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Applied Letters 24 (2011) 559–565

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Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

Integrals of Bernstein : An application for the solution of high even-order differential equations

E.H. Doha a, A.H. Bhrawy b,∗, M.A. Saker c a Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt b Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt c Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, Egypt article info a b s t r a c t

Article history: A new explicit formula for the of Bernstein polynomials of any degree for Received 6 August 2010 any order in terms of Bernstein polynomials themselves is derived. A fast and accurate Received in revised form 12 November algorithm is developed for the solution of high even-order boundary value problems 2010 (BVPs) with two point boundary conditions but by considering their integrated forms. The Accepted 16 November 2010 Bernstein–Petrov–Galerkin method (BPG) is applied to construct the numerical solution for such problems. The method is then tested on examples and compared with other methods. Keywords: It is shown that the BPG yields better results. Bernstein polynomials Spectral method © 2010 Elsevier Ltd. All rights reserved. High order differential equations

1. Introduction

Bernstein polynomials play a prominent role in various areas of mathematics. These polynomials have been frequently used in both the solution of differential equations and approximation theory; see, e.g., [1–6]. With the advent of computer graphics, Bernstein restricted to the interval x ∈ [0, 1] becomes important in the form of Bézier curves [7,8]. Many properties of the Bézier curves and surfaces come from the properties of the Bernstein polynomials. Due to the increasing interest in Bernstein polynomials the question arises as to how to describe their properties in terms of their coefficients when they are given in the Bernstein , Farouki and Goodman [9] proved that the computation in the Bernstein bases are more stable than computations in other polynomial bases. Moreover change of bases algorithms are themselves often numerically unstable, therefore when polynomials are represented in the Bernstein bases, it is better to perform all calculations using only the Bernstein coefficients. Formulae for multiplication, differentiation, integration and subdivision of polynomials in the Bernstein form are given in [10]. In the solution of differential equations, an alternative approach to differentiating solution expansions is to integrate the differential equation q times, where q is the order of the equation. An advantage of this approach is that the general equation in the algebraic system then contains a finite number of terms; see [11]. Phillips and Karageorghis [12] and Doha [13] have followed this approach to obtain a formula relating the expansion coefficients of an infinitely differentiable function that have been integrated an arbitrary number of times in terms of the expansion coefficients of the function when the expansion functions are ultraspherical polynomials. The corresponding formula when the expansion functions are Jacobi polynomials is given in [14]. Up to now, and to the best of our knowledge, many formulae corresponding to those mentioned previously are unknown and are traceless in the literature for Bernstein polynomials. This partially motivates our interest in such polynomials.

∗ Corresponding author. Tel.: +20 0105812028. E-mail addresses: [email protected] (E.H. Doha), [email protected] (A.H. Bhrawy), [email protected] (M.A. Saker).

0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.11.013 560 E.H. Doha et al. / Applied Mathematics Letters 24 (2011) 559–565

Another motivation is concerned with the direct solution techniques for solving the integrated forms of high even-order differential equations, using the BPG approximations. In this paper, the proposed differential equations are integrated q times, where q is the order of the equation and we make use of the formulae relating the expansion coefficients of integration appearing in this integrated forms of the proposed differential equations to the Bernstein polynomials itself, and then we apply Bernstein–Petrov–Galerkin (BPG) approximations. Numerical results are presented in which the usual exponential convergence behavior of spectral approximations is exhibited. The remainder of this paper is organized as follows. In Section 2, we give an overview of Bernstein polynomials and the relevant properties needed in what follows, and in Section 3, we prove the main result of the paper which is an explicit expression for the integrals of Bernstein polynomials of any degree and for any order in terms of the Bernstein polynomials themselves. In Section 4, we discuss the BPG method and describe how they are used to solve high even- order differential equations. Section 5 gives some numerical results exhibiting the accuracy and efficiency of our proposed numerical algorithms. We end the paper with few concluding remarks in Section 6.

