_{1}

^{*}

In this article, we derive a block procedure for some K-step linear multi-step methods (for
*K *= 1, 2 and 3), using Legendre polynomials as the basis functions. We give discrete methods used in block and implement it for solving the non-stiff initial value problems, being the continuous interpolant derived and collocated at grid and off-grid points. Numerical examples of ordinary differential equations (ODEs) are solved using the proposed methods to show the validity and the accuracy of the introduced algorithms. A comparison with fourth-order Runge-Kutta method is given. The ob-tained numerical results reveal that the proposed method is efficient.

Numerous problems in Physics, Chemistry, Biology and Engineering science are modeled mathematically by ordinary differential equations (ODEs), e.g., series circuits, mechanical systems with several springs attached in series lead to a system of differential equations [

In this paper, we present an efficient numerical method to solve numerically the ODEs. The proposed method is a block procedure for some K-step linear multi-step methods (for K = 1, 2 and 3) [

The plan of the paper is as follows: In Section 2, the derivation of the proposed methods is presented. In Section 3, stability and convergence analysis of the block schemes is given. In Section 4, two numerical examples are considered. The paper ends with summary and conclusions in Section 5.

In this section, we derive discrete methods to solve (1) at a sequence of nodal points

In the first we consider the approximate solution of the perturbed form of (1) in the following power series [

where

Substituting from Equation (2) in Equation (1) we have

where

The well-known Legendre polynomials are defined on the interval

where the first four polynomials are

In order to use these polynomials on the interval

Case 1: If K = 1

In this case, we take the polynomial

In addition, from Equation (3) we can deduce that

We now collocate Equation (7) at

Using a suitable method to solve the above system to obtain

From (2), we have

Now, the required numerical scheme of the proposed method will be obtained if we collocate the above Equation (9) at

Which is the well-known trapezoidal rule.

Case 2: If K = 2

In this case, we take the polynomial

In addition, from Equation (3) we can deduce that

We now collocate Equation (11) at

Using a suitable method to solve the above system to obtain

From (2), we have

Now, the required numerical scheme of the proposed method will be obtained if we collocate the above Equation (13) at

Case 3: If K = 3

For case K = 3, we collocate the continuous scheme

at grid and off-grid points

In this section, we present a summary on the order, the error constant and the convergence of the proposed block schemes. This summary in given in the following table.

In this section, we implement the proposed method with K = 2 and K = 3 to solve two first order initial value problems, and compare the obtained numerical results with those obtained from using the fourth-order Runge- Kutta method (RK4).

Example 1.

Consider the following IVP

With the exact solution

The numerical results of this example are presented in

Example 2.

Consider the following IVP

With the exact solution

The numerical results of this example are presented in

x | Block Scheme K = 2 | Exact Solution | RK4 |
---|---|---|---|

0.0 | 1.000000 | 1.000000 | 1.000000 |

0.2 | 0.818712 | 0.818730 | 0.818781 |

0.4 | 0.670345 | 0.670320 | 0.670541 |

0.6 | 0.548848 | 0.548811 | 0.548848 |

0.8 | 0.449215 | 0.449328 | 0.449345 |

1.0 | 0.367822 | 0.367879 | 0.367836 |

x | Block Scheme K = 3 | Exact Solution | RK4 |
---|---|---|---|

0.0 | 1.000000 | 1.000000 | 1.000000 |

0.2 | 0.818735 | 0.818730 | 0.818738 |

0.4 | 0.670328 | 0.670320 | 0.670321 |

0.6 | 0.548817 | 0.548811 | 0.548813 |

0.8 | 0.449322 | 0.449328 | 0.449325 |

1.0 | 0.367871 | 0.367879 | 0.367873 |

In this paper, we presented three new block-schemes (K = 1, K = 2 and K = 3) that are convergent and absolutely stable. We used the proposed method to solve numerically a wide-range of linear initial value problems. The results of the presented examples show that our method was capable for solving such problems of IVPs and generates the convergence analysis, and closed to their exact solutions. This method is very simple and effective for a wide-range of ODEs. All computations are made by Matlab.

We thank the Editor and the referee for their comments.