Approximate Solutions of Differential Equations by Using the Bernstein Polynomials

Abstract

A numerical method for solving differential equations by approximating the solution in the Bernstein polynomial basis is proposed. At first, we demonstrate the relation between the Bernstein and Legendre polynomials. By using this relation, we derive the operational matrices of integration and product of the Bernstein polynomials. Then, we employ them for solving differential equations. The method converts the differential equation to a system of linear algebraic equations. Finally some examples and their numerical solutions are given; comparing the results with the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

1. Introduction

In recent years, the Bernstein polynomials (B-polynomials) have attracted the attention of many researchers. These polynomials have been utilized for solving different equations by using various approximate methods. For instance, B-polynomials have been used for solving Fredholm integral equations [1, 2], Volterra integral equations [3], differential equations [4–7], and integro-differential equations [8]. Singh et al. [6] and Yousefi and Behroozifar [7] have proposed an operational matrix in different ways for solving differential equations. In [6], the B-polynomials have been first orthonormalized by using Gram-schmidt orthonormalization process, and then the operational matrix of integration has been obtained. By the expansion of B-polynomials in terms of Taylor basis, Yousefi and Behroozifar have found the operational matrices of integration and product of B-polynomials. In this paper, firstly, we present operational matrices of integration 𝑃_𝑏 and product 𝐶 for the B-polynomials, by the expansion of B-polynomials in terms of Legendre polynomials. Then, we use them for solving differential equation^𝑠_𝑗=0𝜌_𝑗(𝑥)𝑦^(𝑗)(𝑥)=𝑔(𝑥),0≤𝑥≤1,(1.1)

with the initial conditions𝑦^(𝑘)(0)=𝑏_𝑘,0≤𝑘≤𝑠−1,(1.2)

where 𝑔(𝑥) and 𝜌_𝑗(𝑥), 𝑗=0,…,𝑠 are given functions and 𝑦(𝑥) is the unknown function to be determined. The main characteristic of this technique is that it reduces these equations to those of an easily soluble algebraic equation, thus greatly simplifying the equations. Special attention has been given to the applications of Legendre wavelets method [9–11], Homotopy perturbation method (HPM) [12], modified decomposition method (MDM) [13], Taylor matrix method [14], and Chebyshev wavelets method [15]. The organization of this paper is as follows: in Section 2, we introduce the B-polynomials and their properties. Section 3 is devoted to the function approximation by using B-polynomials basis. Section 4 introduces the expansion of B-polynomial in terms of Legendre basis, and vice versa. The operational matrices of integration and product will be derived in Section 5. Section 6 is devoted to the solution method of differential equations. In section 7, we present some numerical examples. Numerical solution of each equation based on the exact and approximate solutions are compared. And Section 8 offers our conclusion.

2. B-Polynomials and Their Properties

The B-polynomials of 𝑚th degree are defined on the interval [0,1] as [4]𝐵_𝑖,𝑚(𝑥)=𝑚𝑖𝑥^𝑖(1−𝑥)^𝑚−𝑖,0≤𝑖≤𝑚,(2.1)

where𝑚𝑖=𝑖!(𝑚𝑚−!𝑖)!.(2.2)

There are 𝑚+1,𝑚th degree B-polynomials. For mathematical convenience, we usually set 𝐵_𝑖,𝑚(𝑥)=0, if 𝑖<0 or 𝑖>𝑚. These polynomials are quite easy to write down: the coefficients can be obtained from Pascal’s triangle. It can easily be shown that each of the B-polynomials is positive and also the sum of all the B-polynomials is unity for all real 𝑥∈[0,1], that is,^𝑚_𝑖=0𝐵_𝑖,𝑚(𝑥)=1,𝑥∈[0,1].(2.3)

See [16] for complete details.

