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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 [0,1] for degree

๐‘š = 5,and 6 to produce anew approach implementing Bernstein Operational matrix 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 exact solution and gives very accurate results even by low value of m. Illustrative examples are included to demonstrate the validity and efficiency of the technique and convergence of method to the exact solution.

Keywords: Bernstein polynomials, Operational Matrix of Derivative, Linear Fredholm Integral Equations of the Second Kind and Volterra Integral Equations.

1. Introduction:

In the Survey of solutions of integral equations, a large number of analytical but afew approximate methods for solving numerically various classes of integra equations [1]. 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 [2,14]. 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

1-

Freholm integral equation: (FIE)

๐‘ข(๐‘ฅ) = ๐‘”(๐‘ฅ) + โˆซ ๐‘˜(๐‘ฅ, ๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘ก ๐‘ฅ โˆˆ [0,1]01 (1)

2-

Volterra integral equation: (VIE)

๐‘ข(๐‘ฅ) = ๐‘“(๐‘ฅ) + โˆซ ๐‘˜(๐‘ฅ, ๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘ก ๐‘ฅ โˆˆ [0,1]0๐‘ฅ (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 approximation 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 matrix of fractional derivatives which is applied together with the Spectral Tau method for the numerical solution of dynamical systems.Yousefi et al.in [9], [10] and [11] have presented Legendre wavelets and Berntein operational matrices and used them to solve miscellaneous systems. Lakestani et al. [12] constracted the operational matrix 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 [0,1] using the method based on the derivatives of orthonormal (B-polynomials) 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) [13], are some usfel polynomials defined on [0,1]. The Bernstein Polynamials of degree m form a basis for the power polynomials of degree m. we can mentioned, B-Polynomials are aset of B-Polynomials

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Note that each of these m+1 Polynomials having degree m is normalization, i.e

โˆ‘๐‘š๐‘˜=0๐ต๐‘˜,๐‘š(๐‘ฅ) = 1, has one root, each of multiplicity k and m-k, at x=0 and x=1 respectively,also ๐ต๐‘˜,๐‘š(๐‘ฅ) in which ๐‘˜๏ƒ{0, ๐‘š}has asingle unique local maximum of ๐‘˜๐‘š๐‘˜โˆ’๐‘š(๐‘š โˆ’ ๐‘˜)๐‘šโˆ’๐‘˜(๐‘š๐‘˜ ), it can provide flexibility applicable to impose boundary conditions at the end points of the interval. First derivatives of the generalized Bernstein basis polynomials.

๐‘‘๐‘ฅ๐‘‘ ๐ต๐‘˜,๐‘š(๐‘ฅ) = ๐‘š[๐ต๐‘˜โˆ’1,๐‘šโˆ’1(๐‘ฅ) + ๐ต๐‘˜,๐‘šโˆ’1(๐‘ฅ)] (4) In this paper, we use ๏™๐‘š(๐‘ฅ) notation to show

๏™๐‘š(๐‘ฅ) = [๐ต0๐‘š(๐‘ฅ), ๐ต1๐‘š(๐‘ฅ), โ€ฆ , ๐ต๐‘š๐‘š(๐‘ฅ)]๐‘‡ (5) where we can have

๏™๐‘š(๐‘ฅ) = ๐ด๐‘š(๐‘ฅ)โˆ†๐‘š(๐‘ฅ) (6)

that A is the matrix and (๐‘˜ + 1)๐‘กโ„Ž row of A is

๐ด๐‘˜+1= [0,0, โ€ฆ๐‘˜ ๐‘ก๐‘–๐‘š๐‘’๐‘ , 0, ๐‘ _(0, ๐‘˜, ๐‘š), ๐‘ _(1, ๐‘˜, ๐‘š), โ€ฆ , ๐‘ _(๐‘š, ๐‘˜, ๐‘š) ] =[0,0, โ€ฆ๐‘˜ ๐‘ก๐‘–๐‘š๐‘’๐‘ , 0, (โˆ’1)0(๐‘š๐‘˜ ) (๐‘š โˆ’ ๐‘˜ 0 ) , (โˆ’1)1( ๐‘š ๐‘˜ ) (๐‘š โˆ’ ๐‘˜1 ) , โ€ฆ , (โˆ’1)๐‘šโˆ’๐‘˜( ๐‘š ๐‘˜ ) (๐‘š โˆ’ ๐‘˜๐‘š โˆ’ ๐‘˜)] (7) where ๐‘†๐‘–,๐‘˜,๐‘š= (โˆ’1)๐‘–(๐‘š๐‘˜ ) (๐‘š โˆ’ ๐‘˜๐‘– ) and โˆ†๐‘›(๐‘ฅ) = [ ๐‘ฅ0 ๐‘ฅ1 โ‹ฎ ๐‘ฅ๐‘š ] (8)