2. Some properties of Bernstein polynomials

The Bernstein polynomials of nth degree form a complete basis over [0, 1] (see, e.g., [15], p. 66), and they are defined by

 n  i n−i Bi,n(x) = x (1 − x) , 0 ≤ i ≤ n, (2.1) i ! where the binomial coefficients are given by  n  = n . i i!(n−i)! In this paper, we will use the generalized Bernstein polynomials of nth degree which form a complete basis over [a, b] and they are defined by

1  n  i n−i Bi,n(x) = (x − a) (b − x) , 0 ≤ i ≤ n. (2.2) (b − a)n i The of the nth degree Bernstein polynomials are polynomials of degree n − 1, and are given by n d DBi,n(x) = (Bi−1,n−1(x) − Bi,n−1(x)), D ≡ . (2.3) (b − a) dx The multiplication of two Bernstein bases gives

 j   m  i k Bi,j(x)Bk,m(x) = Bi+k,j+m(x). (2.4)  j+m  i+k Indefinite of Bernstein basis is given by

+ ∫ (b − a) −n 1 Bi,n(x)dx = Bj,n+1(x), (2.5) n + 1 j=i+1 and all Bernstein basis functions of the same order have the same definite integral over the interval [a, b], namely

∫ b (b − a) Bi,n(x)dx = . (2.6) a n + 1

3. Integrals of Bernstein polynomials

The main objective of this section is to prove the following theorem for the integrals of Bi,n(x).

Theorem 3.1. ! n+q  − −  q q n − j i 1 I Bi,n(x) = (b − a) Bj,n+q +π ¯ q−1(x) (3.1) n + q ! q − 1 ( ) j=i+q

q where I f (x) is the qth integral of f (x) with respect to x and π¯ q−1(x) is a polynomial of order q − 1. Proof. We use the induction principle to prove this theorem. Farouki and Rajan [10] have shown that

+ ∫ (b − a) −n 1 Bi,n(x)dx = Bj,n+1(x) +π ¯ 0(x), (3.2) n + 1 j=i+1 E.H. Doha et al. / Applied Mathematics Letters 24 (2011) 559–565 561 which may be written in the form + + n! −n 1  j − i − 1  (b − a) −n 1 IBi,n(x) = (b − a) Bj,n+1(x) = Bj,n+1(x) +π ¯ 0(x), n + 1 ! 0 n + 1 ( ) j=i+1 j=i+1 and this in turn shows that (3.1) is true for q = 1. Proceeding by induction, assuming that (3.1) is valid for q, we want to prove that ! n+q+1  − −  q+1 q+1 n − j i 1 I Bi,n(x) = (b − a) Bj,n+q+1(x) +π ¯ q(x). n + q + 1 ! q ( ) j=i+q+1 Using (2.5) and assuming the validity of (3.1) for q, we have ∫ ! n+q  − −  q+1  q  q n − j i 1 I Bi,n(x) = I I Bi,n(x) = ((b − a) Bj,n+q(x) +π ¯ q−1(x))dx n + q ! q − 1 ( ) j=i+q ! n+q  − −  ∫ q n − j i 1 = (b − a) Bj,n+q(x)dx +π ¯ q(x) n + q ! q − 1 ( ) j=i+q ! n+q  − −  − n+q+1 q n − j i 1 (b a) − = (b − a) Bk,n+q+1(x) +π ¯ q(x) n + q ! q − 1 n + q + 1 ( ) j=i+q k=j+1 ! n+q  − −  n+q+1 q+1 n − j i 1 − = (b − a) Bk,n+q+1(x) +π ¯ q(x). n + q + 1 ! q − 1 ( ) j=i+q k=j+1 ∑n+q ∑n+q+1 + ≤ ≤ + If we expand the two summations j=i+q m=j+1 and collect the similar terms of Bj,n+q+1 for i q j n q with the help of Chu’s theorem (see [16], p. 45), we get ! n+q+1  − −  q+1 q+1 n − j i 1 I Bi,n(x) = (b − a) Bj,n+q+1(x) +π ¯ q(x), n + q + 1 ! q ( ) j=i+q+1 which completes the induction and proves the theorem.  We need the following corollary in the following sections.