3. Function Approximation

B-polynomials defined above form a complete basis [1] over the interval [0,1]. It is easy to show that any given polynomial of degree 𝑚 can be expressed in terms of linear combination of the basis functions. A function 𝑓(𝑥) defined over [0,1] may be expanded as𝑓(𝑥)≃𝑃_𝑚(𝑥)=^𝑚_𝑖=0𝑐_𝑖𝐵_𝑖,𝑚(𝑥),𝑚≥1.(3.1)

Equation (3.1) can be written as𝑃_𝑚(𝑥)=𝐶^𝑇𝜙(𝑥),(3.2)

where 𝐶 and 𝜙(𝑥) are (𝑚+1)×1 vectors given by𝜙(𝑥)=𝐵𝐶_0,𝑚=𝑐₀,𝑐₁,…,𝑐_𝑚^𝑇,(3.3)(𝑥),𝐵_1,𝑚(𝑥),…,𝐵_𝑚,𝑚(𝑥)^𝑇.(3.4)

The use of an orthogonal basis on [0,1] allows us to directly obtain the least-squares coefficients of 𝑃_𝑚(𝑥) in that basis, and also ensures permanence of these coefficients with respect to the degree 𝑚 of the approximant, that is, all the coefficients of 𝑃_𝑚+1 agree with those of 𝑃_𝑚(𝑥), except for that of the newly introduced term. The B-polynomials are not orthogonal. But, these can be expressed in terms of some orthogonal polynomials, such as the Legendre polynomials. The Legendre polynomials constitute an orthogonal basis that is well suited [17, 18] to least-squares approximation.

4. Expansion of B-Polynomials in Terms of Legendre Basis and Vice Versa

To use the Legendre polynomials for our purposes, it is preferable to map this to [0,1]. A set of shifted Legendre polynomials, denoted by {𝐿_𝑘(𝑥)} for 𝑘=0,1,…, is orthogonal with respect to the weighting function 𝑤(𝑥)=1 over the interval [0,1]. These polynomials satisfy the recurrence relation [19](𝑘+1)𝐿_𝑘+1(𝑥)=(2𝑘+1)(2𝑥−1)𝐿_𝑘(𝑥)−𝑘𝐿_𝑘−1(𝑥),𝑘=1,2,…,(4.1)

with𝐿𝐿₀(𝑥)=1,₁(𝑥)=2𝑥−1.(4.2)

The orthogonality of these polynomials is expressed by the relation¹⁰𝐿_𝑗(𝑥)𝐿_𝑘(𝑥)𝑑𝑥=⎧⎪⎨⎪⎩02,𝑘1+1,𝑗𝑗≠=𝑘𝑘,,𝑗,𝑘=0,1,2,….(4.3)

When the approximant (3.1) is expressed in the Legendre form𝑃_𝑚(𝑥)=^𝑚_𝑗=0𝑙_𝑗𝐿_𝑗(𝑥),(4.4)

by using (4.3), we can obtain the Legendre coefficients as𝑙_𝑗=(2𝑗+1)¹⁰𝐿_𝑗(𝑥)𝑓(𝑥)𝑑𝑥,𝑗=0,…,𝑚.(4.5)

Lemma 4.1. The Legendre polynomial 𝐿_𝑘(𝑥) can be expressed in the kth degree Bernstein basis 𝐵_0,𝑘(𝑥),𝐵_1,𝑘(𝑥),…,𝐵_𝑘,𝑘(𝑥) as [20] 𝐿_𝑘(𝑥)=^𝑘_𝑖=0(−1)^𝑘+𝑖𝑘𝑖𝐵^𝑖,𝑘(𝑥).(4.6)

Now consider a polynomial 𝑃_𝑚(𝑥) of degree 𝑚, expressed in the 𝑚th degree Bernstein and Legendre bases on 𝑥∈[0,1]𝑃_𝑚(𝑥)=^𝑚_𝑗=0𝑐_𝑗𝐵_𝑗,𝑚(𝑥)=^𝑚_𝑘=0𝑙_𝑘𝐿_𝑘(𝑥).(4.7)