using MATHEMATICA code, the first six (B-Polynomials) of degree five over the interval [0,1], are given

๐ต05(๐‘ฅ) = (1 โˆ’ ๐‘ฅ)5 ๐ต15(๐‘ฅ) = 5๐‘ฅ(1 โˆ’ ๐‘ฅ)4 ๐ต25(๐‘ฅ) = 10๐‘ฅ2(1 โˆ’ ๐‘ฅ)3 ๐ต35(๐‘ฅ) = 10๐‘ฅ3(1 โˆ’ ๐‘ฅ)2 ๐ต45(๐‘ฅ) = 5๐‘ฅ4(1 โˆ’ ๐‘ฅ) ๐ต55(๐‘ฅ) = ๐‘ฅ5

and the first seven (B-Polynomials) of degree six over [0,1] are given

๐ต06(๐‘ฅ) = (1 โˆ’ ๐‘ฅ)6 ๐ต16(๐‘ฅ) = 6๐‘ฅ(1 โˆ’ ๐‘ฅ)5 ๐ต26(๐‘ฅ) = 15๐‘ฅ2(1 โˆ’ ๐‘ฅ)4 ๐ต36(๐‘ฅ) = 20๐‘ฅ3(1 โˆ’ ๐‘ฅ)3 ๐ต46(๐‘ฅ) = 15๐‘ฅ4(1 โˆ’ ๐‘ฅ)2 ๐ต56(๐‘ฅ) = 6๐‘ฅ5(1 โˆ’ ๐‘ฅ) ๐ต66(๐‘ฅ) = ๐‘ฅ6

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3-

(B-Polynomials) Approximation:

A function u(x) equation, square integrable in (0,1), many be expressed in terms of Bernstein basis [7]. In practice, only the first (m+1)-terms Bernstein polynomials are considered. Hence if we write

๐‘ข(๐‘ฅ) โ‰ˆ โˆ‘๐‘š๐‘˜=0๐‘๐‘˜๐ต๐‘˜๐‘š(๐‘ฅ) = ๐ถ๐‘‡๏ƒ†(๐‘ฅ) (9) where ๏ƒ†๐‘‡(๐‘ฅ) = [๐ต0๐‘š(๐‘ฅ), ๐ต1๐‘š(๐‘ฅ), โ€ฆ , ๐ต๐‘š๐‘š(๐‘ฅ)] and

๐ถ๐‘‡= [๐‘0, ๐‘1, โ€ฆ , ๐‘๐‘š] can be calculated by:

๐ถ๐‘‡= (โˆซ ๐‘ข(๐‘ฅ)โˆ…01 ๐‘‡(๐‘ฅ)๐‘‘๐‘ฅ) ๐‘„โˆ’1, (10)

where ๐‘„ is an (๐‘š + 1) ร— (๐‘š + 1) matrix and is said dual matrix of ๏ƒ†(๐‘ฅ)

๐‘„(๐‘ฅ) = (๏ƒ†(๐‘ฅ),๏ƒ†(๐‘ฅ)) = โˆซ โˆ…(๐‘ฅ) โˆ…1 ๐‘‡(๐‘ฅ)๐‘‘๐‘ฅ 0 =โˆซ (๐ดโˆ†๐‘›(๐‘ฅ))(๐ดโˆ†๐‘›(๐‘ฅ)) ๐‘‡ ๐‘‘๐‘ฅ 1 0 = ๐ด [โˆซ โˆ†01 ๐‘›(๐‘ฅ)โˆ†๐‘›๐‘‡(๐‘ฅ)๐‘‘๐‘ฅ] ๐ด๐‘‡ = ๐ด๐ป๐ด๐‘‡, (11) A is defined by eqs.(7) and H is a Hilbert matrix

