Numerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets

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1 Australia Joural of Basic ad Applied Scieces, 5(): 98-5, ISSN Numerical Solutio of the Two Poit Boudary Value Problems By Usig Wavelet Bases of Hermite Cubic Splie Wavelets Mehdi Yousefi, Hesam-Aldie Derili Gheralar ad Asghar Arzhag Departmet of Mathematics, Kara Brach, Islamic Azad Uiversity, Kara, Ira. Abstract: I this paper, compactly supported Hermite cubic splie wavelets are developed to approximate the solutios of the liear two poit boudary value problems. These wavelets costructed ad properties of these wavelets utilized to reduce the computatio of itegral equatios to some algebraic equatios. The orrespodig stiffess matrix arises from the discretizatio of the problem is sparse ad the coditio umber of this matrix is uiformly bouded. Key words: Two Poit Boudary Value Problem, Wavelet, Approximatio, Hermite cubic splie Wavelet bases. INTRODUCTION Wavelets theory is a relatively ew ad emergig area i mathematical research. It has bee applied i a wide rage of egieerig disciplies, particularly; wavelets are very successfully used i sigal aalysis for waveform represetatios ad segmetatios, time frequecy aalysis ad fast algorithms for easy implemetatio. Wavelet umerical method is suited to applicatios such as (Cauto, C., A. Tabacco ad K. Urba, 999; Cohe, A., ). Several umerical methods for approximatig the solutio of itegral ad differetial equatios are kow. I recet years, the applicatio of methods based o splie wavelets have ifluece o may areas of applied mathematics. Splie wavelets are used as a powerful tool i the umerical aalysis of differetial ad itegral equatios (Lakestai, M., et al., 5; Malekead, K., et al., ). I preset paper, we apply compactly supported Hermite cubic splie wavelets to solve the two poit boudary value problems. The paper is orgaized as follows. I sectio, we describe the formulatio of the splie scalig fuctios ad wavelets required for our subsequet developmet. I sectios ad 4 we will apply the wavelets for umerical solutios of the two poit boudary value problems with the Dirichlet boudary coditio. The correspodig stiffess matrix arises from the discretizatio of the problem is sparse ad the coditio umber of this matrix is uiformly bouded (idepedet of the umber of bases fuctios) ad fially i sectio 5, we report our umerical fidig ad computatioal results demostrate the accuracy of the proposed umerical scheme. Wavelets of Hermite Cubic Splies o The Iterval: Let x ad x be the cubic splies give by x x x x,,, xx x, x,,,,, ad x x x, x, ;,,, x x x, x,, I [7], two wavelets x ad x are itroduced as follows: x x x x x x 4, x x x x x x 9 9. Correspodig Author: Mehdi Yousefi, Departmet of Mathematics, Kara Brach, Islamic Azad Uiversity, Kara, Ira. Tel: ; m-yousefi@kiau.ac.ir 98

2 Aust. J. Basic & Appl. Sci., 5(): 98-5, These wavelets ca be easily adapted to the iterval,. For reader's coveiece, we briefly recall the basic ideas. By L, we deote the space of all square itegrable real valued fuctios o,. The ier product i L, is defied as uv, u x v x dx, uv, L,. Let H, be the space of all fuctios u i L, for which u L,. Let H, be the closure of the set uc, C, : uu i the space H,. H, has the followig decompositio H, VWW, where W is the liear spa of :,, :,,, Ad,, :,, :,,, is a basis for V. For,, ad x,, let x x, 79.6,, 4,,,, x x,, 5,,, 5.6,,,, x x x x.,, For x, let 5 4 x x, 5 4, x x,,, 5, 8 5 x. 4 x x,, x,4, Clearly, H, is spaed by x,,,, 4 x,,, &,,,.,, Two Poit Boudary Value Problem: Cosider the secod order equatio v x f x, v x, v x, x,, () with the iitial coditios 99

