Investigation of Haar Wavelet Collocation Method to Solve Ninth Order Boundary Value Problems

Transcription

1 Global Jornal of Pre and Applied Mathematics. ISSN Volme 13, Nmber 5 (017), pp Research India Pblications Investigation of Haar Wavelet Collocation Method to Solve Ninth Order Bondary Vale Problems A. Padmanabha Reddy 1*, C. Sateesha 1 and Manla S.H. 1 1 Department of Stdies in Mathematics, V. S. K. University, Ballari, INDIA Abstract In this paper, nmerical scheme is described to approximate the soltion of ninth order bondary problems by Haar wavelet collocation method (HWCM). The validation and comparison of the scheme is done throgh the simlation of three test problems, which are arise in the stdy of astrophysics, hydrodynamics and hydro magnetic stability. Estimation of error bond and order of convergence are discssed. Efficient soltions obtained by increasing the level of resoltions are seems to be in better agreement with the exact and other nmerical methods sch as Modified decomposition method(mdm), Homotopy pertrbation method (HPM), Petrov-Galerkin Method with Qintic B-splines as Basis Fnctions (PGM). Keywords: Haar wavelets, Ninth order bondary vale problems, Collocation method, Convergence analysis. INTRODUCTION In the recent years the wavelet approach is becoming more poplar in the domain of nmerical estimation. The several types of wavelets and approximate fnctions have been sed for this prpose. A short introdction to the Haar wavelets and applications can be fond in [1]. The Haar wavelet method has some preferences as mathematical simplicity, fast convergence, possibility to implement standard algorithms and high accracy for small nmber of grid points [1, ]. Alfred Haar [3] demonstrated the concept of wavelets and they placed a imported role for the nmerical soltion of differential and integral eqations. At present there are two approaches to applying the Haar wavelet for integrating ordinary differential eqations (ODE). In case of the first method for integrating ODE concept of operational matrix is introdced by Chen and Hsiao [4, 5]. Another approach is called direct method de to Lepik [6] where the Haar fnctions are integrated directly.

2 1416 A. Padmanabha Reddy, C. Sateesha and Manla S.H. The bondary vale problems of ninth order have been developed de to their mathematical importance and the potential applications in the stdy of astrophysics, hydrodynamics and hydro-magnetic stability[7]. It s not so easy to determine the analytical soltion for sch type of BVPs bt existence and niqeness of soltion for these type of problems have been discssed in the book written by Agarwal[8]. The soltions of differential eqations have a maor role in the field of science and engineering. Many mathematical formlations of physical phenomena contain ninth order BVPs. Over the years many researchers have worked on ninth order BVPs by sing different methods for nmerical soltions. Chawla and Katti [9] have sed finite difference method for solving two-point bondary vale problems involving higher order differential eqations. Wazwaz [10] had employed Modified decomposition method (MDM) for solving higher order BVPs. Abdel and Vedat [11] have fond the soltion of different types of linear and nonlinear higher order BVPs by differential transformation method (DTM). Syed and Ahmet[1, 13] sed homotopy pertrbation method(hpm) and variational iteration method (VIM) to find the soltion of tenth and ninth order BVPs interms of convergent series. Lma and Samaher[14] have solved higher order bondary vale problems sing Semi-Analytic techniqes(sat). Bellal and Shafiql [15] have established a novel nmerical approach for odd higher order BVPs. Samir kmar[16] has sed Tchebychev polynomial approximations for order BVPs. Many researchers have worked on Haar wavelet method to solve varios orders of ODEs. Sira l-i et al.[17] have solved second order BVPs. Fazal et al.[18, 19] have fond the soltions for forth order and sixth order BVPs. Reddy et al.[0, 1] have approximated the soltions for fifth and seventh order ODEs. This motivated s to solve ninth order bondary vale problems arise in the astrophysics, hydrodynamics and hydro magnetic stability. The following form of ninth order BVP is considered (9) ( x) f ( x,, (1), (), (3), (4), (5), (6), (7), (8) ), x( c, d), (1) th m sbect to the following type of bondary conditions: (1) () (3) (4) (1) ( c), ( c), ( c), ( c), ( c), ( d), ( d), () (3) 8 9 ( d), ( d). () Where ni ' s, c and d are real constants for i 1,,...,9. The organization of this article is as follows. In section, Haar wavelets and their integrals are introdced. In section 3, a general formlation for the nmerical algorithm based on Haar wavelets is presented. Convergence of Haar wavelet discretization method(hwdm) is briefly presented in section 4. Few problems are solved in section 5 to test the effectiveness of the method and finally conclsion has been discssed in the last part of this paper.