๐ป = [ 1 12 โ€ฆ 1 ๐‘š+1 1 2 1 3 โ€ฆ 1 ๐‘š+2 โ‹ฎ โ‹ฎ โ‹ฑ โ‹ฎ 1 ๐‘š+1 1 ๐‘š+2 โ€ฆ 1 2๐‘š+1] and ๐‘„ = [ ๐‘„1 0 โ€ฆ 0 0 ๐‘„2 โ€ฆ 0 โ‹ฎ โ‹ฎ โ‹ฑ โ‹ฎ 0 0 โ€ฆ ๐‘„๐‘š ]

The elements of the dual matrix ๐‘„, are given explicity by

(๐‘„๐‘š)๐‘˜+1,๐‘–+1= โˆซ ๐ต01 ๐‘˜๐‘š(๐‘ฅ)๐ต๐‘–๐‘š(๐‘ฅ)๐‘‘๐‘ฅ

= (๐‘š๐‘˜ ) (๐‘š๐‘– ) โˆซ (1 โˆ’ ๐‘ฅ)01 2๐‘šโˆ’(๐‘˜+๐‘–)๐‘ฅ๐‘˜+๐‘–๐‘‘๐‘ฅ (12) where ๐‘˜, ๐‘– = 0,1, โ€ฆ , ๐‘š

4-

The Derivative for Orthonormal (B-Polynomials):

The representation of the orthonormal Bernstein Polynomials, denoted by ๐‘๐‘–5(๐‘ฅ), ๐‘๐‘–6(๐‘ฅ) here, was discovered by analyzing the resulting orthonormal polynomials after applying the Gram-Schmidt process on sets of Bernstein polynomials of degree five and six.

We get the following sets of orthonormal polynomials [10],[11].

๐‘05(๐‘ฅ) = โˆš11(1 โˆ’ ๐‘ฅ)5

๐‘15(๐‘ฅ) = 6 [5๐‘ก(1 โˆ’ ๐‘ฅ)4โˆ’12(1 โˆ’ ๐‘ฅ)5]

๐‘25(๐‘ฅ) =18โˆš75 [10(1 โˆ’ ๐‘ฅ)3๐‘ก2โˆ’ 5(1 โˆ’ ๐‘ฅ)4๐‘ก +185 (1 โˆ’ ๐‘ฅ)5]