Corollary 3.2.

 j−i−1  n  n+q  + n+q   ∫ b − q 1 ! − q (b a) n − q 1 k j I Bi,n(x)Bk,n(x)dx = . (3.3) 2n + q + 1 n + q !  2n+q  a ( )( ) j=i+q k+j

Proof. ∫ b − q ! n+q  − −  ∫ b q (b a) n − j i 1 I Bi,n(x)Bk,n(x)dx = Bj,n+q(x)Bk,n(x)dx n + q ! q − 1 a ( ) j=i+q a

 j−i−1   n   n+q  q n+q b (b − a) n! − q−1 k j ∫ = Bj+k,2n+q(x)dx n + q !  2n+q  ( ) j=i+q a k+j

 j−i−1   n   n+q  q+1 n+q (b − a) n! − q−1 k j = .  2n + q + 1 n + q !  2n+q  ( )( ) j=i+q k+j

4. High even-order differential equations

Bernstein polynomials [17] have many useful properties, such as, positivity, continuity and unity partition of the basis set over the interval [0, 1]. The Bernstein polynomial bases [18] vanish except for the first polynomial at x = 0, which is equal to 1 and the last polynomial at x = 1, which is also equal to 1 over the interval [0, 1]. This provides greater flexibility in imposing boundary conditions at the end points of the interval. 562 E.H. Doha et al. / Applied Mathematics Letters 24 (2011) 559–565

We consider such bases to numerically solve the 2mth-order ordinary differential equation

u(2m)(x) + αu(x) = g(x), in I = (a, b), (4.1) subject to the following boundary conditions

u(q)(a) = 0, u(q)(b) = 0, 0 ≤ q ≤ m − 1, (4.2) where α is a real constant and g(x) is a given source function. There is no loss of generality in considering only homogeneous boundary conditions because BVPs with nonhomogeneous boundary conditions can be easily reduced to the treated case by applying suitable transformations. We consider the integrated form of (4.1) and (4.2), namely,

∫ (2m) 2m−1 (2m) − u(x) + α u(x)(dx) = f (x) − diBi,i(x), (4.3) i=0 where di are constants of integration and   q times ∫ (q) ∫ ∫   q times ∫ (2m) u(x)(dx)(q) = ··· u(x) dx dx ··· dx, f (x) = g(x)(dx)(2m).

2 := 2 =  We denote by L (I)(I (a, b)) the L space with inner product: (u, v) I u(x)v(x)dx, and the associated norm 1/2 2 ‖u‖ = (u, u) . It is clear that {Bi,n(x) : n ≥ 0, 0 ≤ i ≤ n} forms a complete system in L (I). Hence, we define   SN = span B0,N (x), B1,N (x), . . . , BN,N (x) ,  (q) (q)  VN = ν ∈ SN : ν (a) = ν (b) = 0; 0 ≤ q ≤ m − 1 , WN = {ν ∈ SN }, then the BPG approximation to (4.3) is to find uN (x) ∈ VN such that

∫ (2m)  2m−1  (2m) − α uN (x)(dx) , ν + (uN (x), ν) + diBi,i(x), ν = (f (x), ν), ∀ ν ∈ WN . (4.4) i=0

We choose the approximate solution uN (x) and the dual of basis ν, to be of the form

− N−m uN (x) = aiBi,N (x), and ν = Bk,N , 0 ≤ k ≤ N. (4.5) i=m It is now clear that (4.4) is equivalent to