3 Investigation of Haar Wavelet Collocation Method to Solve Ninth Order Bondary 1417 HAAR WAVELETS AND THEIR INTEGRALS In this section, we obtain orthogonal basis for the sbspaces of L [ c, d] called Haar wavelet family. For this notations introdced in Ref. [] are sed. The interval [ cd, ] 1 ( d - c) is divided into J sbintervals of eqal length t 1 J, where J is called maximal level of resoltion. We have coarser resoltion vales 0,1,,..., J 1 and translation parameter k 0, 1,,..., 1. With these two parameters in Haar family is defined as th i Haar wavelet 1, for t [ 1(i), (i)), hi ( t) 1, for t [ (i), 3(i)), 0, otherwise, (3) here i m k 1, 1( i) c kt, ( i) c (k 1) t and 3 ( i ) c ( k 1) t, where J. Above eqations are valid for i. h 1 () t and h () t are called father and mother wavelets in Haar wavelet family and are given by c d where, p. 1, for t [ c, d), h1 ( t) (4) 0, otherwise, 1, for t c, p, h ( t) 1, for t p, d, (5) 0, otherwise, Any fnction which is having finite energy on[ cd, ], i.e. decomposed as infinite sm of Haar wavelets: i1 L [ c, d] can be ( x) b h ( x), (6) i i where bi ' s are called Haar coefficients. If f is either piecewise constant or wish to approximate by piecewise constant on each sbinterval then the above infinite series will be terminated at a finite nmber of terms. Since, we have explicit expression for each member of Haar family (3-5 ). We can integrate as many times depend pon the context. The following notations are sed for times of integration of members in the family defined on [ cd, ) :

4 1418 A. Padmanabha Reddy, C. Sateesha and Manla S.H. t t t P ( t)... h ( x) dx, (7), i i c c c d G, i P, i ( t ) dt. (8) c For i 1, (7) becomes P 1,1( t ) ( ), (9)! t c for i, we have P 0, if t [ c, 1( i)), 1 ( 1 ( )) t i, if t [ 1 ( i ), ( i )),! ( t) 1 (10) ( t ( i )) ( t ( i )), if t [ ( i ), ( i )),! 1 ( t ( )) ( ( )) ( ( )) 1 i t i t 3 i, if t [ ( ), ). 3 i d!, i 1 3 METHOD OF SOLUTION 3.1 Haar Wavelet collocation method: The proposed method is as follows [, 0, 1] Approximate highest order derivative by piecewise constant on each sbinterval J 1 (9) ( x) bihi( x). (11) i1 (8) (7) Decompose ( x), ( x),..., ( x ) in terms of integrated Haar fnctions and replace these in to the given linear differential eqation. Discritize eqations obtained in above at collocation points ( x l 1 x x l ), l 1,,..., J 1, where is the grid point given by ( d l x x c n c), n n J 1 n 0,1,,..., J 1 J1 J1. Reslting into linear algebraic system.

