Bernstein Operational Matrix Approach for Integro-Di Erential

DOI 10.1515/nleng-2013-0024 Ë Nonlinear Engineering 2014; 3(2): 117–123

Nagma Irfan, Sunil Kumar*, and Saurabh Kapoor Bernstein Operational Matrix Approach for Integro-Dierential Equation Arising in Control theory

Abstract: The aim of this paper is to propose an e- simple bases polynomial i.e. Bernstein-polynomials. In cient numerical method for solving the integro-dierential recent years, numerous works have been focussing on equations arising in many braches of sciences using Bern- the development of more advanced and ecient meth- stein polynomials. This algorithm based on Bernstein ods for integro-dierential equations, integral and dier- polynomials approximation for integro-dierential equa- ential equations such as Wavelet-Galerkin method [5], La- tions. First, Bernstein operational matrix of dierentiation grange interpolation method [6], Tau method [7], Taylor is derived using Bernstein polynomials and then applied polynomial approach [8] and semi analytical-numerical to solve integro-dierential equations. The solutions ob- techniques such as, homotopy perturbation method [9, 10] tained by proposed method indicate that the approach is homotopy perturbation transform method [11–13], homo- easy to implement and computationally very attractive. A topy analysis transform method [14, 15], Laplace decom- good agreement between the obtained solution and some position method [16–19]. well-known results has been obtained. Two numerical ex- In recent years, there has been an increase usage amples are provided to show the advantage of using Bern- among scientists and engineers to apply wavelet tech- stein operational matrix. This method is capable of greatly nique as well as a numerical solution to solve both lin- reducing the size of calculations while still maintaining ear and non-linear problems. Bernstein polynomials have high accuracy of the numerical solution. attracted the attention of many researchers. These polynomials have been utilized for solving dierent equations Keywords: Bernstein Polynomial, Bernstein Operational by using various approximate methods. These polynomi- matrix, Integro-dierential equation, Control theory, As- als have been used recently to solve some linear as well as tronomy non-linear dierential equations approximately by Bhatti ËË and Bracken [20], Bhatta and Bhatti [21]. These polyno- Nagma Irfan: Sharda University, School of Engineering and Technol- mials dened on an interval form a complete basis over ogy, Knowledge Park III, Greater Noida-201306,Delhi, NCR, India the interval. For, instance-polynomials have been used to *Corresponding Author: Sunil Kumar: Department of Mathematics, National Institute of Technology, Jamshedpur, 831014, Jharkhand, solve integral equations of various types such as dier- India, E-mail: [email protected], [email protected], ential equations [22–24]. Recently, many researchers [25– [email protected], Tel. +91-7870102516/9415846185 28] have applied to obtain the solutions of the dierential Saurabh Kapoor: Regional Institute of Education, Bhubaneswar equation and integral equations arising in physics by us- (NCERT), Odisha, India ing Bernstein operational matrices method. The aim of the present paper is to apply new computational method to solve the some integro-dierential 1 Introduction equations. This method reduces the integro-dierential equations to a set of algebraic equations by using Bern- Since many physical problems are modelled by integro- stein polynomials. As illustrative examples of integro- dierential equation arises in the eld of engineering, dierential equations whose exact solutions are known mechanics, physics, chemistry, astronomy, biology, eco- have been considered for appropriate numerical solutions. nomics, potential theory & electrostatics, for example in Though there exist some other numerical methods for han- control theory and nancial mathematics [1–4]. Integro- dling the integro-dierential equations [29–44], but the dierential equations are important equations, but they advantage of Bernstein operational matrix method to that are hard to solve even numerically and analytically. In method is simplicity of implication beside very better an- recent years, many dierent basic functions have used swers. to estimate the solution of system, such as orthonormal The Bernstein polynomial, named after Sergei bases and wavelets. In this paper, we are going to use Natanovich Bernstein, is a polynomial in the Bernstein 118 Ë Nagma Irfan, Sunil Kumar, and Saurabh Kapoor, Bernstein Operational Matrix Approach ...

