Bernoulli Wavelet Based Numerical Method for Solving Fredholm Integral Equations of the Second Kind

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1 ISSN , England, UK Journal of Inforation and Coputing Science Vol., No., 6, pp.-9 Bernoulli Wavelet Based Nuerical Method for Solving Fredhol Integral Equations of the Second Kind S. C. Shiralashetti*, R. A. Mundewadi Departent of Matheatics, Karnatak University Dharwad-583, Karnataka, India, E-ail: Mob: , Phone: (O), Fax: b884. (Received Septeber 3, 5, accepted February, 6) Abstract. In this paper, a Bernoulli wavelet based nuerical ethod for the solution of Fredhol integral equations of the second kind is proposed. he ethod is based upon Bernoulli wavelet approxiations. he Bernoulli wavelet (BW) is first presented and the resulting Bernoulli wavelet atrices are utilized to reduce the Fredhol integral equations into algebraic equations. Solving these equations using MALAB to obtain Bernoulli coefficients. he nuerical results of the proposed ethod through the illustrative exaples is presented in coparison with the exact and existing ethods (Haar wavelet ethod (HWM) [3], Herite cubic splines (HCS) []) of solution fro the literature are shown in tables and figures, which show that the validity and applicability of the technique with higher accuracy even for the saller values of N. Keywords: Bernoulli Wavelet, Haar wavelet, Herite cubic splines, Bernoulli Polynoials, Bernoulli nubers, Fredhol Integral equations.. Introduction Wavelets theory is a relatively new and an eerging tool in applied atheatical research area. It has been applied in a wide range of engineering disciplines; particularly, signal analysis for wavefor representation and segentations, tie-frequency analysis and fast algoriths for easy ipleentation. Wavelets perit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast nuerical algoriths. Since fro 99 the various types of wavelet ethod have been applied for nuerical solution of different kinds of integral equation, a detailed survey on these papers can be found in [-6]. Consider the Fredhol integral equation of the second kind: f() t y( t) f ( t) K( t, s) y( s) ds ts,, () where and the kernels k( t, s) are assued to be in L (R) on the interval ts,. We assue that Eq.() has a unique solution y to be deterined. Integral equations find its applications in various fields of science and engineering. here are several nuerical ethods for approxiating the solution of Fredhol integral equations are known and any different basic functions have been used. Such as Galerkin ethods for the constructions of orthonoral wavelet bases approached by Liang et.al [7], Maleknejad et al. [8] used the continuous Legendre wavelets, a cobination of Hybrid taylor and block-pulse functions [9], Rationalized haar wavelet [], Herite Cubic splines [], Coifan wavelet as scaling functions []. Lepik et al. [3] applied the Haar Wavelets, Yousefi et al. [4] have introduced a new CAS wavelet, Babolian et al. [5] derived the operational atrix for the product of two triangular orthogonal functions, Muthuvalu et al. [6] applied Half-sweep arithetic ean ethod with coposite trapezoidal schee for the solution of Fredhol integral equations. Keshavarz et al. [7] applied Bernoulli wavelet operational atrix for the approxiate solution of fractional order differential equations. In this paper, we introduced the nuerical ethod based on Bernoulli wavelets approxiations for solving Fredhol integral equations. he article is organized as follows: In Section, the basic forulation of Bernoulli wavelets and the function approxiation is presented. Section 3 is devoted to the ethod of solution. In section 4, we report our nuerical findings and deonstrated the accuracy of the proposed schee by considering illustrative exaples. Conclusion of the proposed ethod is discussed in section 5. Published by World Acadeic Press, World Acadeic Union

