Global Journal of Pure and Appled Mathematcs. ISS 0973-768 Volume 3, umber 2 (207), pp. 463-474 Research Inda Publcatons http://www.rpublcaton.com Haar Wavelet Collocaton Method for the umercal Soluton of onlnear Volterra-Fredholm-Hammersten Integral Equatons S. C. Shralashett* a, R. A. Mundewad *b, S. S. aregal *c and B. Veeresh *d, *a, b, c P. G. Department of Studes n Mathematcs, Karnatak Unversty, Dharwad 580003 *d R. Y. M. Engg. College, Bellary-58304, Karnataka, Inda *Correspondng author Abstract In ths paper, we proposed Haar wavelet collocaton method for the numercal soluton of nonlnear Volterra-Fredholm-Hammersten ntegral equatons. Propertes of Haar wavelet and ts operatonal matrces are utlzed to convert the ntegral equaton nto a system of algebrac equatons, solvng these equatons usng MATLAB to compute the Haar coeffcents. umercal results are compared wth exact soluton through error analyss, whch shows the effcency of ths technque. Keywords: onlnear Volterra-Fredholm-Hammersten ntegral equatons, Haar wavelet collocaton method, Operatonal matrx, Lebntz rule. ITRODUCTIO Integral equatons fnd ts applcatons n varous felds of mathematcs, scence and technology have motvated a large amount of research work n recent years. In partcular, ntegral equatons arse n flud mechancs, bologcal models, sold state
464 S. C. Shralashett, R. A. Mundewad, S. S. aregal and B. Veeresh physcs, knetcs n chemstry etc. In most of the cases, t s dffcult to solve them, especally analytcally. Antcpatng exact soluton for ntegral equatons s not possble always. Due to ths fact, several numercal methods have been developed for fndng solutons of ntegral equatons. onlnearty s one of the most nterestng topcs among the physcsts, mathematcans, engneers, etc. Wavelets theory s a relatvely new and an emergng tool n appled mathematcal research area. It has been appled n a wde range of engneerng dscplnes; partcularly, sgnal analyss for waveform representaton and segmentatons, tmefrequency analyss and fast algorthms for easy mplementaton. Wavelets permt the accurate representaton of a varety of functons and operators. Moreover, wavelets establsh a connecton wth fast numercal algorthms [, 2]. Snce 99 the varous types of wavelet method have been appled for the numercal soluton of dfferent knds of ntegral equatons. The solutons are often qute complcated and the advantages of the wavelet method get lost. Therefore any knd of smplfcaton s welcome. One possblty for t s to make use of the Haar wavelets, whch are mathematcally the smplest wavelets. In the prevous work, system analyss va Haar wavelets was led by Chen and Hsao [3], who frst derved a Haar operatonal matrx for the ntegrals of the Haar functon vector and put the applcatons for the Haar analyss nto the dynamc systems. Recently, Haar wavelet method s appled for dfferent type of problems. amely, Sraj-ul-Islam et al. [4] proposed for the numercal soluton of second order boundary value problems. Shralashett et al. [5-8] appled for the numercal soluton of Klen Gordan equatons, mult-term fractonal dfferental equatons, sngular ntal value problems, Rccat and Fractonal Rccat Dfferental Equatons. Shralashett et al. [9] have ntroduced the adaptve grd Haar wavelet collocaton method for the numercal soluton of parabolc partal dfferental equatons. Also, Haar wavelet method s appled for dfferent knd of ntegral equatons, whch among Lepk et al. [0-3] presented the soluton for dfferental and ntegral equatons. Babolan et al. [4] and Shralashett et al. [5] appled for solvng nonlnear Fredholm ntegral equatons. Azz et al. [6] have ntroduced a new algorthm for the numercal soluton of nonlnear Fredholm and Volterra ntegral equatons. Some of the author s have approached for the numercal soluton of nonlnear Volterra-Fredholm-Hammersten ntegral equatons from varous methods. Such as Legendre collocaton method [7], Legendre approxmaton [8], CAS wavelet [9]. In ths paper, we appled the Haar wavelet collocaton method for the numercal soluton of nonlnear Volterra-Fredholm-Hammersten ntegral equatons. The artcle s organzed as follows: In Secton 2, propertes of Haar wavelets and ts operatonal matrx s gven. In Secton 3, the method of soluton s dscussed. In secton 4, we report our numercal results and demonstrated the accuracy of the proposed scheme. Lastly, the concluson of the proposed work s gven n secton 5.