form that is a linear combination of Bernstein basis poly- 4. For i ̸= 0, Bi n has a unique local maximum in [0,1] at , ! nomials. B-polynomials have many useful properties. The n t = i/n and the maximum value ii n−n(n−i)n−i . procedure takes advantage of the continuity and unity par- i tition properties of the basis set of Bernstein polynomials 5. The Bernstein polynomials form a partition of unity n over an interval [0,R]. The Bernstein polynomials basis P i.e. Bi,n(t) =1. vanish except the rst polynomial at t=0, which is equal i=0 to 1 over the interval [0, R]. This provides greater exibility 6. It has a degree raising property in the sense that any n to impose boundary conditions at the end points of the of the lower-degree polynomials (degree < ) can be interval. It also ensures that the sum at any pointx of all expressed as a linear combinations of polynomials of n Bernstein polynomials is unity. These polynomials have degree . We have, attracted the attention of many researchers. These polyno- n i i B t − B t + 1 B t mials have been utilised for solving dierential equations i,n−1( ) = n i,n( ) + n i+1,n( ). (4) by using various approximation methods. These polyno- 7. Let f (x) ∈ C[0, 1] (the class of continuous functions mials have recently been used to solve some linear as well n P i as non-linear dierential equationns, ordinary and par- on [0, 1]), then Bn(f)(x) = f n Bi,n(x) converges to i tial, approximately by Bhatta and Bhatti [21] some integro- =0 f(x) uniformly on [0, 1] as n → ∞. dierential equations by Mandal and Bhattacharya [25]. f x ∈ C(k) 8. Let ( ) [0, 1], (the class of k-times dierentiable The Bernstein polynomials of nth degree are dened on (k) (k) function with f continuous), then Bn(f ) ≤ the interval [0, 1] as [21] ∞ (n)k f (k) f (k) B f (k) k ! nk and − n( ) → 0 as → n ∞ ∞ i n−i k k (n)k Bi,n(t) = t (1 − t) , i = 0, 1, 2, . . . n. (1) ∞, where . ∞ is the sup. norm and nk = i 0 1 k−1 1 − n 1 − n ··· 1 − n is an eigen value of Bn; ! the corresponding eigen function is a polynomial of n where = n! . There are (n+1),nth degree Bern- degree k. i i!(n−i)! stein polynomials forming a basis for the linear space Vn The Bernstein polynomial, named after Sergei Natanovich consisting of all polynomials of degree less than or equal Bernstein, is a polynomial in the Bernstein form that is to n in R[x]-the ring of polynomials over the eld R. For a linear combination of Bernstein basis polynomials. The mathematical convenience, we usually set Bi n(x) = 0, if , Bernstein basis polynomials of degree n are dened by i < 0 or i > n. It can easily be shown that each of Bernstein ! polynomials is positive and also sum of all the Bernstein n i n−i Bi n(t) = t (1 − t) , for i = 0, 1, 2, ... , n. polynomials is unity for all real x ∈ [0, 1], that is , i n (5) X Bi,n(t) = 1, x ∈ [0, 1]. (2) Using Gram- Schmidt ortho-normalization process on Bi,n i=0 and normalizing, we obtain a class of orthonormal polyno- Any polynomial B(t) in R[t] may be written as mials from Bernstein polynomials. We call them orthonormal Bernstein polynomials of order n and denote them by n X bon , b1n , ··· , bnn. For n = 5 the ve orthonormal polyno- B(t) = βi Bi n(t), for some n. (3) , mials are given by i=0 √ 5 Then B(t) is called a polynomial in Bernstein form or b05(t) = 11(1 − t) , Bernstein polynomial of degree n. The coecients βi are 4 1 5 b15(t) = 6 5(1 − t) t − (1 − t) called Bernstein or Bezier coecients. But several math- 2 √ B t ematicians call Bernstein basis polynomials i.n( ) as the 18 7 3 2 4 5 5 b25(t) = 10(1 − t) t − 5 (1 − t) t + (1 − t) , Bernstein polynomials. We will follow this convention as 5 18 well. These polynomials have the following properties: 28 2 3 3 2 30 4 b35(t) = √ 10(1 − t) t − 15(1 − t) t + (1 − t) t 1. Bi,n(0) = δi0 and Bi,n(1) = δin, where δ is the Kro- 5 7 necker delta function. 5 − (1 − t)5 2. Bi,n(t) has one root, each of multiplicity i and n − i, at 28 t = 0 and t = 1, respectively.