2 S. C. Shiralashetti et al.: Bernoulli Wavelet Based Nuerical Method for Solving Fredhol Integral Equations of the Second Kind. Bernoulli Wavelets and Function Approxiation Bernoulli wavelets are b ( ) (, ˆ,, ) n, t b k n t k have four arguents; nˆ n, n,,3,...,, k is any positive integer, is the order of Bernoulli polynoials and t is the noralized tie. hen it can be defined [7] on the interval [, ) as follows, with ( ) k b.n (t) = { k β ( k n t n ), k t< n k, else,, () ( ),, t t ( ) (!) ( )! where =,,,...,M- and n =,,..., k-. he coefficient ˆ k ( ) n () is for norality, ( ) (!) ( )! i is the dilation paraeter, is the translation paraeter and () t it are the i i well-known Bernoulli polynoials of order. Where i, i,,..., are Bernoulli nubers. hese nubers are a sequence of signed rational nubers which arise in the series expansion of trigonoetric functions and can be defined by the identity, i t t t i e i i!. he first few Bernoulli nubers are 5,,, 4, 6, 8,, With i, i,,3,... he first few Bernoulli Polynoials are ( t), ( t) t, ( t) t t, ( t) t t t, ( t) t t t, ( t) t t t t, ( t) t 3 t t t, he six basis functions are given by: b( t) b( t) 6 (4t ) ; t, b( t) (4t t ) b( t) b( t) 6 (4t 3) ; t b( t) (4t 36t 3) JIC eail for contribution: editor@jic.org.uk

3 Journal of Inforation and Coputing Science, Vol. (6) No., pp -9 3 For k = iplies n =, and M=3 iplies =,, then Eq.() gives the Bernoulli wavelet atrix of k order ( N M) 6x6 as, B For k = and M = 4 of order 8x8 as, B A function f ( x) L [,] ay be expanded as: where f ( t) c b ( t), (3) n n, n, n, n, c ( f ( t), b ( t)). In (4), (.,.) denotes the inner product. If the infinite series in (3) is truncated, then (3) can be rewritten as: where C and and Bt () k (4) M n, n, (5) n f ( t) c b ( t) C B( t), are N atrices given by: C [ c, c,..., c, c,..., c,..., c,..., c ], M, M k k,, M [ c, c,..., c ], k M B( t) [ b ( t), b ( t),..., b ( t), b ( t),..., b ( t),..., b ( t),..., b ( t)], M, M k k,, M [ b ( t), b ( t),..., b ( t)]. k M Siilarly a function k( t, s) L ([,] [,]) ay be approxiated as: where K is N k( t, s) B ( t) KB( s ), (8) N atrix, with K ( b ( t),( k( t, s), b ( s))). ij i j he integration of the product of two Bernoulli wavelet function vectors is obtained as: I B( t) B ( t) dt, (9) (6) (7) JIC eail for subscription: publishing@wau.org.uk

4 4 S. C. Shiralashetti et al.: Bernoulli Wavelet Based Nuerical Method for Solving Fredhol Integral Equations of the Second Kind where I is an identity atrix. And also, we approxiate y( t) Y B( t), () where, Y is a vector of Bernoulli Coefficients and is the unknown function. 3. Method of solution In this section, we present a Bernoulli wavelet ethod (BWM) for solving Fredhol integral equation of the second kind of the for (), where ( ) [,), (, ) ([,) [,)) y( t) f ( t) k( t, s) y( s) ds, f t L k t s L and Let us approxiate f() t,, and k( t, s) SEP : f ( t) C B( t ), y( t) Y B( t) SEP : k( t, s) B ( t) KB( s ) SEP 3: Substituting f() t and k( t, s) is an unknown function. by using the procedure as follows: in Eq.(), we have: B ( t) Y B ( t) C B ( t) KB( s) B ( s) Yds B ( t) Y B ( t) C B ( t) K B( s) B ( s) ds Y B ( t) Y B ( t)( C KY ), then ( I K) Y C. By solving this syste we obtain the vector Y and substituting this Y in step 4. SEP 4: y( t) Y B( t ) which is the required approxiate solution of Eq. (). 4. Illustrative Exaples In this section we consider the soe of the exaples to deonstrate the capability of the ethod and error function is presented to verify the accuracy and efficiency of the following nuerical results: where n Error function y ( t ) y ( t ) y ( t ) y ( t ) e i a i e i a i i y e and y a are the exact and approxiate solution respectively. Exaple. Let us consider the Fredhol integral equation of the second kind. y( t) t ( t s) y( s) ds () he solution of () with the help of Bernoulli coefficients Y as [-.7875, -.74,.3,.56e-6, -.85, ,.3, 3.e-6, -.785, -.353,.3, 3.5e-6, , -.77,.3, 4.e-6] is obtained using the ethod described in section 3. he coputational results for k = 3, M = 4 (N = 6) are copared with the exact solution y( t) t 5t 7 / 6 and HWM [3] are given in table & for k = 4, M = 4 (N = 3) in Fig.. Error analysis is shown in table. able. Coparison of BWM and HWM solutions with exact solution and errors for N=6. JIC eail for contribution: editor@jic.org.uk

