) ad (3.). Usig the iverse t trasformatio, the solutio of the give boudary value problem, y ( ) is obtaied as

8 y 3 ( ) (4.7) The approimatio polyomial, y ( ) is ot give for the other eamples for the sake of brevity. The compariso of the results with the variatioal iteratio method [6] ad Chebyshev fiite differece method is give i Table 3.. As it ca be see from the Table, the absolute error betwee the aalytical solutio ad the result obtaied by CFDM is less tha the absolute error of the give method i [6]. Furthermore, the default precisio i Mathematica, which is used here is 6, if the precisio is setup to 00, the results is much more covicig. Table 3.: The compariso of eample, the aalytical solutio is si. 0.0 Absolute Error by VIM Absolute Error by CFDM E E E E E E E E E E E E E E E E E E E E E E 3 Eample : Cosider the oliear boudary value problem

9 (6) 36 y ( ) 6 y ( ) 0 e 40 ( ), 0 < <, (4.8) with the give boudary coditios y(0) 0, y(0), y (0), 6 6 y() l, y(), y (). 6 4 (4.9) The eact solutio for the above problem is y( ) l( ). Similar to Eample, if the 6 liear trasformatio t is used, the followig equatios are obtaied 6 (6) 36 yt ( ) t 3 y ( t) 0e 40, t, 6 (4.0) y( ) 0, y( ), y ( ), 4 (4.) y() l, y(), y () (4.0) ad (4.) are reduced to the system of algebraic equatios with the help of (3.3) as yt ( ) t 3 k k dk,y( t ) e 40, k,,..., 0 (4.) y( t ) 0, d y( t ), d y( t ), 4,, 0 0 y( t ) l, d y( t ), d y( t ) , 0, 0 0 (4.3) Usig the solutio of the system (4.) ad (4.3) i (3.) ad (3.) after the iverse trasformatio gives the approimatio Chebyshev polyomial which is the solutio of the give problem. The compariso of the results with the quitic B-splie collocatio [7] ad Chebyshev fiite differece methods is give i Table 3.. Table 3.: The compariso of eample, the aalytical solutio is l( ). 6 Absolute Error by QBSCM Absolute Error by CFDM E E 4

10 E E E.9360 E E E E.830 E E E E E E E E E 4 Eample 3: Cosider the oliear boudary value problem (6) y ( ) e y ( ), 0 < <, (4.4) with the give boudary coditios y(0), y(0), y (0), y e y e y e (), (), (). (4.5) The eact solutio for the above problem is y( ) e. With the same trasformatio above, the followig equatios are obtaied t 6 (6) y ( t) e y( t), t (4.6) y( ), y( ), y ( ), 4 e e y() e, y(), y (). 4 Similarly, (4.6) ad (4.7) are trasformed to (4.7) t k 6 6 k, k 0 d y( t ) e y( t ), k,,..., (4.8) y( t ), d y( t ), d y( t ), 4,, 0 0 e e y( t ) e, d y( t ), d y( t ). 0 0, 0, (4.9)

11 Same process is doe to fid the approimatio polyomial. The compariso of the results with the Adomia [8], homotopy perturbatio [9], variatioal iteratio [30], ew iterative, Daftardar Jafari [3] ad Chebyshev fiite differece method is give i Table 3.3. Table 3.3: The compariso of eample 3, the aalytical solutio is Absolute Error[ADM] Absolute Error [HPM] Absolute Error [VIM] Absolute Error [ITM] e. Absolute Error [DJM] Absolute Error [CFDM] 0..4E 7.4E 7.4E 7.4E 7 3.E 4 3.3E E 6.4E 6.4E 6.4E 6.9E 3.0E E 6 3.3E 6 3.3E 6 3.3E 6 4.8E 3 5.4E E 6 6.E 6 5.8E 6 4.E E 6 6.E 6 5.8E 6 4.E 6 5.E 6 6.E 6 5.8E 6 4.E 6 5.E 6 6.E 6 5.8E 6 4.E 6 8.0E 3.0E.0E 8.E 3 9.E 4.E 3.E 3.0E 3 E.9E 6.9E 6.9E 6 4.3E 3 6.3E 4 3.6E 7 3.6E 7 3.6E 7 3.6E 7 9.E 3.6E E 5.0E 0 5.0E 0 5.0E 0 5.6E 4 0 Eample 4: Cosider the problem (7) (4) y ( ) ( e ( )cos ) y ( ) y ( ) e y ( ) e (( 4 ( ) e )cos 8(5 )si ), 0, (4.0) with the coditios give y(0), y(0) 0, y (0), y (0), y() 0, y() ecos, y () ecos si. (4.) The aalytical solutio for the problem is y( ) e ( )cos( ). The same procedure gives (4.0) ad (4.) as t t t ( cos( )) 7 (7) 4 (4) ( ) e y t t t y ( t) y ( t) e y ( t) e ((0 t e )cos( ) t t 8 si( )), t, t (4.)