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๐‘45(๐‘ฅ) = 7โˆš3 [5(1 โˆ’ ๐‘ฅ)๐‘ฅ4โˆ’ 20(1 โˆ’ ๐‘ฅ)2๐‘ฅ3+ 18(1 โˆ’ ๐‘ฅ)3๐‘ฅ2โˆ’ 4(1 โˆ’ ๐‘ฅ)4๐‘ฅ +17(1 โˆ’ ๐‘ฅ)5] ๐‘55(๐‘ฅ) = 6 [๐‘ฅ5โˆ’255 (1 โˆ’ ๐‘ฅ)๐‘ฅ4+1003 (1 โˆ’ ๐‘ฅ)2๐‘ฅ3โˆ’ 25(1 โˆ’ ๐‘ฅ)3๐‘ฅ2+ 5(1 โˆ’ ๐‘ฅ)4๐‘ฅ โˆ’16(1 โˆ’ ๐‘ฅ)5] and ๐‘06(๐‘ฅ) = โˆš13(1 โˆ’ ๐‘ฅ)6 ๐‘16(๐‘ฅ) = โˆš44 [6๐‘ก(1 โˆ’ ๐‘ฅ)5โˆ’12(1 โˆ’ ๐‘ฅ)6] ๐‘26(๐‘ฅ) = 11 [15(1 โˆ’ ๐‘ฅ)4๐‘ฅ2โˆ’ 6(1 โˆ’ ๐‘ฅ)5๐‘ฅ +113 (1 โˆ’ ๐‘ฅ)6] ๐‘36(๐‘ฅ) = โˆš252 [20(1 โˆ’ ๐‘ฅ)3๐‘ฅ3โˆ’452(1 โˆ’ ๐‘ฅ)4๐‘ฅ2+ 5(1 โˆ’ ๐‘ฅ)5๐‘ฅ โˆ’1166(1 โˆ’ ๐‘ฅ)6] ๐‘46(๐‘ฅ) =โˆš542[15(1 โˆ’ ๐‘ฅ)2๐‘ฅ4โˆ’ 40(1 โˆ’ ๐‘ฅ)3๐‘ฅ3+1807 (1 โˆ’ ๐‘ฅ)4๐‘ฅ2โˆ’307(1 โˆ’ ๐‘ฅ)5๐‘ฅ +425 (1 โˆ’ ๐‘ฅ)6] ๐‘56(๐‘ฅ) =โˆš328[6๐‘ก5(1 โˆ’ ๐‘ฅ) โˆ’752(1 โˆ’ ๐‘ฅ)2๐‘ฅ4+ 60(1 โˆ’ ๐‘ฅ)3๐‘ฅ3โˆ’ 30(1 โˆ’ ๐‘ฅ)4๐‘ฅ2+307 (1 โˆ’ ๐‘ฅ)5๐‘ฅ โˆ’283 (1 โˆ’ ๐‘ฅ)6] ๐‘66(๐‘ฅ) = 7 [๐‘ฅ6โˆ’ 18(1 โˆ’ ๐‘ฅ)๐‘ฅ5+ 75(1 โˆ’ ๐‘ฅ)2๐‘ฅ4โˆ’ 100(1 โˆ’ ๐‘ฅ)3๐‘ฅ3+ 45(1 โˆ’ ๐‘ฅ)4๐‘ฅ2โˆ’ 6๐‘ก(1 โˆ’ ๐‘ฅ)5+ 1 7(1 โˆ’ ๐‘ฅ) 6]

In addition, we have determind the explicit representation for the orthonormal Bernstein polynomials as [16]

๐‘๐‘˜๐‘š(๐‘ฅ) = (โˆš2(๐‘š โˆ’ ๐‘˜) + 1) (1 โˆ’ ๐‘ฅ)๐‘šโˆ’๐‘˜โˆ‘๐‘˜๐‘–=0(โˆ’1)๐‘–(2๐‘š + 1 โˆ’ ๐‘–๐‘˜ โˆ’ ๐‘– ) (๐‘˜๐‘–) ๐‘ฅ๐‘˜โˆ’๐‘– (13)

The eqs(13) can be written in terms of the Bernstein basis functions as

๐‘๐‘˜๐‘š(๐‘ฅ) = (โˆš2(๐‘š โˆ’ ๐‘˜) + 1) โˆ‘ (โˆ’1)๐‘– (2๐‘š+1โˆ’๐‘– ๐‘˜โˆ’๐‘– )(๐‘˜๐‘–) (๐‘šโˆ’๐‘– ๐‘˜โˆ’๐‘–) ๐ต๐‘˜โˆ’๐‘–,๐‘šโˆ’๐‘–(๐‘ฅ) ๐‘˜ ๐‘–=0 (14)

Any generalized Bernstein basis polynomials of degree m can be wretten as a linear combination of the generalized Bernstein basis polynomials of degree m+1

๐ต๐‘˜,๐‘š(๐‘ฅ) =๐‘šโˆ’๐‘˜+1๐‘š+1 ๐ต๐‘˜,๐‘š+1(๐‘ฅ) +๐‘š+1๐‘˜+1๐ต๐‘˜+1,๐‘š+1(๐‘ฅ) (15)

By utilizing eqs(15),the following functions can be written as

๐ต๐‘˜,๐‘šโˆ’1(๐‘ฅ) =๐‘šโˆ’๐‘˜๐‘š ๐ต๐‘˜,๐‘š(๐‘ฅ) +๐‘˜+1๐‘š ๐ต๐‘˜+1,๐‘š(๐‘ฅ) (16)

and

๐ต๐‘˜โˆ’1,๐‘šโˆ’1(๐‘ฅ) =๐‘šโˆ’๐‘˜+1๐‘š ๐ต๐‘˜โˆ’1,๐‘š(๐‘ฅ) +๐‘š๐‘˜๐ต๐‘˜,๐‘š(๐‘ฅ) (17)

