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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.

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 [

with the initial conditions

where

The B-polynomials of

where

There are

See [

B-polynomials defined above form a complete basis [

Equation (

where

The use of an orthogonal basis on

To use the Legendre polynomials for our purposes, it is preferable to map this to

with

The orthogonality of these polynomials is expressed by the relation

When the approximant (

by using (

The Legendre polynomial

Now consider a polynomial

We write the transformation of the Legendre polynomials on

The elements

We now replace (

The integrals of the products of Bernstein basis functions can be found using

as follows:

Therefore, we have the elements of

Now, we write the transformation of the B-polynomials on

The elements

Since we can express each

replacing (

of the matrix

using (

Let

By using (

where the

and, therefore, by using (

In this subsection, we present a general formula for finding the operational matrix of product of

Using (

Now, we approximate all functions

by (

Using (

where

If we define a

and, therefore, we have the operational matrix of product as

Consider the linear differential equation (

where

Let

where

Substituting (

Replacing (

Using (

Therefore, we get

The unknown vector

Consider the eighth-order linear differential equation given in [

Numerical results for Example

Exact solution | Method of [ | Method of [ | Method of [ | Presented method | |

for | for | for | for | ||

0 | 1 | 1 | 1 | 1 | 0.9999999992 |

0.1 | 0.9946538263 | 0.9946538263 | 0.9946538262 | 0.9946538266 | 0.9946538261 |

0.2 | 0.9771222065 | 0.9771222065 | 0.9771222014 | 0.9771222093 | 0.9771222065 |

0.3 | 0.9449011653 | 0.9449011653 | 0.9449010769 | 0.9449011752 | 0.9449011655 |

0.4 | 0.8950948186 | 0.8950948186 | 0.8950941522 | 0.8950948487 | 0.8950948184 |

0.5 | 0.8243606354 | 0.8243606356 | 0.8243574386 | 0.8243607328 | 0.8243606353 |

0.6 | 0.7288475202 | 0.728847522 | 0.7288359969 | 0.7288478604 | 0.7288475204 |

0.7 | 0.6041258122 | 0.6041258211 | 0.6040917111 | 0.6041269662 | 0.6041258121 |

0.8 | 0.4451081857 | 0.4451082201 | 0.4450208387 | 0.4451117669 | 0.4451081857 |

0.9 | 0.2459603111 | 0.2459604249 | 0.2457599482 | 0.2459703618 | 0.2459603113 |

1 | 0 |

Absolute difference between exact and approximate solutions of Example

Consider the Lane-Emden equation given in [

Numerical results for Example

Exact solution | Method of [ | Presented method | |

for | for | ||

0.01 | 1.00010001 | 0.9958 | 1.00010001 |

0.3 | 1.09417428 | 1.0942 | 1.09417428 |

0.5 | 1.28402542 | 1.2843 | 1.28402542 |

0.75 | 1.75505466 | 1.7551 | 1.75505466 |

0.9 | 2.24790799 | 2.2480 | 2.24790799 |

0.95 | 2.46575981 | 2.4658 | 2.46575981 |

1 | 2.71828183 | 2.7184 | 2.71828183 |

Approximate and exact solution of Example

Consider the Bessel differential equation of order zero given in [

Numerical results for Example

Method of [ | Method of [ | Presented method | |

for | for | for | |

0.0 | |||

0.1 | |||

0.2 | |||

0.3 | |||

0.4 | |||

0.5 | |||

0.6 | |||

0.7 | |||

0.8 | |||

0.9 | |||

1.0 |

Approximate and exact solution of Example

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.

The work was supported by Alzahra university.