3. Bi,n(t) ≥ 0 for t ∈ [0, 1] and Bi,n(1 − t) = Bn−i,n(t). Nagma Irfan, Sunil Kumar, and Saurabh Kapoor, Bernstein Operational Matrix Approach ... Ë 119

√ h 4 2 3 3 2 b45(t) = 7 3 5(1 − t)t − 20(1 − t) t + 18(1 − t) t 2. Subsequently, the various signals involved in the integral equation are approximated by representing them 1 −4(1 − t)4t + (1 − t)5 , as linear combinations of the orthonormal basis func- 7 tions and truncating them at optimal levels. 25 100 b (t) = 6 t5 − (1 − t)t4 + (1 − t)2t3 − 25(1 − t)3t2 3. Finally, the integral equation is converted to an alge- 55 2 3 braic equation by introducing the operational matrix 1 +5(1 − t)4t − (1 − t)5 . of integration of the basis functions. 6

The key idea of the technique depends on the following integral property of the basis vector ϕ(t).

2 Function Approximation t t Z Z k k ··· ϕ(σ)(dσ) ≈ Pm+1ϕ(t), Bernstein polynomials form a complete basis [1] over the a a interval [0, 1].It as easy to show that any given polynomial ϕ t ϕ t ϕ t ... ϕ t T of degree n can be expressed in terms of linear combina- where ( ) = [ 0( ), 1( ), , m( )] in which the ele- ϕ t ϕ t ... ϕ t tion of the basis functions. A function f ∈ L2[0, 1] may be ments 0( ), 1( ), , m( ) are the basis functions, or- a b P written as thogonal on a certain interval [ , ] and m+1 is the oper- n X ational matrix for integration of ϕ(t). Note that Pm+1 is a f(t) = lim cin bin(t), (6) n→∞ constant matrix of order (m + 1) × (m + 1). i=0 Using the operational matrix of an orthonormal sys- c hf b i h i where, in = , in and , is the standard inner product tem of functions to perform integration for solving, identi- L2 n m on [0, 1]. If the series (6) is truncated at = , then fying and optimizing a linear dynamic system has several m advantages: ∼ X T f = cim bim = C B(t), (7) 1. The method is computer oriented, thus solving higher i=0 order dierential equation becomes a matter of dimen- where, sion increasing. T C = [c0m , c1m , ··· , cmm] (8) 2. The solution is a multi-resolution type. 3. The solution is convergent, even though the size of in- and T crement may be large. B(t) = [b0m(t), b1m(t), ··· , bmm(t)] . (9)

3.1 The Operational Matrix of Integration: 3 Bernstein Operational Matrix The orthonormal Bernstein polynomials operational ma- Approximations by orthonormal family of functions have trix of integration of order (m + 1) × (m + 1) will be derived played a vital role in the development of physical sciences, now. To achieve this, consider the following integral t engineering and technology in general and mathematical Z analysis in particular since long. In the last three decades, bim(x)dx = ϕi(t), 0 ≤ t < 1, i = 0, 1, ··· , m. they have been playing an important part in the evalua- 0 tion of new techniques to solve problems such as identi- m X i cation, analysis and optimal control. The aim of these = cjm bjm(t), techniques has been to obtain eective algorithms that are j=0 i i i suitable for the digital computers. The motivation and phi- = [c0m , c1m , ··· , cmm]B(t), for 0 ≤ i ≤ m. losophy behind this approach is that it transforms the un- (10) derlying dierential equation of the problem to an alge- Using equations (9) and (10), we obtain braic equation, thus simplifying the solution process of the t problem to a great extent. The basic idea of this technique Z is as follows: B(x)dx = Pm+1B(t), (11) 1. The dierential equation is converted to an integral 0