5 Journal of Inforation and Coputing Science, Vol. (6) No., pp -9 5 t Exact BWM HWM Error(BWM) Error(HWM) e e e e e e e e e e e e e e e e BWM EXAC HWM y x Fig.. Coparison of BWM and HWM solutions with exact solution for N=3. able. Error analysis of exaple. N L ( BWM ) L ( HWM ) 8.48 e e e e e e e-4.78 e e e-4 Exaple. Next, consider y( t) sin( t) cos( t) y( s) ds () JIC eail for subscription: publishing@wau.org.uk

6 6 S. C. Shiralashetti et al.: Bernoulli Wavelet Based Nuerical Method for Solving Fredhol Integral Equations of the Second Kind which has the exact solution y( t) sin( t). We applied the Bernoulli wavelets approach and solved Eq. (). able 3 for k = 3 and M = 4 (N = 6) presents values of with the help of the Bernoulli coefficients Y as [.386,.443, -.347, -.77,.386, -.443, -.347,.77, -.386, -.443,.347,.77, -.386,.443,.347, -.77] is obtained using the present ethod together with the exact values and presented in Fig. for k = 3 and M = 4 (N = 3). Error analysis is copared with ethod [] are shown in table 4. able 3. Coparison of BWM solutions with exact solutions for N=6. t Exaple Exaple 3 Exaple 4 BWM Exact BWM Exact BWM Exact BWM EXAC.6.4. y x Fig.. Coparision of BWM with exact solution for N=3. Exaple 3.hirdly, consider y( t) sin( t) ( t t s s) y( s) ds (3) JIC eail for contribution: editor@jic.org.uk

7 Journal of Inforation and Coputing Science, Vol. (6) No., pp -9 7 has the exact solution y( t) sin( t). By using the ethod presented in section 3 and solved Eq. (3). he solution with the Bernoulli coefficients Y as [.386,.443, -.347, -.77,.386, -.443, -.347,.77, -.386, -.443,.347,.77, -.386,.443,.347, -.77] is obtained and copared with the exact solution and ethod [] given in table 3 for k = 3 and M = 4 (N = 6) and Fig. 3 for k = 3 and M = 4 (N = 3). Error analysis is shown in table 4. Exaple 4. Finally, consider 3 y( t) t 3 t t ( t t s s) y( s) ds (4) he approxiate solution of (4) with the help of Bernoulli coefficients Y as [-.355, -.353,.54, -.53, -.734,.353,.74, -.53,.734,.353, -.74, -.53,.355, -.353, -.54, -.53] is obtained using the proposed ethod. he nuerical values are copared with 3 the exact solution y( t) t 3t t and ethod [] are shown in table 3 for k = 3 and M = 4 (N = 6) and in Fig. 4 for k = 3 and M = 4 (N = 3). Error analysis is presented in table 4. N able 4. Error analysis of BWM with the ethod []. Exaple Exaple 3 Exaple 4 L ( BWM ) L ( Method[]) L ( BWM ) L ( Method[]) L ( BWM ) L ( Method[]) e e-. e e e e e e e e e e e-4.4 e-6.99 e e e e-5.48 e e-6.78 e e e-.8 BWM EXAC.6.4. y x Fig. 3. Coparision of BWM with exact solution for N=3. JIC eail for subscription: publishing@wau.org.uk