Substituting these eqs(16) and (17) into the right hand side of the eqs(4), we get the following derivatives of Bernstein basis polynomials

๐‘‘

๐‘‘๐‘ฅ๐ต๐‘˜,๐‘š(๐‘ฅ) = (๐‘š โˆ’ ๐‘˜ + 1)๐ต๐‘˜โˆ’1,๐‘š(๐‘ฅ) + (2๐‘˜ โˆ’ ๐‘š)๐ต๐‘˜,๐‘š(๐‘ฅ) โˆ’ (๐‘˜ + 1)๐ต๐‘˜+1,๐‘š(๐‘ฅ) (18) In [10], the derivative of the orthonormal (B-Polynomials)of degree five are introducedas given

๐‘05โ€ฒ = [โˆ’5โˆš11 ๐ต05โˆ’ โˆš11๐ต15]

๐‘15โ€ฒ = [45๐ต05โˆ’ 15๐ต15โˆ’ 2๐ต25]

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๐‘35โ€ฒ = [145โˆš5๐ต05โˆ’235โˆš5๐ต15+โˆš578๐ต25+154โˆš5๐ต35โˆ’113โˆš5๐ต45]

๐‘45โ€ฒ = [โˆ’33โˆš3๐ต05+331โˆš35 ๐ต15โˆ’2175 ๐ต25126โˆš35 ๐ต35+ 77โˆš3๐ต45โˆ’ 35โˆš3๐ต55]

๐‘55โ€ฒ = [35๐ต05โˆ’ 55๐ต15+ 63๐ต25+ 35๐ต35โˆ’ 119๐ต45+ 105๐ต55]

and we introduce the derivative of the orthonormal (B-Polynomials) for degree six

๐‘06โ€ฒ = [โˆ’6โˆš11 ๐ต06โˆ’ โˆš13๐ต16] ๐‘16โ€ฒ = [8โˆš11๐ต06โˆ’ 7โˆš11๐ต16โˆ’ 4โˆš11๐ต26] ๐‘26โ€ฒ = [โˆ’84 ๐ต06+ 96๐ต16+ ๐ต26โˆ’ 3๐ต36] ๐‘36โ€ฒ = [36โˆš7๐ต06โˆ’ 64โˆš7๐ต16+ 32โˆš7๐ต26+ 27โˆš7๐ต36โˆ’ 24โˆš7๐ต46] ๐‘46โ€ฒ = [โˆ’42โˆš5๐ต06+ 95โˆš5๐ต16โˆ’ 150.263681๐ต26โˆ’ 40.24922359๐ต36+ 187.8297101๐ต46โˆ’ 42โˆš5๐ต56] ๐‘56โ€ฒ = [46โˆš3๐ต06โˆ’ 206.1140461๐ต16+ 235.5589098๐ต26โˆ’ 14โˆš3๐ต36โˆ’ 242.4871131๐ต46+ 266.7358244๐ต56โˆ’ 56โˆš3๐ต66] ๐‘66โ€ฒ = [โˆ’48๐ต06+ 132๐ต16โˆ’ 168๐ต26+ 42๐ต36+ 168๐ต46โˆ’ 252๐ต56+ 168๐ต66] 5. Second kind integral equations:

In this section, we use Orthonormal Polynomials for solving second kind Fredholm and Volterra integral equations.

1- Fredholm integral equation (FIE):

Where in eq(1) ๐‘”(๐‘ฅ) โˆˆ ๐ฟ2[0,1], ๐‘˜(๐‘ฅ, ๐‘ก) โˆˆ ๐ฟ2([0,1] ร— [0,1]) are known and ๐‘ข(๐‘ก) is unknown function to be determined.