We write the transformation of the Legendre polynomials on [0,1] into the 𝑚th degree Bernstein basis functions as [21]𝐵_𝑘,𝑚(𝑥)=^𝑚_𝑖=0𝑤_𝑘,𝑖𝐿_𝑖(𝑥),𝑘=0,…,𝑚.(4.8)

The elements 𝑤_𝑘,𝑖, 𝑘,𝑖=0,1,…,𝑚, form an (𝑚+1)×(𝑚+1) basis conversion matrix 𝑊. To compute them, we multiply (4.8) by 𝐿_𝑗(𝑥), integrate over 𝑥∈[0,1], and use (4.3) to obtain𝑤_𝑘,𝑗=(2𝑗+1)¹⁰𝐵_𝑘,𝑚(𝑥)𝐿_𝑗(𝑥)𝑑𝑥.(4.9)

We now replace (4.6) into (4.9) and obtain𝑤_𝑘,𝑗=(2𝑗+1)^𝑗_𝑖=0(−1)^𝑗+𝑖𝑗𝑖¹⁰𝐵_𝑘,𝑚(𝑥)𝐵_𝑖,𝑗(𝑥)𝑑𝑥.(4.10)

The integrals of the products of Bernstein basis functions can be found using¹⁰(1−𝑥)^𝑟𝑥^𝑖𝑑𝑥=1(𝑟+𝑖+1)^{𝑖𝑟+𝑖},𝑖,𝑟∈𝑁∪{0},(4.11)

as follows:¹⁰𝐵_𝑘,𝑚(𝑥)𝐵_𝑖,𝑗(𝑥)𝑑𝑥=𝑚𝑘𝑗𝑖¹⁰𝑥^𝑘+𝑖(1−𝑥)^{𝑚+𝑗−𝑘−𝑖}𝑑𝑥=(^𝑚𝑘)^𝑗𝑖(𝑚+𝑗+1)^{𝑚+𝑗𝑘+𝑖}.(4.12)

Therefore, we have the elements of 𝑊 as𝑤_𝑘,𝑗=(2𝑗+1)𝑚+𝑗+1𝑚𝑘^𝑗_𝑖=0(−1)^𝑗+𝑖^𝑗𝑖^𝑗𝑖^{𝑚+𝑗𝑘+𝑖},𝑘,𝑗=0,…,𝑚.(4.13)

Now, we write the transformation of the B-polynomials on [0,1] into 𝑚th degree Legendre basis functions as [21]𝐿_𝑘(𝑥)=^𝑚_𝑗=0Λ_𝑘,𝑗𝐵_𝑗,𝑚(𝑥),𝑘=0,…,𝑚.(4.14)

The elements Λ_𝑘,𝑗 form an (𝑚+1)×(𝑚+1) basis conversion matrix Λ. Replacing (4.14) into (4.7) and rearranging the order of summation, we obtain𝑐_𝑗=^𝑚_𝑘=0𝑙_𝑘Λ_𝑘,𝑗,𝑗=0,…,𝑚.(4.15)

Since we can express each 𝑘th degree Bernstein basis function in the 𝑚th degree Bernstein basis as [21]𝐵_𝑖,𝑘(𝑥)=^{𝑚−𝑘+𝑖}_𝑗=𝑖^𝑘𝑖^{𝑚−𝑘𝑗−𝑖}^𝑚𝑗𝐵^𝑗,𝑚(𝑥),𝑖=0,…,𝑘,(4.16)

replacing (4.16) into (4.6) and rearranging the order of summation, we find that the basis transformation (4.14) is defined by the elementsΛ_𝑘,𝑗=1^𝑚𝑗^{min{𝑗,𝑘}}_𝑖=𝑟(−1)^𝑘+𝑖𝑘𝑖𝑘𝑖𝑚𝑗−−𝑖𝑘,𝑟=max{0,𝑗+𝑘−𝑚},(4.17)

of the matrix Λ for 𝑘,𝑗=0,…,𝑚. If we denote the Legendre basis vector as𝐿(𝑥)=𝐿₀(𝑥),𝐿₁(𝑥),…,𝐿_𝑚(𝑥)^𝑇,(4.18)

using (3.4), (4.8), (4.14), and (4.18), we have𝜙𝐿((𝑥𝑥))==𝑊Λ𝜙𝐿((𝑥𝑥)).,((44..1290))

