Collocation Orthonormal Berntein Polynomials method for Solving Integral Equations.

Transcription

1 ISSN (Paper) ISSN (Online) Vol.5, No.2, 25 Collocation Orthonormal Berntein Polynomials method for Solving Integral Equations. Suha. N. Shihab; Asmaa. A. A.; Mayada. N.Mohammed Ali University of Technology Applied Science Department Abstract: In this paper, we use a combination of Orthonormal Bernstein functions on the interval [,] for degree m = 5,and 6 to produce anew approach implementing Bernstein Operational matri of derivative as a method for the numerical solution of linear Fredholm integral equations of the second kind and Volterra integral equations. The method converges rapidly to the eact solution and gives very accurate results even by low value of m. Illustrative eamples are included to demonstrate the validity and efficiency of the technique and convergence of method to the eact solution. Keywords: Bernstein polynomials, Operational Matri of Derivative, Linear Fredholm Integral Equations of the Second Kind and Volterra Integral Equations.. Introduction: In the Survey of solutions of integral equations, a large number of analytical but afew approimate methods for solving numerically various classes of integra equations []. Orthogonal functions and polynomial series have received considerable attention in dealing with various problems of dynamical Systems. The main Characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem [2,4]. While in recent years interest in the solution of integral and differential equations, such as Fredholm, Volterra, and integro-differential equations [3]. The general form of Fredholm, Volterra integral equations respectively are - Freholm integral equation: (FIE) u() = g() + k(, t)u(t)dt [,] 2- Volterra integral equation: (VIE) u() = f() + k(, t)u(t)dt [,] () (2) Integral equations are widely used for solving many problems in mathematical physics and engineering. In recent years, many different basic functions have been used to estimate the solution of integral equations, such as Block-Pulse functions [4,5], Hybrid Legendre and Block-Pulse functions. Bernstein polynomials play a promineut role in various areas of mathematics. These polynomials, have been frequently used in the solution of integral equations,differential equations and approimation theory,see [6]. Recently the various operational matrices of the polynomials have been developed to cover the numerical solution of differential, integral and integro-differential equations. In [7] the operational matrices of Bernstein polynomials are introduced. Doha [8] has drived the shifted Jacobi operational matri of fractional derivatives which is applied together with the Spectral Tau method for the numerical solution of dynamical systems.yousefi et al.in [9], [] and [] have presented Legendre wavelets and Berntein operational matrices and used them to solve miscellaneous systems. Lakestani et al. [2] constracted the operational matri of fractional derivatives using B-Spline functions. Another motivation is concerned with the direct solution techniques for solving the Fredholm and Volterra integral equations respectively on the interval [,] using the method based on the derivatives of orthonormal (Bpolynomials) sense for m=5 and 6. Finally, the accuracy of the proposed algorithm is demonstrated by test problems. 2-Bernstein Polynomials (B-Polynomials): The Bernstein polynomias (B-Polynomials) [3], are some usfel polynomials defined on [,]. The Bernstein Polynamials of degree m form a basis for the power polynomials of degree m. we can mentioned, B- Polynomials are aset of Polynomials B k,m () = ( m k ) k ( ) m k, k m, (3)

2 ISSN (Paper) ISSN (Online) Vol.5, No.2, 25 Note that each of these m+ Polynomials having degree m is normalization, i.e m B k,m () =, has one root, each of multiplicity k and m-k, at = and = respectively,also B k,m () in k= which k {, m}has asingle unique local maimum of k k m m (m k) m k ( m ), it can provide fleibility k applicable to impose boundary conditions at the end points of the interval. First derivatives of the generalized Bernstein basis polynomials. d B d k,m() = m[b k,m () + B k,m ()] (4) In this paper, we use m () notation to show where we can have m () = [B m (), B m (),, B mm ()] T (5) m () = A m () m () (6) that A is the matri and (k + ) th row of A is A k+ = [,, k times,, s_(, k, m), s_(, k, m),, s_(m, k, m) ] =[,, k times,, ( ) ( m k ) (m k ), ( ) ( m k ) (m k ),, ( )m k ( m k ) (m )] (7) k m k where S i,k,m = ( ) i ( m k k ) (m ) i and n () = [ ] (8) m using MATHEMATICA code, the first si (B-Polynomials) of degree five over the interval [,], are given B 5 () = ( ) 5 B 5 () = 5( ) 4 B 25 () = 2 ( ) 3 B 35 () = 3 ( ) 2 B 45 () = 5 4 ( ) B 55 () = 5 and the first seven (B-Polynomials) of degree si over [,] are given B 6 () = ( ) 6 B 6 () = 6( ) 5 B 26 () = 5 2 ( ) 4 B 36 () = 2 3 ( ) 3 B 46 () = 5 4 ( ) 2 B 56 () = 6 5 ( ) B 66 () = 6 2