First we assume the unknown functions

๐‘ข๐‘–(๐‘ฅ) = ๐ถ๐‘–๐‘‡๐ต(๐‘ฅ), ๐‘– = 1,2, โ€ฆ , ๐‘› (19) by substituting (19)in (1) we have:

๐ถ๐‘–๐‘‡๐ต(๐‘ฅ) = ๐‘”๐‘–(๐‘ฅ) + โˆซ ๐‘˜๐‘–,๐‘—(๐‘ฅ, ๐‘ก) 1 0 ๐ถ๐‘–๐‘‡๐ต(๐‘ก)๐‘‘๐‘ก ๐ถ๐‘–๐‘‡๐ต(๐‘ฅ) โˆ’ โˆซ ๐‘˜01 ๐‘–,๐‘—(๐‘ฅ, ๐‘ก)๐ถ๐‘–๐‘‡๐ต(๐‘ก)๐‘‘๐‘ก = ๐‘”๐‘–(๐‘ฅ) (20) Pick distinct node points ๐‘ก1, ๐‘ก2, โ€ฆ , ๐‘ก๐‘›โˆˆ [0,1]

This leads to determining { ๐‘1, ๐‘2, โ€ฆ , ๐‘๐‘›} as the solution of linear system

โˆ‘๐‘›๐‘–=1๐ถ๐‘—[๐ต(๐‘ฅ๐‘–) โˆ’ โˆซ ๐‘˜(๐‘ฅ01 ๐‘—, ๐‘ก๐‘–)๐ต(๐‘ก๐‘–)๐‘‘๐‘ก] = ๐‘”(๐‘ฅ๐‘–) (21)

In this paper Collocation points are ๐‘ก๐‘–=๐‘›๐‘– ๐‘“๐‘œ๐‘Ÿ ๐‘– = 1,2, โ€ฆ , ๐‘› so that we have a system of linear equations ๐ฟ๐‘›๐‘‹ = ๐‘™๐‘› where ๐ฟ๐‘›= |๐ต(๐‘ฅ๐‘–) โˆ’ โˆซ ๐‘˜(๐‘ฅ01 ๐‘—, ๐‘ก๐‘–)๐ต(๐‘ก๐‘–)๐‘‘๐‘ก| ๐‘–=0 ๐‘› ๐‘— = 1,2, โ€ฆ , ๐‘› ๐‘™๐‘›= [๐‘”(๐‘ฅ๐‘–)], ๐‘– = ๐‘œ, 1, โ€ฆ , ๐‘›

2- Volterra integral equation (VIE):

Similarly above section by using Collocation points ๐‘ก๐‘–= ๐‘–

(6)

๐ฟ๐‘›= |๐ต(๐‘ฅ๐‘–) โˆ’ โˆซ ๐‘˜(๐‘ฅ0๐‘ฅ ๐‘—, ๐‘ก๐‘–)๐ต(๐‘ก๐‘–)๐‘‘๐‘ก|๐‘–=0 ๐‘›

๐‘— = 1,2, โ€ฆ , ๐‘›

๐‘™๐‘›= [๐‘“(๐‘ฅ๐‘–)], ๐‘– = ๐‘œ, 1, โ€ฆ , ๐‘› 6.Numerical Results:

In this section VIE, FIE is considered and solved by the introduced method. Example 1: Consider the following FIE

๐‘ข(๐‘ฅ) = sin ๐‘ฅ + โˆซ (1 โˆ’ ๐‘ฅ cos ๐‘ฅ๐‘ก) ๐‘ข(๐‘ก)๐‘‘๐‘ก01 (22) the exact solution u(x)=1. Solving eqs(20) and (21) we get the values of ๐ถ ๐‘‡

๐ถ=[0.91402903 0.10091438 3.48867166 โˆ’2.7619172 3.51373945 0. 15379915 0.98896685]๐‘‡ Table1 shows the numerical results for this example.

Table 1:some numerical results for example 1 x Approximat solution ๐‘๐‘›(๐‘ฅ) Approximat solution ๐ต๐‘›(๐‘ฅ) ๐ด๐‘๐‘ ๐‘œ๐‘ข๐‘ก๐‘’๐ธ๐‘Ÿ๐‘Ÿ๐‘œ๐‘Ÿ |๐‘’๐‘ฅ๐‘Ž๐‘๐‘ก โˆ’ ๐ต๐‘›6| 0 0.91402333 0.99993256 6.7440e-005 0.1 0.93422123 0.99993267 6.7330e-005 0.2 0.96878356 0.99994532 5.4680e-005 0.3 0.96878320 0.99994444 5.5560e-005 0.4 0.97843933 0.99995324 4.6760e-005 0.5 0.93270466 0.99995417 4.5830e-005 0.6 0.99388042 0.99996618 3.3820e-005 0.7 0.99963715 0.99997654 2.3460e-005 0.8 0.98888323 0.99998790 1.2100e-005 0.9 0.99999668 0.99999999 1.0000e-008 1 0.99999998 1.00000000 0.00000000 ๐‘€. ๐‘†. ๐ธ =6.7440e-005 ๐ฟ. ๐‘†. ๐ธ=2.1286e-008