5. Operational Matrices of Integration and Product of B-Polynomials

5.1. B-Polynomials Operational Matrix of Integration

Let 𝑃_𝑏 be an (𝑚+1)×(𝑚+1) operational matrix of integration, then^𝑥0𝜙(𝑡)𝑑𝑡⋍𝑃_𝑏𝜙(𝑥),0≤𝑥≤1.(5.1)

By using (4.19), we have^𝑥0𝜙(𝑡)𝑑𝑡=𝑊^𝑥0(𝑡)𝑑𝑡=𝑊𝑃𝐿(𝑥),(5.2)

where the (𝑚+1)×(𝑚+1) matrix 𝑃 is the operational matrix of integration of the shifted Legendre polynomials on the interval [0,1] and can be obtained as [22]𝑃=12⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−11100⋯00030−01130⋯000⋮0⋮50⋮00⋮015⋱⋯⋯−⋮01⋮0⋮00000⋯2𝑚0−1−012𝑚1−12𝑚+10⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(5.3)

and, therefore, by using (4.20)–(5.2), we have the operational matrix of integration as𝑃_𝑏=𝑊𝑃Λ.(5.4)

5.2. B-Polynomials Operational Matrix of Product

In this subsection, we present a general formula for finding the operational matrix of product of 𝑚th degree B-polynomials. Suppose that 𝐶 is an arbitrary (𝑚+1)×1 vector, then 𝐶 is an (𝑚+1)×(𝑚+1) operational matrix of product whenever𝐶^𝑇𝜙(𝑥)𝜙^𝑇(𝑥)⋍𝜙^𝑇(𝑥)𝐶.(5.5)

Using (4.19) and since 𝐶^𝑇𝜙(𝑥)=∑^𝑚𝑖=0𝑐_𝑖𝐵_𝑖,𝑚, we have𝐶^𝑇=𝜙(𝑥𝐶)𝜙^𝑇(𝑥)_𝑇=𝐿₀𝜙(𝑥)𝐿^𝑇(𝑥)𝑊^𝑇(𝑥)𝐶^𝑇𝜙(𝑥),𝐿₁(𝑥)𝐶^𝑇𝜙(𝑥),…,𝐿_𝑚(𝑥)𝐶^𝑇𝜙(𝑥)𝑊^𝑇=^𝑚_𝑖=0𝑐_𝑖𝐿₀(𝑥)𝐵_𝑖,𝑚(𝑥),^𝑚_𝑖=0𝑐_𝑖𝐿₁(𝑥)𝐵_𝑖,𝑚(𝑥),…,^𝑚_𝑖=0𝑐_𝑖𝐿_𝑚(𝑥)𝐵_𝑖,𝑚(𝑥)𝑊^𝑇.(5.6)

Now, we approximate all functions 𝐿_𝑘(𝑥)𝐵_𝑖,𝑚(𝑥) in terms of {𝐵_𝑖,𝑚(𝑥)}^𝑚𝑖=0 for 𝑖,𝑘=0,1,…,𝑚. Let𝜂_𝑘,𝑖=⎡⎢⎢⎢⎢⎢⎣𝜂^0𝑘,𝑖𝜂^1𝑘,𝑖𝜂⋮_{𝑚𝑘,𝑖}⎤⎥⎥⎥⎥⎥⎦,(5.7)

by (3.2), we have𝐿_𝑘(𝑥)𝐵_𝑖,𝑚(𝑥)≃𝜂^{𝑇𝑘,𝑖}𝜙(𝑥),𝑖,𝑘=0,1,…,𝑚.(5.8)