12 y( ), y( ) 0, y ( ), y ( ), 4 ecos ecossi y() 0, y(), y (). Usig (3.3) reduces (4.) ad (4.3) to t k tk tk tk ( cos( )) ( ) e y t t k k dk, y t dk, y t e y tk e tk e 0 0 ( ) ( ) ( ) ((0 ) tk tk tk cos( ) 8 si( )), k,,..., (4.3) (4.4) y( t ), d y( t ) 0, d y( t ), d y( t ), 4 3,,, ecos ecossi y( t ) 0, d y( t ), d y( t ). 0 0, 0, 0 0 (4.5) The compariso of the reproducig kerel space [3] ad Chebyshev fiite differece method is give i Table 3.4. Table 3.4: The compariso of eample 4, the aalytical solutio is e ( )cos( ). 0.0 Absolute Error by RKS with 30 Absolute Error by RKS with 50 Absolute Error by CFDM E 6.43 E E E E E E 09.9 E E E E E E E E E E E E E E E 09.0 E E E E E 3 Eample 5: Cosider the problem y ( ) y( ), 0 < < (4.6) 4 (0) 475

13 with the coditios (3) (4) y(0) =0, y (0) =, y (0) =, y (0) =, y (0) =, (3) (4) y() =0, y () =, y () = 4, y () =, y () = 48. (4.7) The aalytical solutio is followig equatios y ( ). With the same procedure oe ca get the 0 (0) 475 t3 y(t) y (t), t, y y y y y (3) 3 (4) y() 0, y(), y (), y (), y () 3. (3) (4) ( ) 0, ( ), ( ), ( ), ( ), These equatios are trasformed to the followig form (4.8) (4.9) ( ) tk y tk dk, y( t ), k,,..., 0 4 (4.30) 3 3 y( t ) 0, d y( t ), d y( t ), d y( t ), d y( t ), ,,,, y( t ) 0, d y( t ), d y( t ), d y( t ), d y( t ) , 0, 0, 0, (4.3) Whe the iverse trasformatio is applied to the umerical solutio of the system (4.30) ad (4.3), ad the substitutig this ito (3.) ad (3.) gives the approimatio polyomial. The compariso of the result with the quitic B-splie collocatio method [33] ad homotopy aalysis method [34] is give i Table 3.5. Table 3.5: The compariso of eample 5, the aalytical solutio is. Absolute Error by QBSCM Absolute Error by HAM Absolute Error by CFDM E E E E E E E E E 0

14 E E E E E E E E E E E E E E E E E E 5. Coclusio A geeral o-recursive formula is preseted for the derivatives of the Chebyshev polyomials of ay order istead of eistig recursive relatioship of the Chebyshev polyomials. By usig these derivatives, a geeral formula is obtaied for the Chebyshev approimatio polyomial. Thus, the Chebyshev fiite differece method is made applicable to ot oly the first ad secod order problems as i the previous studies, but also higher order iitial or boudary value problems. The obtaied results show that this geeralizatio of the Chebyshev method ca solve boudary value problems efficietly, ad the better accuracy is observed i compariso with the preseted method ad eistig techiques. For future prospects, the method is epected to be applied to partial differetial equatios for higher order problems. 6. Refereces [] A.C. Erige, E.S. Suhubi, oliear theory of simple micro-elastic solids, It. J. Eg. Sci. (964) [] A.C. Erige, Theory of micropolar fluids, J. Math. Mech. 6 (966) 8. [3] M. Aouadi, Fiite elemet ad Chebyshev fiite differece methods for micropolar flow past a stretchig surface with heat ad mass trasfer, It. J. Comp. Math. 85 (008) 05-. doi:0.080/ [4] A.R. Davis, A. Karageorghis, T.. Phillips, Spectral Galerki methods for the primary two-poit boudary-value problem i modelig viscoelastic flows, It. J. umer. Methods Eg. 6 (988) [5] A.R. Davis, A. Karageorghis, T.. Phillips, Spectral collocatio methods for the primary two-poit boudary-value problem i modellig viscoelastic flows, It. J. umer. Methods Eg. 6 (988) [6] M.S. Kha, Fiite differece solutios of fifth order boudary value problems, PhD thesis, Bruel Uiversity, Lodo, 994.