3 Aust. J. Basic & Appl. Sci., 5(): 98-5, v, v. () Equatios () ad () defie a two poit boudary value problem [8]. Note that the restrictio of the iterval ito the iterval, i the relatio () has o loss of geerality, sice a problem o ay fiite iterval may be coverted to the oe o the iterval,, (See remark.). Remark.: Cosider the boudary value problem u z = gz, uz, u z, with ua ( )= ad ub ( )= chage of variable t = bax a, this problem is equivalet to () ad () with =,, =,, f xv x v x ba g ba xav x b a v x.. By the v x u ba x a ad I this sectio we treat oly liear problems, i which the equatio () may be writte i the form v x = b x v x c x v x d x, x [,], () where b, c ad d are give fuctios. The boudary coditios that we cosider first are the coditios give i (). Later, we will treat other types of the boudary coditios. Equatios () ad () defie a liear two poit boudary value problem for the ukow fuctio v ad our task is to develop procedures to approximate the solutio. We assume that the problem has a uique solutio that is the result of beeig at least two times cotiuously differetiable of the ukow fuctio. We cosider the special case of () i which bx ( ), so that the equatio is v x = c x v x d x, x [,]. (4) Assume that cx ( ) for x [,]. This is a sufficiet coditio for the problem () ad (4) to have a uique solutio. Cosider the followig liear two poit boudary value problem v x q x v = f x, x [,], (5) with the coditios v =, v =, (6) where, for simplicity, we have take the iterval to be [,] ad the boudary coditios to be zero (see remark.). Remark.: The boudary value problem vx pxvxqxvx= f x, v =, v, ca be coverted to a problem with zero boudary coditios as follows. Let ux= vx x, the u = u = ad u satisfies the differetial equatio ux pxuxqxux= f xq x p x q x. Assume that there exist fiite costat C such that qx C, x[,], by a result i theory of ODE, there is a uique fuctio v satisfyig equatio (5) ad the boudary coditios (6). The Galerki Method to Solve the Liear Two Poit Boudary Value Problem by Usig Wavelet Bases of

4 Aust. J. Basic & Appl. Sci., 5(): 98-5, Hermite Cubic Splies: A fuctio vx ( ) defied over [,] ca be represeted by Hermite cubic splie wavelets as 4 i k, k i, i, k= i= = (7) v x = c d, where,k ad i, are scalig ad wavelet fuctios, respectively. If the ifiite series i (7) is trucated, the (7) ca be writte as 4 i = k, k i, i, =, (8) v x u x c d c g x k= i= = = where the basis fuctios g satisfy the boudary coditios g = g =, =,,. (9) If (9) holds, the the approximate solutio u give by (8) satisfies the boudary coditios. We use the wavelet set G = g,, g as a basis for V, where g, for =,,4 ad g =, =,, ad =,,. Let the residual fuctio for ux ( ) be defied by rx= uxqxux f x, x [,]. If ux ( ) were the exact solutio of (5), the the residual fuctio would be idetically zero. Obviously, the residual is orthogoal to every fuctio ad, i particular, it is orthogoal to the set of basis fuctios. However, we ca t expect ux ( ) to be the exact solutio because we restrict ux ( ) to be a liear combiatio of the basis fuctios. By the Galerki method the residual fuctio is orthogoal to all of the basis fuctios g,, g u x q x u x f x gi x dx =, i =,,. () If we put (8) ito () ad iterchage the summatio ad itegratio, we obtai c g xqxg xgixdx f xgixdx i = =, =,,. This leads to a system of liear equatios of the form AC = F with for C = c, c,, c, T T F = f, f,, f, where fi = f x gi x dx, A = a = i g x qxg x gi xdx. If we itegrate the first term i this itegral by parts, i i i g x g x dx = g x g x g x g x dx, ad ote that the first term vaishes because a = g x g x dx q x g x g x dx. i i i g i is zero at the ed poits, we ca rewrite a i as () Let L be a operator defied as Lu t = u t q t u t, u t L,, the,

5 Aust. J. Basic & Appl. Sci., 5(): 98-5, a = Lg x, g x = g x g x dx q x g x g x dx, =,,. i i i i Theorem 4.: The sequece Proof: (See [7]). Corollary 4.: g k x k =,, 4 m is a Riesz sequece i L,. For every g t = c, t bm, m, t L,, there exist two positive costats such that = m= = C ad C CA g t dtca, () where 4 m A= c bm,. = m= = (4) Theorem 4.: Let L be a operator defied as Lu t = ut qtu t, u t L,, satisfyig C costat q t C t ad g L, satisfies C A Lg, g CC A, ( ):,,, where A is defied i (4). g = g =, the Proof: Note that by itegratio by parts, Lg t, g t = g t g t dt q t g t dt = g t dt q t g t dt,,, g t g t q t g t g t ad g t dt g t g t dt = g t, g t, also, q t g t dt = q t g t, g t, by addig these two iequalities, g t dt Lg, g. (5) Also, ote that g t, g t g t dt. (6) Because g = the g t = g s ds. Hece by the Cauchy Schwartz iequality t t ( )( ),,, g t g s ds ds g s ds t therefore,

