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A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form

In recent times, the integration of Ordinary Differential Equations (ODEs) are carried out using some kind of block methods. In particular, this paper discusses the general second-order ODEs which arise frequently in the area of science, engineering and mechanical systems and are generally written in the form,

Development of LMM for solving ODEs can be generated using methods such as Taylor's Series, numerical integration, and collocation method, which are restricted by an assumed order of convergence. In this paper, we will follow suite from the previous papers of Okunuga and Ehigie [

Block methods for solving Ordinary Differential Equations have initially been proposed by Milne [

Hybrid methods, using collocation technique, were discussed by Ehigie et al. [

The procedure for the derivation of our methods in a multistep collocation technique is discussed by the methods in previous papers by Okunuga and Ehigie [

Consider the second-order equation

The numerical solution to (

Most of the problems encountered in solving the general second-order equation (

To derive an

The coefficients

This is evaluated for at

The

To derive a 1 block 3-point Explicit Block Method (EBM) that is,

Substituting the values

On evaluating (

Differentiating (

Expressing the schemes (

Equation (

To derive a 1 block 3-point Implicit Block Method (IBM), we also define the following terms:

The 3- point IBM will be generally represented as

Substituting the

On evaluating (

On differentiating (

The derivative formulae will be used to obtain the first derivative term in (

This scheme is also of the form (

The methods (

The method (

Applying this definition to the individual methods (

A Linear Multistep Method of the form (

Since the methods derived in (

From literature, it is known that stability of a linear multistep method determines the manner in which the error is propagated as the numerical computation proceeds. Hence, it would be necessary to investigate the stability properties of the newly developed methods. In this paper, the 0-stability and the Region of Absolute Stability (RAS) of the methods are discussed.

The first characteristic polynomial,

The block method of form (

The Region of Absolute Stability (RAS) of methods of (

The stability property of the 3-point EBM is determined by applying the scheme (

Similarly, this is extended to the 3-point implicit block method (IBM) given in (

The LMM (

The proof is given in Fatunla [

Since the consistency and 0-stability of the methods have been established, then the explicit block method (

The Region of Absolute Stability (RAS) of the block methods in this paper are drawn based on the third scheme of the block. The RAS of the linear multistep methods in the EBM (

RAS for the explicit scheme.

RAS for implicit scheme.

It is observed that the RAS of the IBM is wider in range than the RAS of the EBM. This means that the implicit schemes will cope with Initial Value Problems better than the EBM in implementation with a higher step length.

A Matlab code was developed for the implementation of the schemes in Sections

Thereafter it generated the values for

In this paper three standard problems are considered and our newly developed methods are used to solve these problems. The problems are presented below.

Consider the test problem for second-order ODE given by

This problem is known to have an analytical solution of

Result of the test problem (

Explicit max. error | Implicit max. error | |
---|---|---|

0.01 | 5.12 | 6.55 |

0.005 | 6.30 | 2.12 |

0.0025 | 7.81 | 1.78 |

0.001 | 4.95 | 9.43 |

0.0005 | 5.52 | 8.00 |

0.00025 | 1.15 | 1.77 |

The Van der Pol equation which describes the Van der Pol oscillator is the second-order ODE

The results are presented using Maximum Error which is given in Table

Result of the Van der Pol problem (

Explicit | Implicit | Explicit | Implicit | Explicit | Implicit | |

max. error | max. error | max. error | max. error | max. error | max. error | |

0.01 | 1.15 | 1.10 | 6.22 | 1.10 | 5.13 | 1.10 |

0.005 | 1.10 | 1.10 | 1.72 | 1.10 | 6.48 | 1.10 |

0.0025 | 1.10 | 1.10 | 1.17 | 1.10 | 8.90 | 1.10 |

0.001 | 1.10 | 1.10 | 1.10 | 1.10 | 1.59 | 1.10 |

0.0005 | 1.10 | 1.10 | 1.10 | 1.10 | 1.16 | 1.10 |

0.00025 | 1.10 | 1.10 | 1.10 | 1.10 | 1.07 | 1.10 |

The third problem is the second order ODE

The results obtained by using the 3-point EBM and IBM are presented in Table

Result of problem (

Explicit max. error | Implicit max. error | |
---|---|---|

0.01 | Failed | 6.90 |

0.005 | Failed | 7.05 |

0.0025 | Failed | 4.34 |

0.001 | 3.40 | 2.79 |

0.0005 | 6.17 | 1.10 |

0.00025 | 9.17 | 5.07 |

The Numerical results for the solution of the problems illustrated in the previous subsection will be presented in form of the Maximum Error.

It would be observed that in Problems

We have been able to derive some 3-point Implicit and Explicit Block Methods via collocation multistep technique. This block schemes derived in this paper have been represented in form of (