15 [7] H.. Caglar, S.H. Caglar, E.H. Twizell, The umerical solutio of fifth-order boudary value problems with sith degree B-splie fuctios, Appl. Math. Lett. (999) [8] A.M. Wazwaz, The umerical solutio of fifth-order boudary value problems by the decompositio method, J. Comp. ad Appl. Math. 36 (00) [9] S.S. Siddiqi, G. Akram, Septic splie solutios of sith order boudary value problems, J. Comp. ad Appl. Math. 5 (008) [0] A.H. Bhrawy, Legedre-Galerki method for sith order boudary value problems, J. Egypt. Math. Soc. 7 (009) [] I. Ullah, H. Kha, M.T. Rahim, umerical solutios of fifth ad sith order oliear boudary value problems by Daftardar Jafari Method, J. Comp. Eg. 04 (04) 3-7. [] A.M. Wazwaz, A reliable modificatio of Adomia s decompositio Method, Appl. Math. ad Comput. 9 (998) -7. [3] G.R. Liu, T.Y. Wu, Differetial quadrature solutios of eighth-order boudary-value differetial equatios, J. Comp. ad Appl. Math. 45 (00) [4] A. Golbabai, M. Javidi, Applicatio of homotopy perturbatio method for solvig eighthorder boudary value problems, Appl. Math. ad Comput. 9 (007) [5] S.S. Siddiqi, E.H. Twizell, Splie solutios of liear eighth order boudary-value problems, Comp. Methods i Appl. Mech. ad Eg. 3 (996) [6] S. Chadrasekhar, Hydrodyamic ad Hydromagetic Stability, The Iteratioal Series of Moographs o Physics, Claredo Press, Oford, 96. [7] R.P. Agarwal, Boudary Value Problems for Higher Ordiary Differetial Equatios, World Scietific, Sigapore, 986. [8] I. Ullah, H. Kha, M.T. Rahim, umerical solutios of higher order oliear boudary value problems by ew iterative method, Appl. Math. Sci. 7 (003) [9] D. Jai, A etetio of the homotopy aalysis method for improvig coverge of the multivariate solutio of oliear algebric equatios as typically ecoutered i aalog circuits, Uiversity of Ciciati, Ciciati, 007. [0] L. Fo, I.B. Parker, Chebyshev Polyomials i umerical Aalysis, Oford Uiversity Press, Lodo, 968. [] E.M.E. Elbarbary, M. El-Kady, Chebyshev fiite differece approimatio for the boudary value problems, Appl. Math. ad Comp. 39 (003) [] A. Saadatmadi, J.A. Farsagi, Chebyshev fiite differece method for a oliear system of secod-order boudary value problems, Appl. Math. ad Comput. 9 (007) [3] A. Saadatmadi, M. Dehgha, The umerical solutio of problems i calculus of variatio usig Chebyshev fiite differece method, Physics Lett. A. 37 (008) [4] C.W. Cleshaw, A.R Curtis, A method for umerical itegratio o a automatic computer, umer. Math. (960) 97. [5] S. Aydılık, The Solutio of Higher Order Iitial ad Boudary Value Problems Usig Chebyshev Fiite Differece Method, MS thesis, Istabul Techical Uiversity, Istabul, 05.

16 [6] M.A. oor, S.D. Mohyud-Di, A efficiet method for fourth-order boudary value problems, Comp. ad Math. with Appl. 54 (007) 0. [7] K..S. Kasi Viswaadham, Y. Showri Rau, Quitic b-splie collocatio method for sith order boudary value problems, Global J. Researc. i Eg. um. Methods (0) -8. [8] A.M. Wazwaz, The umerical solutio of sith-order boudary value problems by the modified decompositio method, Appl. Math. ad Comput. 8 (00) [9] S. Momai, M.A. oor, S.T. Mohyud-Di, umerical methods for solvig a special sith-order boudary value problem, Comp. ad Math. with Appl. 55 (008) [30] M.A. oor, K.I. oor, S.T. Mohyud-Di, Variatioal iteratio method for solvig sithorder boudary value problems, Comm. i o. Sci. ad um. Simu. 4 (009) [3] I. Ullah, H. Kha, M.T. Rahim, umerical solutios of fifth ad sith order oliear boudary value problems by Daftardar Jafari Method, J. Comp. Eg. 04 (04) -8. [3] G. Akram, H.U. Rehma, umerical solutio of seveth order boudary value problems usig the reproducig kerel space, Research J. Appl. Sci. Eg. ad Techo. 7 (04) [33] K..S. Kasi Viswaadham, Y.S. Rau, Quitic B-splie collocatio method for teth order boudary value problems, It. J. Comp. Appl. 5 (0) 7 3. [34] S.S. Siddiqi, M. Iftikhar, umerical solutio of higher order boudary value problems, Abstract ad Appl. Aalysis 03 (04)