5 Investigation of Haar Wavelet Collocation Method to Solve Ninth Order Bondary 1419 Calclate the wavelet coefficients bi ' s and obtain the Haar soltion for nknown fnction. The proposed method is frther simplified with the help of particlar bondary conditions for BVPs: c0, d The following type of bondary conditions are considered: (1) () (3) (4) (1) (0), (0), (0), (0), (0), (1), (1), () (3) 8 9 (1), (1). (1) The soltion (x) can be derived as x x x (5) x (6) x (7) x ( x) x (0) (0) (0) J 1 (8) x (0) bip9, i( x). (13) 4030 i1 (5) (6) (7) (8) Where the nknowns (0), (0), (0), (0) can be fond sing bondary conditions (1) and given by (5) (0) J 1 i b [ 670G 50G 360G 0 G ]. (14) i 9, i 8, i 7, i 6, i (6) (0) J b [ G 4030G 610G 3600 G ]. (15) i 9, i 8, i 7, i 6, i i1 (7) (0) J 1 50 b [ G 5000G 4030G 50 G ]. (16) 9 i 9, i 8, i 7, i 6, i i1 (8) (0) J b (141100G G G 670 G ). (17) 9 i 9, i 8, i 7, i 6, i i Where, G P ( x) dx, G P ( x) dx, G P ( x) dx, G P ( x) dx. (18) 6, i 6, i 7, i 7, i 8, i 8, i 9, i 9, i

6 140 A. Padmanabha Reddy, C. Sateesha and Manla S.H. CONVERGENCE ANALYSIS OF HAAR WAVELET DISCRETIZATION METHOD The accracy isses of the HWDM open from year This isse is clarified by J. Maak et al. [] in 015. The following reslts are de to notations introdced by J. Maak et al. [3]. General form of ninth order ODE is (1) () (3) (4) (5) (6) (7) (8) (9) f x,,,,,,,,,, 0 (19) Expand ninth order derivative into Haar wavelets as 9 d ( x) b ( ) (0) 9 ihi x dx i1 1 b h b h 1 1 k1 k1 0 k0. (1) In (0) and (1) k 1 i, k 0,1,..., 1. Integrating (1) 9 times we obtain the soltion of DE (19) as 1 b1 ( x) b P ( x) B( x). () k1 9, k1 9! 0 k0 Here P ( ) x represents the ninth order integrals of the Haar fnctions (9,10) and 9, 1 k Bxis ( ) a bondary term.let s assme that J 1 9 d ( x) L ( R) 9 dx its next derivative is bonded on [0,1], 10 ( ) : d x 10 dx is a continos fnction and b1 Let J 1 ( x) b P ( x) B( x) k1 9, k1 9! be the approximation to the nknown 0 k0 by integrated Haar wavelets. The absolte error at the and given by 1 E J 1 ( x) J 1( x) b P ( x). k1 9, k1 J 1 k0 Norm of the error in Hilbert space L ( R )[3] is defined as 1 1 J 1 k1 9, k1 0 J 1 k0 E b P ( x) dx th J resoltion is denoted E J 1

7 Investigation of Haar Wavelet Collocation Method to Solve Ninth Order Bondary 141 r 1 1 r r k1 s1 9, k1 9, s1 J 1 k0 rj 1 s0 0 1 b b P ( x) P ( x) dx, (3) where, P9, ( x ) are the integrals of Haar fnctions J. Maak et al.[] have shown that i, for bi i k 1 and P 1 9, i ( x) are monotonically increasing on [0,1). Therefore, E J 1 r r J 1 k 0 rj 1 s r r1 160 r r1 8, (4) Above eqation can be simplified as factrization and, r1 m J 1 rj1 1 m 1,,3,4. E J 1 J 1 J 1 J 1 J 1, (5) E 1 O. (6) J 1 J 1 From above eqation (6), we can conclde that the convergence is of order two. NUMERICAL STUDIES In this section, three nmerical experiments are given. The approximate soltion for each problem is devised by the HWCM. To show the accracy of the present method, approximate soltions of the problems are compared with the exact and other nmerical methods are available in the literatre. All comptations are carried ot by MATLAB software. Example 1: Consider the linear bondary vale problem [10, 4], (9) ( x) 9e x ( x) 0, x(0,1), (7)

8 14 A. Padmanabha Reddy, C. Sateesha and Manla S.H. with bondary conditions: (1) () (3) (4) (1) () (0) 1, (0) 0, (0) 1, (0), (0) 3, (1) 0, (1) e, (1) e, (3) (1) 3 e. (8) Its exact soltion is(1 xe ) x Figre 1: Comparison of exact and approximate soltion for J=3 of example 1 Figre : Comparison of absolte errors