N−m [ ∫ (2m)  ] 2m−1 − (2m)   −   ai α Bi,N (x)(dx) , Bk,N (x) + Bi,N (x), Bk,N (x) + di Bi,i(x), Bk,N (x) i=m i=0   = f (x), Bk,N (x) , ∀ν ∈ WN , k = 0, 1,..., N. (4.6) Let us denote   T fk = f , Bk,N (x) , f = (f0, f1,..., fN ) , T a = (d0,..., dm−1, am,..., aN−m, dm,..., d2m−1) ,

A = (akj)0≤k,j≤N , B = (bkj)0≤k,j≤N , D = (dkj)0≤k,j≤N ; then (4.6) is equivalent to the matrix equation (A + αB + D)a = f, (4.7) where the elements of (N + 1) × (N + 1) matrices A, B and D for 0 ≤ k, j ≤ N are given explicitly in the following theorem.

Theorem 4.1. If we take ν as defined in (4.5), and if we denote  B (x), B (x) , 0 ≤ k ≤ N, m ≤ j ≤ N − m, a = j,N k,N kj 0, otherwise, ∫ (2m)   (2m) Bj,N (x)(dx) , Bk,N (x) , 0 ≤ k ≤ N, m ≤ j ≤ N − m, bkj = 0, otherwise, E.H. Doha et al. / Applied Mathematics Letters 24 (2011) 559–565 563 and   ≤ ≤ ≤  Bj,j(x), Bk,N (x) , 0 k N, 0 j < m, d = 0, 0 ≤ k ≤ N, m ≤ j ≤ N − m, kj    Bj−N+2m−1,j−N+2m−1(x), Bk,N (x) , 0 ≤ k ≤ N, N − m < j ≤ N, then the nonzero elements of the matrices A, B and D in system (4.7) are given explicitly by

 N   N  +  N   N+2m   i−j−1  (b − a)2m 1N! N+2m (b − a) j k k − i 2m−1 akj = , bkj = , 2N + 1  2N  N + 2m ! 2N + 2m + 1  2N+2m  ( ) ( ) ( ) i=j+2m k+j k+i

  N  −  k b a   , 0 ≤ j < m,  N+j N + j + 1  k+j dkj =    N  k b − a   , N − m < j ≤ N.  j+2m−1 j + 2m k+j−N+2m−1

Proof. Using (2.4) and (2.6), we get

 N   N  b ∫ (b − a) j k = = akj Bj,N (x)Bk,N (x)dx   , a (2N + 1) 2N k+j and from Corollary 3.2 with q = 2m, we get

   +   − −  2m+1 N + N 2m i j 1 ∫ b ∫ (2m)  (b − a) N! N 2m − (2m) k − i 2m 1 bkj = Bj,N (x)(dx) Bk,N (x)dx = . N + 2m ! 2N + 2m + 1  2N+2m  a ( ) ( ) i=j+2m k+i Finally from (2.4) and (2.6), we get

  N  −  k b a   , 0 ≤ j < m,  N+j N + j + 1  k+j d = kj  N   −  k b a   , N − m < j ≤ N.   j+2m−1 j + 2m k+j−N+2m−1

5. Numerical results

In this section three examples are considered for the numerical illustrations of the method developed. A comparison between BPG approximation and other methods proposed in [19–22] are made. All calculations are obtained by using MATHEMATICA 5.1. Example 1. Consider the boundary value problem (see, [20]):

u(2)(x) − u(x) = 4 − 2x2 sin x + 4x cos x, x ∈ [0, 1], (5.1) subject to the boundary conditions u(0) = u(1) = 0, with the exact solution u(x) = (x2 − 1) sin(x).

Table 1 lists the maximum pointwise error (Ep) and maximum relative error (Er ) of u − uN using the BPG method with various choices of N. Table 1 shows that our method has better accuracy compared with the quintic nonpolynomial method developed in [20], it is shown that, in the case of solving linear system of order 14, we obtain a maximum absolute error of order 10−16. In the mentioned method [20] the maximum absolute error is 6.5 × 10−14 obtained by using linear system of order 64.