Haar Wavelet Collocaton Method for the umercal Soluton of onlnear 465 2. PROPERTIES OF HAAR WAVELETS 2.. Haar wavelets The scalng functon h ( ) x for the famly of the Haar wavelet s defned as The Haar Wavelet famly for x [0,) s defned as, for x 0, h ( x) (2.) 0 otherwse for x [, ), h ( x) for x [, ), 0 elsewhere, (2.2) where k k 0.5 k where,,, m m m m 2 l, l 0,,..., J, J s the level of resoluton; and k 0,,..., m s the translaton parameter. Maxmum level of resoluton s J. The ndex n (2.2) s calculated usng m k. In case of mnmal values m, k 0 then 2. The maxmal value of s 2 J. j 0.5 Let us defne the collocaton ponts x j, j,2,...,, Haar coeffcent matrx H, j h ( x ) whch has the dmenson. j For nstance, J 3 6, then we have - - - - - - - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - H6,6-0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -
466 S. C. Shralashett, R. A. Mundewad, S. S. aregal and B. Veeresh Any functon f( x ) whch s square ntegrable n the nterval (0, ) can be expressed as an nfnte sum of Haar wavelets as, f ( x) a h ( x) (2.3) The above seres termnates at fnte terms f f( x) s pecewse constant or t can be approxmated as pecewse constant durng each subnterval. Gven a functon 2 f ( x) L ( R) a mult-resoluton analyss (MRA) of L 2 ( R ) produces a sequence of subspaces Vj, Vj,... such that the projectons of f( x ) onto these spaces gves fner approxmaton of the functon f( x ) as j. 2.2. Operatonal Matrx of Haar Wavelet The operatonal matrx P whch s an square matrx s defned by often, we need the ntegrals x P ( x) h ( t) dt, (2.4) x x x x r r r, ( )... ( ) ( ) ( ) ( r )! A A A A P x h t dt x t h t dt rtmes r,2,..., n and,2,...,. 0 (2.5) For r, corresponds to the functon P ( ), x, wth the help of (2.2) these ntegrals can be calculated analytcally; we get, x α for x [α, β) P, (x)={ γ x for x [β, γ) 0 Otherwse (2.6) P 2, (x)= { 2 (x α)2 for x [α, β) 4m 2 2 (γ x)2 for x [β, γ) for x [γ, ) 4m2 0 Otherwse (2.7)
Haar Wavelet Collocaton Method for the umercal Soluton of onlnear 467 In general, the operatonal matrx of ntegraton of P r, (x)= { r! (x α)r for r! {(x α)r 2(x β) r } th r order s gven as x [α, β) for x [β, γ) r! {(x α)r 2(x β) r + (x γ) r } for x [γ, ) 0 Otherwse For nstance, J=3 = 6, then we have (2.8), (6,6) 32 P 3 5 7 9 3 5 7 9 2 23 25 27 29 3 3 5 7 9 3 5 5 3 9 7 5 3 3 5 7 7 5 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5 7 7 5 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 and 2, ( 6,6) 2048 P 9 25 49 8 2 69 225 289 36 44 529 625 729 84 96 9 25 49 8 2 69 225 287 343 39 43 463 487 503 5 9 25 49 79 03 9 27 28 28 28 28 28 28 28 28 0 0 0 0 0 0 0 0 9 25 49 79 03 9 27 9 23 3 32 32 32 32 32 32 32 32 32 32 32 32 0 0 0 0 9 23 3 32 32 32 32 32 32 32 32 0 0 0 0 0 0 0 0 9 23 3 32 32 32 32 0 0 0 0 0 0 0 0 0 0 0 0 9 23 3 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 0 0 7 8 8 8 8 8 8 8 8 8 8 8 8 0 0 0 0 7 8 8 8 8 8 8 8 8 8 8 0 0 0 0 0 0 7 8 8 8 8 8 8 8 8 0 0 0 0 0 0 0 0 7 8 8 8 8 8 8 0 0 0 0 0 0 0 0 0 0 7 8 8 8 8 0 0 0 0 0 0 0 0 0 0 0 0 7 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7
468 S. C. Shralashett, R. A. Mundewad, S. S. aregal and B. Veeresh 3. METHOD OF SOLUTIO In ths secton, we present a Haar wavelet collocaton method (HWCM) based on Lebntz rule for the numercal soluton of nonlnear Volterra-Fredholm-Hammersten ntegral equaton of the form, x u( x) f ( x) K ( x, t) F( t, u( t)) dt K ( x, t) G( t, u( t)) dt, 2 0 0 (3.) where K(x, t) and K2 (x, t) are known functons whch are called kernels of the ntegral equaton and f( x ) s also a known functon, whle the unknown functon u( x ) represents the approxmate soluton of the ntegral equaton. Basc prncple s that for converson of the ntegral equaton nto equvalent dfferental equaton wth ntal condtons. The converson s acheved