6 Aust. J. Basic & Appl. Sci., 5(): 98-5, g t dt ( g s ds)( ds)= g s ds. Hece, q t g t, g t = q t g t dt C g t dt, (7) by addig (6) ad (7), we have Lg, g C g t dt. (8) The proof of theorem is completed by the relatios (), (5) ad (8). Corollary 4.: The coditio umber of the stiffess matrix A is uiformly bouded (idepedet of ). Let us recall that the coditio umber of a ivertible square matrix A is defied by cod( A)= A A, where. is a matrix orm. The spectral coditio umber of A is defied as max, where = max max : is a eigevalue of A, ad = mi : mi is a eigevalue of A. If A is a (real) symmetric matrix, the its coditio umber with respect to the orm is equal to its spectral coditio umber (Ciarlet, P.G., 989). The coditio umber of a matrix A measures the stability of the liear system AC = F uder perturbatios of F. Thik of perturbig F by F to obtai F F. Let C be the solutio of AC = F ad C be the solutio of AC = F, the by liearity, A( CC)= F F. The stability of the liear system is most aturally described by comparig the relative size C / C of the chage i the solutio to the relative size F / F of the chage i the give data. The coditio umber is the maximum value of this ratio. Theorem 4.: Let A is a ivertible matrix, C, AC = F ad AC = Fthe mi C C F cod( A). (9) F Moreover, cod( A ) caot be replaced by ay smaller umber i relatio (9). Test Problems: I order to verify the efficiecy of the method itroduced i the previous sectio, the results of several umerical experimets are preseted ad compared with either exact solutios. I the followig tables we applied the orm = v x u x v x u x dt The programs are compiled i Maple. Example : Cosider the problem =, v =. v x = si x, x[,], v

7 Aust. J. Basic & Appl. Sci., 5(): 98-5, The exact solutio of this problem is v x =si x. Table shows the absolute values of errors of the approximatio solutios. I this example, stiffess matrix A= g, g i i, =,, is block diagoal ad each block is a baded matrix. By corollary 4., the coditio umber of the matrix A is uiformly bouded. I table, the coditio umber of A for =6,, are show. Table : The error i v for example ux v x Table : Coditio umber of the stiffess matrix A for example cod( A ) Example : Cosider the liear two poit boudary value problem v x v x = x x si x x cos x, v =, v =. The exact solutio of this problem is v x = x x si x. Table shows the absolute values of errors of the approximatio solutios. The stiffess matrix A= gi, g gi, g the coditio umber of A is uiformly bouded (see Table 4). i, =,, Table : The error i v for example ux v x is still a sparse matrix ad Table 4: Coditio umber of the stiffess matrix A for example cod( A ) Coclusio: I the preset work, a techique has bee developed for solvig some two poit boudary value problems. The method is based o Hermite cubic splie wavelets. The problem has bee reduced for solvig a system of liear algebraic equatios ad the stiffess matrix of the system is sparse ad coditio umber of this matrix is uiformly bouded(idepedet of the umber of bases fuctios). The computatioal results demostrate the advatage of this wavelet basis. Ackowledgemets The authors would like to thak Islamic Azad Uiversity(Kara Brach) for supportig this work. REFERENCES Cauto, C., A. Tabacco ad K. Urba, 999. The wavelet elemet method part I: costructio ad aalysis, appl. Comput. Harmo. A., 6: -5. Cauto, C., A. Tabacco ad K. Urba,. The wavelet elemet method part II: realizatio ad additioal feauture i D ad D, appl. Comput. Harmo. A., 8: -65. Ciarlet, P.G., 989. Itroductio to umerical liear algebra ad optimizatio, Cambridge Uiversity press, Cambridge. Cohe, A.,. Numerical Aalysis of Wavelet Method, Elsevier Press, Amsterdam. Golub, G.H. ad J.M. Ortega, 99. Scietific Computig ad Differetial Equatios: A Itroductio to Numerical Methods, Academic press. Lakestai, M. ad M. Dehgha, 6. The solutio of a secod-order oliear differetial equatio with Neuma boudary coditios usig semi-orthogoal B-splie wavelets, Iteratioal Joural of Computer Mathematics, 8(8-9):

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