Example 2. We consider the fourth-order two point boundary value problem (see, [19]): u(4)(x) − 3u(x) = −2ex, x ∈ [0, 1], ′ ′ u(0) = 1, u(1) = e, u (0) = 1, u (1) = e, with the analytical solution u(x) = ex. 564 E.H. Doha et al. / Applied Mathematics Letters 24 (2011) 559–565

Table 1

Ep and Er for N = 2, 4,..., 14.

N (Ep)(Er ) 2 4.12549 × 10−2 4.48265 × 10−1 4 6.47323 × 10−4 1.52743 × 10−2 6 2.58405 × 10−6 1.07868 × 10−4 8 4.53272 × 10−9 2.95069 × 10−7 10 4.44754 × 10−12 4.15409 × 10−10 12 2.85882 × 10−15 3.52716 × 10−13 14 2.77556 × 10−16 4.30838 × 10−14

Table 2

Ep and Er for N = 4, 6,..., 14.

NEp Er 4 1.44117 × 10−4 8.11976 × 10−5 6 5.12228 × 10−7 3.58542 × 10−7 8 7.31859 × 10−10 5.81123 × 10−10 10 1.77636 × 10−13 5.37474 × 10−13 12 4.51154 × 10−15 8.72459 × 10−16 14 1.33227 × 10−15 6.52387 × 10−16

Table 3 Comparison between different methods for Example 2. Error BPG Sinc-Galerkin in [19] Modified decomposition in [19]

−15 −9 −8 Ep 1.33227 × 10 3.7 × 10 2.5 × 10

Table 4

Ep and Er for N = 6, 8,..., 16.

NEp Er 6 5.01225 × 10−6 8.22534 × 10−6 8 2.12730 × 10−8 3.80401 × 10−8 10 3.93391 × 10−11 7.22777 × 10−11 12 4.16334 × 10−14 7.68740 × 10−14 14 4.44089 × 10−16 1.66256 × 10−14 16 4.44089 × 10−16 1.37344 × 10−14

Table 2 lists the maximum pointwise error and maximum relative error of u − uN using the BPG method with various choices of N. In Table 3, we introduce a comparison between the error obtained by using the BPG method and the sinc-Galerkin method and modified decomposition method developed in [19]. It is shown that our method is more accurate.

Example 3. Consider the sixth-order BVP (see, [19,21,22]) u(6)(x) − u(x) = −6ex, x ∈ [0, 1], ′ ′′ u(0) = 1, u (0) = 0, u (0) = −1, ′ ′′ u(1) = 0, u (1) = −e, u (1) = −2e, with the exact solution u(x) = (1 − x)ex.

Table 4 lists the maximum pointwise error and maximum relative error of u − uN using the BPG method with various choices of N. Table 5 exhibits a comparison between the error obtained by using the BPG method and sinc-Galerkin [19], septic splines [21] and modified decomposition [22]. It is shown that our method is more accurate.

6. Conclusion

In this paper we derived an explicit formula for the integrals of Bernstein polynomials of any degree for any order in terms of Bernstein polynomials themselves. An efficient and accurate numerical scheme based on the Bernstein–Petrov–Galerkin method is developed for solving the integrated form of the high even-order differential equation. The problem is reduced E.H. Doha et al. / Applied Mathematics Letters 24 (2011) 559–565 565

Table 5 Comparison between the errors of different methods in Example 3. Error BPG Sinc-Galerkin [19] Septic spline [21] Modified decomposition [22]

−16 −6 −4 −4 Ep 4.44089 × 10 9.2 × 10 2.1 × 10 1.3 × 10 −14 −3 −3 Er 1.37344 × 10 0.1 × 10 1.77 × 10 – to the solution of a system of linear algebraic equations. Numerical examples were given to demonstrate the validity and applicability of the presented method.

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