by the well-known Lebntz rule [20]. umercal computatonal Procedure s as follows, Step : Dfferentatng (3.) twce w.r.t x, usng Lebntz rule, we get dfferental equatons wth subject to ntal condtons u(0), u'(0). Step 2: Applyng Haar wavelet collocaton method, Let us assume that, u( x) a h ( x) (3.2) Step 3: By ntegratng (3.2) twce and substtutng the ntal condtons, we get, u( x) a p ( x) (3.3), u( x) x a p ( x) (3.4) Step 4: Substtutng (3.2) - (3.4) n the dfferental equaton, whch reduces to the nonlnear system of equatons wth unknowns and then the ewton s method s used to obtan the Haar coeffcents a, =,2,...,. Substtutng Haar coeffcents n (3.4) to obtan the requred approxmate soluton of equaton (3.). 2, 4. ILLUSTRATIVE EXAMPLES In ths secton, we consder the some of the examples to demonstrate the capablty of the present method and error functon s presented to verfy the accuracy and effcency of the followng numercal results,
Haar Wavelet Collocaton Method for the umercal Soluton of onlnear 469 n Error E u ( x ) u ( x ) u ( x ) u ( x ) max 2 e a max e a where u e and u a are the exact and approxmate soluton respectvely. Example 4. Consder the onlnear Volterra-Fredholm-Hammersten ntegral equaton [7], 4 x x x 2 ( ) ( ) ( ) ( ) ( ), 0, (4.) u x x t u t dt x t u t dt x t 2 2 3 0 0 Intal condton s: u(0) 0, u'(0). Whch has the exact soluton u( x) x. Dfferentatng Eq. (4.) twce w.r.t x and usng Lebntz rule whch reduces to the dfferental equaton, Assume that, Integratng Eq. (4.4) twce, x 3 2 u( x) x u( t) dt u ( t) dt 2 3 (4.2) 0 0 2 2 ( ) [ ( )] 0 (4.3) u x u x x u( x) a h ( x) (4.4) u( x) a p ( x), (4.5) 2, (4.6) u( x) a p ( x) x Substtutng Eq. (4.4) Eq. (4.6) n Eq. (4.3), we get the system of equatons wth unknowns, 2 2 ah ( x) a p2, ( x) x x 0. (4.7) Solvng Eq. (4.7) usng ewton s method to fnd Haar wavelet coeffcents a s for = 6,.e., [-.93e-.93e- 8.48e-9 3.70e- -2.56e-8.08e-2 6.99e- -5.24e-20 -.59e-8-7.87e-8 8.86e-8 4.92e-3.53e-2 2.60e-2 9.36e-]. Substtutng a s, n Eq. (4.6) and obtaned the requred HWCM soluton wth exact soluton s presented n table 2. Error analyss s shown n table. Hence, justfes the effcency of the HWCM.
470 S. C. Shralashett, R. A. Mundewad, S. S. aregal and B. Veeresh Table. Error analyss of example 4.. E (HWCM) max 4.93e-3 8.68e-3 6 5.29e-3 32 6.66e-3 64 6.55e-3 28 9.38e-3 Table 2. Comparson of exact and approxmate soluton of example 4.. x(/32) Exact (HWCM) Error (HWCM) 0.0325 0.0325 0 3 0.09375 0.09375 0 5 0.5625 0.5625 0 7 0.2875 0.2875 0 9 0.2825 0.2825 0 0.34375 0.34375 0 3 0.40625 0.40625 0 5 0.46875 0.46875 0 7 0.5325 0.5325 0 9 0.59375 0.59375 4.44e-6 2 0.65625 0.65625 4.44e-5 23 0.7875 0.7875.4e-4 25 0.7825 0.7825 3.82e-4 27 0.84375 0.84375 7.70e-4 29 0.90625 0.90625.67e-3 3 0.96875 0.96875 5.29e-3
Haar Wavelet Collocaton Method for the umercal Soluton of onlnear 47 Example 4.2. ext, consder the onlnear Volterra-Hammersten Integral equaton [8], x 3 2x 2 ( ) [( ( )) ( )], 0, u x e u t u t dt x 2 2 (4.8) 0 wth ntal condtons u(0). Whch has the exact soluton ( ) x u x e. Successvely dfferentatng Eq. (4.8) w.r.t x and usng Lebntz rule reduces to the dfferental equaton, 2x 2 u '( x) e ( u( x) u( x)) (4.9) 2 2x u '( x) ( u( x) u( x)) e 0 (4.0) Assume that, 2M u '( x) a h ( x) (4.) Integratng Eq. (4.), 2M, (4.2) u( x) a p ( x) Substtutng Eqs. (4.) and (4.2) n Eq. (4.9), we get the system of equatons wth unknowns. 2M 2M 2 2M 2x ah ( x) a p, ( x) a p, ( x) e 0 (4.3) solvng (4.3) usng Matlab to fnd Haar wavelet coeffcents a s, for = 6.e, [- 0.63-0.6-0.0-0.06-0.06-0.04-0.03-0.03-0.03-0.03-0.02-0.02-0.02-0.02-0.0-0.0]. Substtutng a s, n Eq. (4.2) and obtaned the requred HWCM soluton compared wth exact solutons s shown n table 3. Error analyss s gven n table 4, whch justfes the effcency of the HWCM.