Example 2: Consider the following FIE

๐‘ข(๐‘ฅ) = ๐‘’โˆ’๐‘ฅโˆ’ โˆซ ๐‘ฅ๐‘’1 ๐‘ก๐‘ข(๐‘ก)๐‘‘๐‘ก 0

the exact solution ๐‘ข(๐‘ฅ) = ๐‘’โˆ’๐‘ฅโˆ’๐‘ฅ

2. Solving eqs(20) and (21) we get the values of ๐ถ ๐‘‡

๐ถ=[1 1.13944655 โˆ’0.05603088 0.85196593 โˆ’0.13491534 0. 11655846 โˆ’ 0.16383600]๐‘‡ Table2 shows the numerical results for this example.

(7)

Table2:some numerical results for example 2 x

Exact solution Approximat solution ๐‘๐‘›(๐‘ฅ) Approximat solution ๐ต๐‘›(๐‘ฅ) ๐ด๐‘๐‘ ๐‘œ๐‘ข๐‘ก๐‘’๐ธ๐‘Ÿ๐‘Ÿ๐‘œ๐‘Ÿ |๐‘’๐‘ฅ๐‘Ž๐‘๐‘ก โˆ’ ๐ต๐‘›6| 0 1 1 1 0.00000000 0.1 0.85483742 0.94188967 0.85488967 0.00005225 0.2 0.71873075 0.76843118 0.71811760 0.00008101 0.3 0.59081822 0.59503874 0.59081874 0.00000052 0.4 0.47032005 0.46240281 0.47032115 0.00000110 0.5 0.35653066 0.35230187 0.35653066 0.00352200 0.6 0.24881164 0.24605093 0.24880053 0.00001111 0.7 0.14658530 0.13907957 0.14658510 0.00000020 0.8 0.04932896 0.04047384 0.04932895 0.00000001 0.9 -0.04343034 -0.04663478 -0.04343155 0.00000121 1 -0.13212056 -0.16383600 -0.13212056 0.00000000 ๐‘€. ๐‘†. ๐ธ =0.00352200 ๐ฟ. ๐‘†. ๐ธ=0.00000000 Fig1 of example1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 exact Bn6

(8)

Fig 2 of example3

Example 3: Consider the following VIE

๐‘ข(๐‘ฅ) = ๐‘ฅ โˆ’ โˆซ (๐‘ฅ โˆ’ ๐‘ก)๐‘ข(๐‘ก)๐‘‘๐‘ก0๐‘ฅ (23) The exact solution ๐‘ข(๐‘ฅ) = sin ๐‘ฅ. Table(3) shows the numerical results for this example(3)

๐ถ=[0 0.15606151 0.37680718 0.40682330 0.73751386 0. 68080388 0.84172197]๐‘‡

Table3:some numerical results for example 3 x

Exact solution Approximat solution ๐‘๐‘›(๐‘ฅ) Approximat solution ๐ต๐‘›(๐‘ฅ) ๐ด๐‘๐‘ ๐‘œ๐‘ข๐‘ก๐‘’๐ธ๐‘Ÿ๐‘Ÿ๐‘œ๐‘Ÿ |๐‘’๐‘ฅ๐‘Ž๐‘๐‘ก โˆ’ ๐ต๐‘›6| 0 0.00000000 0.00000000 0.00000000 0.00000000 0.1 0.09983342 0.09924030 0.09983389 0.00000059 0.2 0.19866933 0.19972478 0.19865878 0.00001055 0.3 0.29552021 0.29617066 0.29552053 0.00000065 0.4 0.38941834 0.38930420 0.38941820 0.00000014 0.5 0.47942554 0.47990931 0.47942560 0.00000048 0.6 0.56464247 0.56604342 0.56604342 0.00140095 0.7 0.64421769 0.64342047 0.64342047 0.00007972 0.8 0.71735609 0.70896119 0.71732785 0.00008394 0.9 0.78332691 0.76751046 0.78331110 0.00001581 1 0.84147098 0.84172197 0.84147073 0.00000025 ๐‘€. ๐‘†. ๐ธ =0.00140095 ๐ฟ. ๐‘†. ๐ธ=0.00000000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 exact Bn6