equation via multiple integration. where the operational matrix Pm+1 of integration associ- ated with orthonormal Bernstein polynomials is given by 120 Ë Nagma Irfan, Sunil Kumar, and Saurabh Kapoor, Bernstein Operational Matrix Approach ...

Theorem 2: Let us consider another integro-dierential i m Pm+1 = (cjm)i,j=0 (12) equation of the type:  and 1 i  y′ t R k t s y s ds y t x t cjm = ϕi , bjm (13) ( ) = ( , ) ( ) + ( ) + ( ), 0 (19)  For m = 5, the matrix P6 is denoted by P.  y(0) = y0

where x ∈ L2[0, 1), k ∈ L2([0, 1) × [0, 1)) and y is and unknown function. 4 Method of solution for Proof. If we approximate x, y′ and k by (7) as follows: integro-dierential equations ∼ T ′ ∼ ′ T ∼ T x(t) = X B(t), y (t) = Y B(t), y(0) = Y0 B(t), k(t, s) T Our aim in this paper is to compute the computational and ∼= B (t)KB(s). numerical solution of dierent integro-dierential equa- Then tions using Mathcad-14. t T Theorem 1: We consider the following linear integro- Z Z ′ ∼ ′ T T dierential equation: y(t) = y (s)ds + y(0) = Y B(s)ds + Y0 B(t)  x 0 0 R  y′ x f x y t dt T T ( ) = ( ) + ( ) ∼= Y” PB(t) + Y B(t). 0 , (14) 0  y(0) = a Putting this value in equation (19), we have where f(x) ∈ L2[0, 1) is known function and yis unknown Z1 function to be determined. Using equation (7), we can ap- ′ T T T ′ T Y B (t) = B (t)kB(s)(P Y + Y0)B (s)ds proximate f (x) as 0 m T T T ∼ X T + B (t)(P Y′ + Y ) + B (t)X. f (x) = Cim bim = F B(x), (15) 0 i=0 where Z1 ′ T T T ′ T h i Y B (t) = B (t)kB(s)(P Y + Y0)B (s)ds F = F1, F2, F3, F4, F5, F6 . 0 T T ′ T Proof. Let us consider the following equation + B (t)(P Y + Y0) + B (t). T y′ = C B(t). (16) ′ T T T 1 T T ′ T Y B (t) = B (t)k(P Y + Y0) + B (t)(P Y + Y0) + B (t)X. t ′ T ′ T ′ We integrate the above equation with respect to , we get Y = k(P Y + Y0) + (P Y + Y0) + X. y t CT PB t y ′ T ′ T ′ ( ) = ( ) + (0). Y I − kP Y − P Y = kY0 + Y0 + X. y ′ T T Since, from initial condition (0) = 0. So above equation Y [I − kP − P ] = kY0 + Y0 + X. becomes ′ T T −1 T Y = [I − kP − P ] [kY + Y + X]. y(t) = C PB(t). (17) 0 0 By solving above linear system we can nd the vector Y′, From the above equations (14)-(16), we get x so Z T ′ T T ∼ T T T T Y = Y P + Y ⇒ y(t) = Y B(t). C B(x) = d B(x) + C PB(t)dt. 0 0  t  Z T T T   C B(x) = d B(x) + C P2B(x) B(x)dx = PB(t) .   0 5 Numerical Examples and Error T T C = d [I − P2]−1. (18) Estimation From Eqs. (17) and (18), we get T To demonstrate the eectiveness of the proposed method y(t) = d [I − P2]−1PB(t). we consider here Integro dierential equation. We use Mathcad-14 to get the numerical results. Nagma Irfan, Sunil Kumar, and Saurabh Kapoor, Bernstein Operational Matrix Approach ... Ë 121

Fig. 3. Comparison between the exact solution y(t) (solid line) Fig. 1. Comparison between the exact solution y(t) (solid line) and approximate solution Y(t) (dotted line) obtained by proposed and approximate solution Y(t) (dotted line) obtained by proposed method for example 2. method for example 1.