3 ISSN (Paper) ISSN (Online) Vol.5, No.2, (B-Polynomials) Approimation: A function u() equation, square integrable in (,), many be epressed in terms of Bernstein basis [7]. In practice, only the first (m+)-terms Bernstein polynomials are considered. Hence if we write u() m k= c k B km () = C T () (9) where T () = [B m (), B m (),, B mm ()] and C T = [c, c,, c m ] can be calculated by: C T = ( u() T ()d) Q, () where Q is an (m + ) (m + ) matri and is said dual matri of () Q() = ( (), ()) = () T ()d = (A n ())(A n ()) T d = A [ n () T n ()d] A T = AHA T, () A is defined by eqs.(7) and H is a Hilbert matri H = [ m+ m+2 m+ m+2 2m+] and Q Q Q = [ 2 ] Q m The elements of the dual matri Q, are given eplicity by (Q m ) k+,i+ = B km ()B im ()d = ( m k ) (m i ) ( )2m (k+i) k+i d (2) where k, i =,,, m 4- The Derivative for Orthonormal (B-Polynomials): The representation of the orthonormal Bernstein Polynomials, denoted by b i5 (), b i6 () here, was discovered by analyzing the resulting orthonormal polynomials after applying the Gram-Schmidt process on sets of Bernstein polynomials of degree five and si. We get the following sets of orthonormal polynomials [],[]. b 5 () = ( ) 5 b 5 () = 6 [5t( ) 4 2 ( )5 ] b 25 () = 8 7 [( 5 )3 t 2 5( ) 4 t + 5 ( 8 )5 ] b 35 () = 28 [( 5 )2 3 5( ) ( 7 )4 5 ( 28 )5 ] 3

4 ISSN (Paper) ISSN (Online) Vol.5, No.2, 25 b 45 () = 7 3 [5( ) 4 2( ) ( ) 3 2 4( ) ( )5 ] b 55 () = 6 [ ( )4 + 3 ( )2 3 25( ) ( ) 4 6 ( )5 ] and b 6 () = 3( ) 6 b 6 () = 44 [6t( ) 5 2 ( )6 ] b 26 () = [5( ) 4 2 6( ) ( )6 ] b 36 () = 252 [2( ) ( 2 ) ( ) 5 ( 66 )6 ] b 46 () = 42 [5( 5 )2 4 4( ) ( 7 )4 2 3 ( 7 )5 + 5 ( 42 )6 ] b 56 () = 28 3 [6t5 ( ) 75 ( 2 ) ( ) 3 3 3( ) ( 7 )5 3 ( 28 )6 ] b 66 () = 7 [ 6 8( ) ( ) 2 4 ( ) ( ) 4 2 6t( ) ( )6 ] [6] In addition, we have determind the eplicit representation for the orthonormal Bernstein polynomials as b km () = ( 2(m k) + ) ( ) m k ( ) i 2m + i ( ) ( k k i i ) k i k i= (3) The eqs(3) can be written in terms of the Bernstein basis functions as (2m+ i)( k i i ) ( m i k i ) k b km () = ( 2(m k) + ) ( ) k i i= B k i,m i () (4) Any generalized Bernstein basis polynomials of degree m can be wretten as a linear combination of the generalized Bernstein basis polynomials of degree m+ B k,m () = m k+ B m+ k,m+() + k+ B m+ k+,m+() (5) By utilizing eqs(5),the following functions can be written as B k,m () = m k B m k,m() + k+ B m k+,m() (6) and B k,m () = m k+ m B k,m() + k m B k,m() (7) Substituting these eqs(6) and (7) into the right hand side of the eqs(4), we get the following derivatives of Bernstein basis polynomials d d B k,m() = (m k + )B k,m () + (2k m)b k,m () (k + )B k+,m () (8) In [], the derivative of the orthonormal (B-Polynomials)of degree five are introducedas given b 5 = [ 5 B 5 B 5 ] b 5 = [45B 5 5B 5 2B 25 ] = [23 7 B B B B 5 35 ] b 25 4

5 ISSN (Paper) ISSN (Online) Vol.5, No.2, 25 b 35 = [ ] = [ 33 3B b 45 5 B B B B B 55 ] b 55 = [35B 5 55B B B 35 9B B 55 ] and we introduce the derivative of the orthonormal (B-Polynomials) for degree si b 6 = [ 6 B 6 3B 6 ] b 6 = [8 B 6 7 B 6 4 B 26 ] b 26 = [ 84 B B 6 + B 26 3B 36 ] b 36 = [36 7B B B B B 46 ] b 46 = [ 42 5B B B B B B 56 ] b 56 = [46 3B B B B B B B 66 ] b 66 = [ 48B B 6 68B B B B B 66 ] 5. Second kind integral equations: In this section, we use Orthonormal Polynomials for solving second kind Fredholm and Volterra integral equations. - Fredholm integral equation (FIE): Where in eq() g() L 2 [,], k(, t) L 2 ([,] [,]) are known and u(t) is unknown function to be determined. First we assume the unknown functions u i () = C i T B(), i =,2,, n (9) by substituting (9)in () we have: C T i B() k i,j (, t) Pick distinct node points t, t 2,, t n [,] C T i B() = g i () + k i,j (, t) This leads to determining { c, c 2,, c n } as the solution of linear system C i T B(t)dt C i T B(t)dt = g i () (2) n i= C j [B( i ) k( j, t i ) B(t i )dt] = g( i ) (2) In this paper Collocation points are t i = i n for i =,2,, n so that we have a system of linear equations L n X = l n where n L n = B( i ) k( j, t i ) B(t i )dt i= j =,2,, n l n = [g( i )], i = o,,, n 2- Volterra integral equation (VIE): Similarly above section by using Collocation points t i = i n for i =,2,, n 5