Using (4.15), we can obtain the elements of vector 𝜂_𝑘,𝑖, for 𝑖,𝑘=0,1,…,𝑚. Therefore,^𝑚_𝑖=0𝑐_𝑖𝐿_𝑘(𝑥)𝐵_𝑖,𝑚(𝑥)≃^𝑚_𝑖=0𝑐_𝑖^𝑚_𝑗=0=𝜂^{𝑗𝑘,𝑖}𝐵_𝑗,𝑚(𝑥)_𝑚_𝑗=0𝐵_𝑗,𝑚(𝑥)^𝑚_𝑖=0𝑐_𝑖𝜂^{𝑗𝑘,𝑖}=𝜙_𝑇(𝑥)⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣^𝑚_𝑖=0𝑐_𝑖𝜂^{0𝑚𝑘,𝑖}_𝑖=0𝑐_𝑖𝜂^1𝑘,𝑖_𝑚⋮_𝑖=0𝑐_𝑖𝜂^{𝑚𝑘,𝑖}=𝜙_𝑇⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=𝜙_𝑇(𝑥)𝜂_𝑘,0,𝜂_𝑘,1,…,𝜂_𝑘,𝑚𝐶(𝑥)𝐶_𝑘,(5.9)

where𝐶_𝑘=𝜂_𝑘,0,𝜂_𝑘,1,…,𝜂_𝑘,𝑚𝐶,𝑘=0,1,…,𝑚.(5.10)

If we define a (𝑚+1)×(𝑚+1) matrix 𝐶=[𝐶₀,𝐶₁,…,𝐶_𝑚], then by using (5.6) and (5.9), we have𝐶^𝑇𝜙(𝑥)𝜙^𝑇(𝑥)≃𝜙^𝑇(𝑥)𝐶₀,𝐶₁,…,𝐶_𝑚𝑊^𝑇=𝜙^𝑇(𝑥)𝐶𝑊^𝑇,(5.11)

and, therefore, we have the operational matrix of product as𝐶=𝐶𝑊^𝑇.(5.12)

6. Solution of the Linear Differential Equation

Consider the linear differential equation (1.1) with the initial conditions (1.2). If we approximate 𝑔(𝑥), 𝜌_𝑗(𝑥), 𝑗=0,…,𝑠 and 𝑦^(𝑠)(𝑥) as follows:𝜌𝑔(𝑥)=𝐺^𝑇𝜙(𝑥),_𝑗(𝑥)=𝑦𝑃^𝑇𝑗𝜙(𝑥),𝑗=0,…,𝑠,(6.1)_(𝑠)(𝑥)=𝐶^𝑇𝜙(𝑥),(6.2)

where 𝐺, 𝑃_𝑗, 𝑗=0,…,𝑠, and 𝐶 are the coefficients which are defined similarly to (3.3). With 𝑠-times integrating from (6.2) with respect to 𝑥 between 𝑥=0 to 𝑥=𝑥, using (5.1) and the initial conditions (1.2), we will have𝑦𝑦^(𝑠−1)(𝑥)=𝑏_𝑠−1+𝐶^𝑇𝑃_𝑏𝜙(𝑥),_(𝑠−2)𝑦′(𝑥)=𝑏₁(𝑥)=𝑏_𝑠−2+⋮𝑏_𝑠−1𝑥+𝐶^𝑇𝑃^2𝑏𝜙(𝑥),+𝑏₂𝑥+𝑏³2!𝑥²+⋯+𝑏^𝑠−1(𝑠−2)!𝑥^(𝑠−2)𝑦(𝑥)=𝑏₀+𝐶^𝑇𝑃^𝑏𝑠−1𝜙(𝑥),+𝑏₁𝑥+𝑏²2!𝑥²+⋯+𝑏^𝑠−1(𝑠−1)!𝑥^(𝑠−1)+𝐶^𝑇𝑃^𝑠𝑏𝜙(𝑥).(6.3)