Fig. 4. Absolute error E(t) between the exact solution and approximate solution obtained by proposed method. Fig. 2. Absolute error E(t) between the exact solution and approximate solution obtained by proposed method. It can be seen from Fig. 1 that the analytical solution obtained by the present method is nearly identical to the Example1: We consider the following Integro-dierential exact solution of the integro-dierential equation. Fig. 2 equation of rst order: shows the absolute error between the well-known exact  Z1 solution and solution obtained by proposed Bernstein op-  1 y′(x) = 1 − x + y(t)dt , (20) erational matrix method.  3 0 Example 2: In this example, Let us consider following Integro-dierential equation: with initial condition y(0) = 0, whose exact solution is  1 y(x) = x. t+1  y′ t R est y s ds y t 1−e ( ) = ( ) + ( ) + t+1 Figs. 1-2 show the evaluation results of the approxi- 0 , (21) mate solution for the example 1. Figs. 1 show the compar-  y(0) = 1 ison between the exact solution and the approximate so- t lution (which is obtained by aforesaid method). It can be with exact solution y(t) = e . seen from Figs. 1 that the numerical solution obtained by Figs. 3 show the comparison between the exact so- the present method is nearly identical to the exact solution lution and the approximate solution (which is obtained of the given integro-dierential equation (20). by aforesaid method). It can be seen from Figs. 3 that The simplicity and accuracy of the proposed method the numerical solution obtained by the present method is is illustrated by computing the absolute error E(x) = nearly identical to the exact solution of the given integro- |y(x)−Y(x)| where y(x) the exact solution and Y(x) are so- dierential equation (21). lution obtained my new proposed method. Figures 2 rep- resent the absolute error between exact and approximate solution obtained by new proposed method. 6 Concluding remarks Here Figs. 1-2 show the evaluation results of the approximate solution, exact solution and absolute error for In this paper, we have proposed a numerical approach the example 1. Fig. 1 show the comparison between the for solving integro-dierential equation arising in sci- exact solution and the approximate solution (which is ob- ence and other physical phenomena by using Bernstein tained by aforesaid Bernstein operational matrix method). polynomial operational matrix. We have introduced Bern- 122 Ë Nagma Irfan, Sunil Kumar, and Saurabh Kapoor, Bernstein Operational Matrix Approach ... stein operational matrices of integration to propose a new [13] S. Kumar, Numerical computation of time-fractional equation and accurate algorithm for numerical solution of integro- arising in solid state physics and circuit theory. Z Natur 2013, dierential equations. First, Bernstein operational matrix 68a, 1-8. [14] M. M. Khader, S. Kumar, S. Abbasbandy, New homotopy anal- of dierentiation is derived using Bernstein polynomials ysis transform method for solving the discontinued problems and then applied to solve integro-dierential equations. arising in nanotechnology. Chinese Phy B 2013, 22. The low order matrices make our method more attrac- [15] S. Kumar, An analytical algorithm for nonlinear frac- tive and simpler to use as illustrated by the given numeri- tional Fornberg-Whitham equation arising in wave break- cal examples. Finally two numerical examples reveal that ing based on a new iterative method. Alex Eng J 2013, DOI:org/10.1016/j.aej.2013.11.004. the present method is very accurate and computationally [16] G. C. Wu, Adomian decomposition method for non-smooth very attractive for solving linear and nonlinear integro- initial value problems. Math and Comp Mod 2011, 54 ,2104– dierential equations arising in science and engineering. 