8 8 S. C. Shiralashetti et al.: Bernoulli Wavelet Based Nuerical Method for Solving Fredhol Integral Equations of the Second Kind..8 BWM EXAC.6.4. y 5. Conclusion Fig. 4. Coparision of BWM with exact solution for N = 3. he proposed work is based on Bernoulli wavelet atrix is applied for the nuerical solution of Fredhol integral equations of the second kind. Our nuerical findings are copared with the solutions obtained by existing ethods [, 3] and exact solutions. For instance in exaple, our results are higher accuracy with exact ones than the other ethod. Subsequently other exaples are also sae in the nature. Error analysis shows the accuracy and effectiveness of the described ethod. he efficiency of the present ethod is approached through the illustrative exaples. 6. Acknowledgeent he authors thank for the financial support of UGC s UPE Fellowship vide sanction letter D.O. No. F. 4-/8(NS/PE), dated-9/6/ and F. No. 4-/(NS/PE), dated //3. 7. References x [] C. K. Chui, Wavelets: A Matheatical ool for Signal Analysis, SIAM, Philadelphia, PA, 997. [] G. Beylkin, R. Coifan, V. Rokhlin, Fast wavelet transfors and nuerical algoriths I, Coun. Pure Appl. Math. 44 (99) [3] M. Shasi, M. Razzaghi, Solution of Hallen s integral equation using ultiwavelets, Coput. Phys. Coun. 68 (5) [4] M. Lakestani, M. Razzaghi, M. Dehghan, Sei orthogonal spline wavelets approxiation for Fredhol integrodifferential equations, Math. Probl. Eng, (6). Article ID [5] I. Aziz, S. Isla, New algoriths for the nuerical solution of nonlinear Fredhol and Volterra integral equations using Haar wavelets, Jour. Cop. Appl. Math. 39 (3) [6] U. Lepik, E. ae, Solution of nonlinear Fredhol integral equations via the Haar wavelet ethod, Proc. Estonian Acad. Sci. Phys. Math. 56,, (7) 7-7. [7] X. Liang, M. Liu, X. Che, Solving second kind integral equations by Galerkin ethods with continuous orthogonal wavelets, J. Cop. App. Math. 36 () [8] K. Maleknejad, M. avassoli Kajani, Y. Mahoudi, Nuerical solution of linear Fredhol and Volterra integral equation of the second kind by using Legendre wavelets, Kybernetes, vol. 3 No. 9/, (3) [9] K. Maleknejad, Y. Mahoudi, Nuerical solution of linear Fredhol integral equation by using hybrid aylor and JIC eail for contribution: editor@jic.org.uk

9 Journal of Inforation and Coputing Science, Vol. (6) No., pp -9 9 Block-Pulse functions, App. Math. Cop. 49 (4) [] K. Maleknejad, F. Mirzaee, Using rationalized Haar wavelet for solving linear integral equations, App. Math. Cop. 6 (5) [] K. Maleknejad, M. Yousefi, Nuerical solution of the integral equation of the second kind by using wavelet bases of herite cubic splines, App. Math. Cop. 83 (6) [] K. Maleknejad,. Lotfi, Y. Rostai, Nuerical coputational ethod in solving fredhol integral equations of the second kind by using coifan wavelet, App. Math. Cop. 86 (7) -8. [3] Ü. Lepik, E. ae, Application of the Haar wavelets for solution of linear integral Equations, Ant. urk Dyna. Sys. Appl. Proce. (5), [4] S. Yousefi, A. Banifatei, Nuerical solution of Fredhol integral equations by using CAS wavelets, App. Math. Cop. 83 (6) [5] E. Babolian, H.R. Marzban, M. Salani, Using triangular orthogonal functions for solving fredhol integral equations of the second kind, App. Math. Cop. (8) [6] M. S. Muthuvalu, J. Sulaian, Half-Sweep Arithetic Mean ethod with coposite trapezoidal schee for solving linear fredhol integral equations, App. Math. Cop. 7 () [7] E. Keshavarz, Y. Ordokhani, M. Razzaghi, Bernoulli wavelet operational atrix of fractional order integration and its applications in solving the fractional order differential equations, App. Math. Model. 38 (4) JIC eail for subscription: publishing@wau.org.uk