Let𝑏𝑥^𝑖=𝑑^𝑇𝑖𝜙(𝑥),𝑖=1,2,…,𝑠−1,_𝑠−𝑖=𝑏_𝑠−𝑖𝐸^𝑇𝜙(𝑥),𝑖=1,2,…,𝑠,(6.4)

where1=𝐸^𝑇𝜙(𝑥).(6.5)

Substituting (6.4) into (6.3), we have𝑦𝑦^(𝑠)(𝑥)=𝐶^𝑇𝜙(𝑥)=𝑄^𝑇𝑠𝜙(𝑥),_(𝑠−1)𝑦(𝑥)=𝑏_𝑠−1𝐸^𝑇+𝐶^𝑇𝑃_𝑏𝜙(𝑥)=𝑄^𝑇𝑠−1𝜙(𝑥),_(𝑠−2)𝑦′(𝑥)=𝑏₁(𝑥)=𝑏_𝑠−2𝐸^𝑇+𝑏_𝑠−1𝑑^𝑇1⋮+𝐶^𝑇𝑃^2𝑏𝜙(𝑥)=𝑄^𝑇𝑠−2𝜙(𝑥),𝐸^𝑇+𝑏₂𝑑^𝑇1+𝑏³2!𝑑^𝑇2+⋯+𝑏^𝑠−1(𝑠−2)!𝑑^𝑇𝑠−2+𝐶^𝑇𝑃^𝑏𝑠−1𝜙(𝑥)=𝑄^𝑇1𝑦(𝑥)=𝑏₀𝜙(𝑥),(6.6)𝐸^𝑇+𝑏₁𝑑^𝑇1+𝑏²2!𝑑^𝑇2+⋯+𝑏^𝑠−1(𝑠−1)!𝑑^𝑇𝑠−1+𝐶^𝑇𝑃^𝑠𝑏𝜙(𝑥)=𝑄^𝑇0𝜙(𝑥).(6.7)

Replacing (6.6) and (6.7) into (1.1), we obtain^𝑠_𝑗=0𝑃^𝑇𝑗𝜙(𝑥)𝜙^𝑇(𝑥)𝑄_𝑗=𝐺^𝑇𝜙(𝑥).(6.8)

Using (5.5), we have^𝑠_𝑗=0𝜙^𝑇(𝑥)𝑃_𝑗𝑄_𝑗=𝜙^𝑇(𝑥)𝐺.(6.9)

Therefore, we get^𝑠_𝑗=0𝑃_𝑗𝑄_𝑗=𝐺.(6.10)

The unknown vector 𝐶 can be obtained by solving (6.10). Once 𝐶 is known, 𝑦(𝑥) can be calculated from (6.7).

7. Illustrative Examples

Example 7.1. Consider the eighth-order linear differential equation given in [14] by 𝑦^{(𝑣𝑖𝑖𝑖)}(𝑥)−𝑦(𝑥)=−8𝑒^𝑥,0≤𝑥≤1,(7.1)with the initial conditions 𝑦𝑦(0)=1,𝑦^(0)=0,𝑦^(0)=−1,_(0)=𝑦−2,𝑦^(𝑖𝑣)(0)=−3,𝑦^(𝑣)(0)=−4,_(𝑣𝑖)(0)=−5,𝑦^{(𝑣𝑖𝑖)}(0)=−6.(7.2)The exact solution for this example is 𝑦(𝑥)=(1−𝑥)𝑒^𝑥. Using the method described in Section 6, we assume that 𝑦^{(𝑣𝑖𝑖𝑖)}(𝑥) is approximated by 𝑦^{(𝑣𝑖𝑖𝑖)}(𝑥)=𝐶^𝑇𝜙(𝑥).(7.3) By using (5.1) and the initial conditions (7.2), we have 𝑦(𝑥)=𝑄₀𝜙(𝑥),(7.4) where 𝑄₀=𝐸^𝑇−12!𝑑^𝑇2−23!𝑑^𝑇3−34!𝑑^𝑇4−45!𝑑^𝑇5−56!𝑑^𝑇6−67!𝑑^𝑇7+𝐶^𝑇𝑃^8𝑏=𝐴^𝑇+𝐶^𝑇𝑃^8𝑏.(7.5) We can express function −8𝑒^𝑥 as −8𝑒^𝑥=𝐺^𝑇𝜙(𝑥).(7.6) Substituting (7.3)–(7.6) into (7.1), we obtain 𝐶^𝑇𝜙(𝑥)−𝐴^𝑇+𝐷^𝑇𝑃^8𝑏𝜙(𝑥)=𝐺^𝑇𝜙(𝑥).(7.7) Therefore, we get 𝐶=𝐼−𝑃^8𝑏^𝑇⁻¹(𝐴+𝐺),(7.8) where 𝐼 is the (𝑚+1)×(𝑚+1) identity matrix and Equation (7.8) is a set of algebraic equations which can be solved for 𝐶. Now, we apply the method presented in this paper with 𝑚=8 to solve (7.1) with the initial conditions (7.2). In Table 1, the numerical results obtained by the present method are compared with the results of the HPM [12] and MDM [13] and method in [14]. As we see from this table, it is clear that the result obtained by the present method is very superior to that by HPM, MDM methods, and method in [14]. The absolute difference between exact and approximate solutions is plotted in Figure 1. It is observed in this figure that the accuracy is of the order of 10⁻¹⁰.