472 S. C. Shralashett, R. A. Mundewad, S. S. aregal and B. Veeresh Table 3. Comparson of Exact and HWCM for =6 of example of 4.2. x=(/32) HWCM Exact 0.9697 0.9692 3 0.909 0.905 5 0.8556 0.8553 7 0.8037 0.8035 9 0.7550 0.7548 0.7092 0.709 3 0.6662 0.666 5 0.6259 0.6258 7 0.5879 0.5879 9 0.5523 0.5523 2 0.588 0.588 23 0.4874 0.4874 25 0.4578 0.4578 27 0.430 0.430 29 0.4040 0.4040 3 0.3795 0.3796 Table 4. Error analyss of the example 4.2. E (HWCM) max 4 5.3e-3 8.6e-3 6 4.38e-4 32.5e-4 64 2.96e-5 28 7.52e-6
Haar Wavelet Collocaton Method for the umercal Soluton of onlnear 473 5. COCLUSIO In the present work, Haar wavelet collocaton method based on Lebntz rule s appled to obtan the numercal soluton of nonlnear Volterra-Fredholm-Hammersten ntegral equaton of the second knd. The Haar wavelet functon and ts operatonal matrx were employed to solve the resultant ntegral equatons. The numercal results are obtaned by the proposed method have been demonstrated n tables and fgures. Illustratve examples are tested wth error analyss to justfy the effcency and possblty of the proposed technque. ACKOWLEDGEMETS The authors thank for the fnancal support of UGC s UPE Fellowshp vde sancton letter D. O. o. F. 4-2/2008(S/PE), dated-9/06/202 and F. o. 4-2/202(S/PE), dated 22/0/203. REFERECES [] Beylkn, G., and Cofman, R., Rokhln, V., 99, Fast wavelet transforms and numercal algorthms I, Commun. Pure. Appl. Math., 44, pp. 4 83. [2] Chu, C. K., 997, Wavelets: A Mathematcal Tool for Sgnal Analyss, SIAM, Phladelpha, PA. [3] Chen, C. F., Hsao, C. H., 997, Haar wavelet method for solvng lumped and dstrbuted parameter systems, IEEE Proc. Pt. D. 44 (), pp. 87-94. [4] Islam, S., Azz, I., Sarler, B., 200, The numercal soluton of second order boundary value problems by collocaton method wth the Haar wavelets, Math. comp. Model. 52, pp. 577-590. [5] Shralashett, S. C., Angad, L. M., Desh, A. B., Kantl, M. H., 206, Haar wavelet method for the numercal soluton of Klen Gordan equatons, Asan-European J. Math. 9(0), 65002. [6] Shralashett, S. C., Desh, A. B., 206, An effcent haar wavelet collocaton method for the numercal soluton of mult-term fractonal dfferental equatons, onlnear Dyn. 83, pp. 293 303. [7] Shralashett, S. C., Desh, A. B., Mutalk Desa, P. B., 206, Haar wavelet collocaton method for the numercal soluton of sngular ntal value problems, An Shams Eng. J. 7(2), pp. 663-670. [8] Shralashett, S. C., Desh, A. B., 206, Haar Wavelet Collocaton Method for Solvng Rccat and Fractonal Rccat Dfferental Equatons, Bulletn. Math. Sc. Appl. 7, pp. 46-56.
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