6 ISSN (Paper) ISSN (Online) Vol.5, No.2, 25 n L n = B( i ) k( j, t i ) B(t i )dt i= j =,2,, n l n = [f( i )], i = o,,, n 6.Numerical Results: In this section VIE, FIE is considered and solved by the introduced method. Eample : Consider the following FIE u() = sin + ( cos t) u(t)dt (22) the eact solution u()=. Solving eqs(2) and (2) we get the values of C T C=[ ] T Table shows the numerical results for this eample. Table :some numerical results for eample Approimat Approimat AbsouteError eact B n6 solution b n () solution B n () e e e e e e e e e e M. S. E =6.744e-5 L. S. E=2.286e-8 Eample 2: Consider the following FIE u() = e e t u(t)dt the eact solution u() = e T. Solving eqs(2) and (2) we get the values of C 2 C=[ ] T Table2 shows the numerical results for this eample. 6

7 ISSN (Paper) ISSN (Online) Vol.5, No.2, 25 Table2:some numerical results for eample 2 Eact solution Approimat solution b n () Approimat solution B n () AbsouteError eact B n M. S. E =.3522 L. S. E=..2 eact Bn Fig of eample 7

8 ISSN (Paper) ISSN (Online) Vol.5, No.2, eact Bn Fig 2 of eample3 Eample 3: Consider the following VIE u() = ( t)u(t)dt (23) The eact solution u() = sin. Table(3) shows the numerical results for this eample(3) C=[ ] T Table3:some numerical results for eample 3 Eact solution Approimat solution b n () Approimat solution B n () AbsouteError eact B n M. S. E =.495 L. S. E=. 8

9 ISSN (Paper) ISSN (Online) Vol.5, No.2, 25 Conclusion: In this work, VIE,FIE have been solved by using Bernstein basis polynomials of degree m in collocation method. Comparison of the approimate solutions and the eact solutions show that the proposed method is efficient tool. Illustrative eamples are included to demonstrate the validity and applicability of the technique. References:. S.Swarup, Integral equations (Krishna Prakanshan Media Prt.Ltd,5 th Editions,27). 2. Shihab. S. N. and Asmaa. A. A. 22. Numerical Solution of Calculus of Variations by using the Second Chebyshev Wavelets, Eng. & Tech. Journal. 3(8): H. Goghary. H. Goghary. M, Tow Computational methods for Solving linear Fredholm fuzzy integral equations of the Second kind, APPl. Math. Comput, Vol 82, PP , K. Maleknejad, S.Sohrab, B. Bevenj-, Application of D-BPFS to nonlinear integral equation, Commun Non linear Sci Number Simulat, 5,PP ,2. 5. K.Maleknejad, M.Mordad, B.Raimi, Anumerical method to solve Fredholm Volterra integral equations two dimensional spaces using Block Pulse Functions and operational Matri, Journal of Computational and Applied Mathematics, E. H.Doha, A.H.Bhrawy, M.A.Saher, On the derivatives of Bernstein Polynomials: An application for the Solution of high even-order differential equations, Boundary Value Problems Vol, 2, Article I D & 29543,6 pages doi:.55/2/829543,2. 7. S. A. Yousefi, M. Behroozifar, Operational matrices of Bernstein Polynomials and their applications, Int. J. Syst.Sci.4 (6) (2) E. H. Doha and A. H. Bhrawy S. S. Ezz-Eldien, Anew Jacobi Operational matri: An Application for Solving frastional differential equations, Appl. Math. Model. 36, pp S. A. Yousefi and H. Jafari and M. A. Firooz jaecard. S. Momani and C.M.Khaliqued, Application of Legendre wavelets for solving fractional differential equations, Comput. Math. Appl. 62, (2).. S. A. Yousefi and M.Behroozifar and M. Dehghan, Numerical Solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials, Appl. Math. Modell. 36, (22).. S.A.Yousefi and M. Behroozifar and M. Dehghan, The Operational matrices of Bernstein Polynomials for Solving the Parabolic equation subject to specification of the ma ss, J. Comput. Appl.Math. 335, (2). 2. Lakestani, M, Dehghan, M, Irandoust-Pakchin, S: The Construction of Operational matri of fractional derivatives using B-Spline functions, Commun Nonlinear Sci Number simulat. 7, (22). 3. Vineet Kumar Singh, Eugene B. Postnikov, Operational matri approach for solution of integrodifferential equations arising in theory of an omalous relaation processes in Vicinity of singular point, Applied Mathematical Modelling 37,(23), Asmaa. A. A. 24. Numerical solution of Optimal problems using new third kind Chebyshev Wavelets Operational matri of integration, Eng. & Tec. Journal. 32():