Table 1 

Numerical results for Example 7.1.

Figure 1 

Absolute difference between exact and approximate solutions of Example 7.1.

Example 7.2. Consider the Lane-Emden equation given in [10] by 𝑦^(𝑥)+2𝑥𝑦^(𝑥)=22𝑥²+3𝑦(𝑥),0≤𝑥≤1,(7.9) with the initial conditions 𝑦(0)=1,𝑦′(0)=0.(7.10) The exact solution of this example is 𝑒^𝑥2. We solve (7.9) with the initial conditions (7.10) by using the method in Section 6 with 𝑚=11. The comparison among the present method, Legendre wavelets solution [10], and analytic solution for 𝑚=11 is shown in Table 2. As we see from this Table, it is clear that the result obtained by the present method is very superior to that by Legendre wavelets method. It is noted that the mean square error for this example, obtained in Legendre wavelets method, is 2.07×10⁻⁵; but in the present method, the mean square error is 2.3174×10⁻¹⁸. We display a plot of the approximate and exact solution of this example for 𝑚=11 in Figure 2.

Table 2 

Numerical results for Example 7.2.

Figure 2 

Approximate and exact solution of Example 7.2 for 𝑚=11.

Example 7.3. Consider the Bessel differential equation of order zero given in [15] by 𝑥𝑦^+𝑦′+𝑥𝑦=0,(7.11) with the initial conditions 𝑦(0)=1,𝑦′(0)=0.(7.12) The exact solution of this example is 𝒥₀(𝑥)=^∞_𝑞=0(−1)^𝑞(𝑞!)²𝑥2^2𝑞.(7.13) Now, we solve (7.11) with the initial conditions (7.12) by using the method in Section 6 with 𝑚=5. Table 3 shows the absolute difference between exact and approximate solutions of the present method and the methods in [11, 15] for equality basis functions. The results of [11] have been given in [15]. As we see from this table, the maximum error for this example, for the methods in [11, 15], is 10⁻⁴; but in the present method, the maximum error is 10⁻⁷. We display a plot of the approximate and exact solution of this example for 𝑚=5 in Figure 3.

Table 3 

Numerical results for Example 7.3.

Figure 3 

Approximate and exact solution of Example 7.3 for 𝑚=5.

8. Conclusion

In this article, at first, we demonstrate the relation between the Bernstein and Legendre polynomials. By using this relation, we derived the operational matrix of integration and product of B-polynomials. They are applied to solve ordinary differential equations. The present method reduces an ordinary differential equations into a set of algebraic equations. We applied the presented method on three test problems and compared the results with their exact solutions and the other methods, revealing that the present method is very effective and convenient.

Acknowledgment

The work was supported by Alzahra university.