9 Investigation of Haar Wavelet Collocation Method to Solve Ninth Order Bondary 143 Figre 3: Absolte errors by HWCM with J=4, 5 and 6 for example 1 Example : Consider the bondary vale problem [5], (9) (7) (4) (3) (1) ( x) ( x) x ( x) ( x) sin( x) ( x) ( x) 5xsin( x) cos( x) x cos( x) xsin ( x) sin( x)cos( x) xcos( x), x (0,1), (9) with bondary conditions: (1) () (3) (4) (1) (0) 0, (0) 1, (0) 0, (0) 3, (0) 0, (1) cos(1), (1) cos(1) sin(1), () (3) (1) sin(1) cos(1), (1) 3cos(1) sin(1). (30) Its exact soltion is x cos(x). Figre 4: Comparison of exact and approximate soltion for J=4 of example

10 144 A. Padmanabha Reddy, C. Sateesha and Manla S.H. Figre 5: Comparison of absolte errors Figre 6: Absolte errors by HWCM with J=3,4 and 5 for example Example 3: Consider the non linear bondary vale problem [5], (9) (1) 3 x x x x x ( ) ( ) ( ) cos ( ) 0, (0,1), (31) with bondary conditions: (1) () (3) (4) (1) (0) 0, (0) 1, (0) 0, (0) 1, (0) 0, (1) sin(1), (1) cos(1), () (3) (1) sin(1), (1) cos(1). (3) ( x) sin( x) Its exact soltion is. With the aid of qasilinearzation techniqe we converted nonlinear linear BVP(31) into a seqence of linear BVP[6] as ( x) ( x) ( x) ( x) ( x) ( x) cos ( x) ( x) ( x), n 0,1,... (33) (9) (1) (1) 3 (1) ( n1) ( n) ( n1) ( n) ( n) ( n1) ( n) ( n)

11 Investigation of Haar Wavelet Collocation Method to Solve Ninth Order Bondary 145 with the bondary conditions: (0) 0, (0) 1, (0) 0, (0) 1, (0) 0, (1) sin(1), (1) cos(1), (1) () (3) (1) (1) ( n1) ( n1) ( n1) ( n1) ( n1) ( n1) ( n1) (1) sin(1), (1) cos(1). (34) () (3) ( n1) ( n1) In this problem we assme that 0 ( x) has Maclarian series expansion and we calclated only one iteration i.e. 1 ( x) ( x). Figre 7: Comparison of exact and approximate soltion for J=5 of example 3 Figre 8: Comparison of absolte errors

12 146 A. Padmanabha Reddy, C. Sateesha and Manla S.H. Figre 9: Absolte errors by HWCM with J=3,4 and 5 for example 3 RESULT AND DISCUSSION: The comparison of approximate and exact soltion at collocation points with J=3, 4 and 5 for Examples 1, and 3 have been demonstrated in Figres 1, 4 and 7 respectively. Here in each figre approximate soltion coincided with the exact soltion, this assres the exactness of HWCM reslts. Figres, 5 and 8 indicate the comparison of absolte errors obtained by HWCM with MDM, HPM and PGM. These graphs exhibit the reslts that HWCM has given least absolte errors at each grid point. Absolte errors obtained for Examples 1, and 3 for varios resoltions are drawn in Figres 3, 6 and 9 conclde that as the resoltion vale increses absolte error crve approaches to x-axies (where the absolte errors are zero). CONCLUSION In this paper, we have employed a Haar wavelet collocation method to solve ninth order bondary vale problems. The proposed method has been tested on two linear and one non-linear BVP. Convergence analysis shown that the HWCM is of order two. The nmerical reslts obtained by this method are compared with MDM, HPM and PGM. The strength of the HWCM lies in its easy applicability, accracy and efficiency to solve ninth order BVPs. ACKNOWLEDGMENT Athor A. Padmanabha Reddy is gratefl to Vision Grop on Science and Technology, Govt. of Karnataka, India, for financial assistance nder the scheme Seed Money to Yong Scientists for Research (SMYSR-FY /GRD-497).

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