2108. [17] X. Y. Qin, Y. P. Sun, Approximate analytical solutions for a Acknowledgement: The author is very grateful to the ref- mathematical model of a tubular packed-bed catalytic reac- erees for carefully reading the paper and for their com- tor using an Adomian decomposition method. Appl Math and Comp 2011, 218, 1990–1996. ments and suggestions which have improved the paper. [18] M. Khan, M. A. Gondal, S. Kumar, A new analytical solution procedure for nonlinear integral equations. Math and Comp Mod 2012, 55, 1892-1897. [19] S. Kumar, Deepak Kumar, S. Abbasbandy, M. M. Rashidi, Ana- References lytical Solution of fractional Navier-Stokes equation by using Modied Laplace Decomposition Method. Ain Sham Eng J [1] R. Cont, P. Tankov, E. Voltchkova, Option Pricing Models with 2013. Jumps: Integro-dierential Equations and Inverse Problems. [20] M. I. Bhatti, P. Bracken, Solutions of dierential equations in Eur Cong Comp Meth Appl Sci Eng 2004, 24, 1-20. a Bernstein polynomial basis. J Comp Appl Math 2007, 205, [2] W. Liu, The exponential stabilization of the high dimensional 272-280. linear system of thermo-viscoelasticity. J Math Pur Appl 1998 , [21] D. D. Bhatta, M. I. Bhatti, Numerical Solution of KdV equa- 77, 355-386. tions using modied Bernstein polynomials. Appl Math Comp [3] R. Natalini, M. Briani ,C.L. Chioma, Finite dierences schemes 2006,174, 1255-1268. for integro-dierential equations in nancial mathematics. [22] S. A. Youse, H. Behrooziyar, Operational matrices of Bern- IAC-CNR and Univ 2003. stein polynomials and their applications. Int. J Sys Sci 2010, [4] T. Zhang, Finite element methods for evolution Integro- 41, 709-716. dierential equations. North-East Univ Press Sheny of China [23] A.K. Singh, V.K. Singh, O.P. Singh, The Bernstein operational 2001. matrix of integration. Appl Math Sci 2009, 49, 2427-2436. [5] G. Ghasemi, S. T. Kajani, E. Babolian, Numerical Solutions of [24] Y. Ordokhani, S. D. Far, Approximate Solutions of Dierential the non-linear Integro-Dierential Equations; Wavelet Galerkin Equations by using Bernstein Polynomials. ISRN Appl Math Method and Homotopy perturbation Method. Appl Math Comp 2011, Article ID 787694, 15. 2007, 188, 450-455. [25] S. Bhattarcharya, B. N. Mandal, Numerical solution of singular [6] M. T. Rashed, Lagrange Interpolation to compute the numer- integro-dierential equations. Appl Math Comp2008, 195,346- ical solutions dierential integral and Integro-dierential 350. equations. Appl Math Comp 2004, 151, 869–878. [26] D. Bhatta, Use of Bernstein Polynomials to solve Kdv-Burger [7] M. Y. Rahimi-Ardabili, S. Shahmorad, Iterative numerical solu- equation numerically. Appl Math Comp 2008 , 206, 456-457. tion of non-linear and integro-dierential equations by the Tau [27] S.A. Youse, M. Behroozifar, Mehdi Dehghan, Numerical so- method. Appl Math Comp 2007, 193, 514-522. lution of the nonlinear age-structured population models by [8] S. Nas, S. Yalcumbas, M. Sezer, A Taylor polynomial approach using the operational matrices of Bernstein polynomials. Appl for solving higher order linear Fredholm integro-dierential Math Modell 2012, 36, 945–963. equations. Int J Math Edu Sci Tech 2000, 31, 213-225. [28] V. K. Singh, E. B. Postnikov, Operational matrix approach for [9] S. Kumar, Om P. Singh, Numerical Inversion of the Abel Inte- solution of integro-dierential equations arising in theory of gral Equation using Homotopy Perturbation Method. Z Natur anomalous relaxation processes in vicinity of singular point. 2010, 65a, 677-682. Appl Math Modell 2013, 37, 6609–6616. [10] S. Kumar, Yasir Khan, Ahmet Yildirim, A mathematical mod- [29] Om P. Singh , V. K. Singh, R. K. Pandey, A stable numerical elling arising in the chemical systems and its approximate inversion of Abel’s integral equation using almost Bernstein numerical solution. Asia Pac J Chem Eng 2012, 7, 835-840. operational matrix. J Quan Spec Rad Tran 2010, 111, 245–252. [11] S. Kumar, H. Kocak, Ahmet Yildirim, A fractional model of gas [30] S.A. Youse, M. Behroozifar, Mehdi Dehghan, The operational dynamics equation by using Laplace transform. Z Natur 2012, matrices of Bernstein polynomials for solving the parabolic 67a, 389 – 396. equation subject to specication of the mass. J Comp Appl [12] S. Kumar, A numerical study for solution of time fractional Math 2011,235, 5272–5283. nonlinear Shallow-water equation in oceans. Z Natur 2013, 68 [31] K. Maleknejad, B. Basirat, E. Hashemizadeh, A Bernstein op- a, 1-7. erational matrix approach for solving a system of high order Nagma Irfan, Sunil Kumar, and Saurabh Kapoor, Bernstein Operational Matrix Approach ... Ë 123