(9)

Conclusion:

In this work, VIE,FIE have been solved by using Bernstein basis polynomials of degree m in collocation method. Comparison of the approximate solutions and the exact solutions show that the proposed method is efficient tool. Illustrative examples are included to demonstrate the validity and applicability of the technique.

References:

1. S.Swarup,โ€Integral equations (Krishna Prakanshan Media Prt.Ltd,15th Editions,2007).

2. Shihab. S. N. and Asmaa. A. A. 2012. Numerical Solution of Calculus of Variations by using the Second Chebyshev Wavelets, Eng. & Tech. Journal. 30(18): 3219-3229.

3. H. Goghary. H. Goghary. M,โ€Tow Computational methods for Solving linear Fredholm fuzzy integral equations of the Second kindโ€, APPl. Math. Comput, Vol 182, PP. 791-794, 2006.

4. K. Maleknejad, S.Sohrab1, B. Bevenj1-,โ€Application of D-BPFS to nonlinear integral equationโ€, Commun Non linear Sci Number Simulat, 15,PP. 527-535,2010.

5. K.Maleknejad, M.Mordad, B.Raimi,โ€Anumerical method to solve Fredholm Volterra integral equations two dimensional spaces using Block Pulse Functions and operational Matrix โ€œ, Journal of Computational and Applied Mathematics, 2010.

6. E. H.Doha, A.H.Bhrawy, M.A.Saher, โ€œOn the derivatives of Bernstein Polynomials: An application for the Solution of high even-order differential equations, Boundary Value Problems Vol, 2011, Article I D & 29543,16 pages doi: 10.1155/2011/829543,2011.

7. S. A. Yousefi, M. Behroozifar, โ€œOperational matrices of Bernstein Polynomials and their applications, Int. J. Syst.Sci.41 (6) (2010) 709-716.

8. E. H. Doha and A. H. Bhrawy S. S. Ezz-Eldien, โ€œAnew Jacobi Operational matrix: An Application for Solving frastional differential equationsโ€, Appl. Math. Model. 36, pp. 4931-4943 2012.

9. S. A. Yousefi and H. Jafari and M. A. Firooz jaecard. S. Momani and C.M.Khaliqued, โ€œApplication of Legendre wavelets for solving fractional differential equations, โ€œComput. Math. Appl. 62, 1038-1045 (2011). 10. S. A. Yousefi and M.Behroozifar and M. Dehghan, โ€œNumerical Solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomialsโ€, Appl. Math. Modell. 36, 945-963 (2012).

11. S.A.Yousefi and M. Behroozifar and M. Dehghan, โ€œ The Operational matrices of Bernstein Polynomials for Solving the Parabolic equation subject to specification of the ma ss, J. Comput. Appl.Math. 335, 5272-5283 (2011).

12. Lakestani, M, Dehghan, M, Irandoust-Pakchin, S: โ€ The Construction of Operational matrix of

fractional derivatives using B-Spline functionsโ€, Commun Nonlinear Sci Number simulat. 17, 1149-1162 (2012). 13. Vineet Kumar Singh, Eugene B. Postnikov, โ€œ Operational matrix approach for solution of integro-differential equations arising in theory of an omalous relaxation processes in Vicinity of singular pointโ€, Applied Mathematical Modelling 37,(2013), 6609-6619.

14. Asmaa. A. A. 2014. Numerical solution of Optimal problems using new third kind Chebyshev Wavelets Operational matrix of integration, Eng. & Tec. Journal. 32(1):145-156.

Figure

Fig 2 of example3

References

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