linear Volterra–Fredholm integro-dierential equations. Math Comp Modell 2012, 55, 1363–1372. [32] K. Maleknejad, E. Hashemizadeh, B. Basirat, Computational method based on Bernstein operational matrices for nonlinear Volterra–Fredholm-Hammerstein integral equations. Commun Nonlinear Sci Numer Simulat 2012, 17, 52-61. [33] S. Yuzbasi, Numerical solutions of fractional Riccati type dierential equations by means of Bernstein operational matrices. Appl Math Comp2013, 219, 6328-6343. [34] A. Avudainayaagam, C. Vani, Wavelet-Galerkin method of Integro-dierential equation. Appl Numb Math2000, 32, 247- 254. [35] K. Maleknejad, M.T. Kajani, Solving Integro Dierential Equa- tion by Hybrid Legendre and Block-Pulse function. J Appl Math2002, 11, 67-76. [36] P. Darania, A. Ebadian, A method for the numerical solution of the Integro-Dierential Equation. Appl Math Comp 2007, 188, 657-668. [37] H. Danfu, S. Xufeng, Numerical solution of Integro-Dierential Equation using CAS Wavelet Operational Matrix of Integration. Appl Math Comp 2007, 194, 460-466. [38] M. T. Kajani, M. Ghasemi, E. Babolian, Numerical solution of linear integro-dierential Equation by using sine-cosine wavelets. Appl Math Comp 2006, 180, 569-574. [39] M. T. Rashed, Numerical Solution of functional dierential, integral and integro-dierential equations. Appl Math Comp 2004, 156, 485-492. [40] A. Karamate, M. Sezer, A Taylor collocation method for the solution of linear integro-dierential equations. Int. J Comp Math 2002, 79, 987-1000. [41] K. Maleknejad, M. T. Kajani, Solving integro dierential equation system by Galerkin methods with hybrid functions. Appl Math Comp 2004, 159, 603-612. [42] J. Rashidinia, M. Zarebnia, The numerical solution of integro- dierential equation by means of the Sinc method. Appl Math Comput 2007,188 , 1124-1130. [43] S. Wang, J. He, Variational iteration method for solving integro-dierential equations. Phy Lett A 2007, 367, 188-191. [44] S. Karimi Vanani, A. Aminataei, Numerical solution of Volterra integro-dierential equations, Journal of Computational Analy- sis and Applications 13 (2011) 654-662.

Note: Nagma Irfan’s aliation at the time of rst online publication was: Department of Basic & Applied Sciences, CMJ University, Shillong, Meghalaya-793 003, India. This has been now replaced by the current aliation.

Received November 25, 2013; accepted February 23, 2014.