^{1,2}

^{1}

^{1,2}

^{1}

^{2}

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in the

In this paper, we are considering solving directly the general second-order initial value problems (IVPs) for systems of ODEs in the form

One-step block method such as the implicit Runge-Kutta method is also being referred to as one previous point to obtain the solution. The multistep block method in the form of Adams type formula is presented in [

The block method of Runge-Kutta type has been explored in [

In [

In order to compute the two approximation values of

Block one-step method.

In Figure

We obtained the approximation values of

By applying the formulae for the constants

For

For

For

For

For

The initial starting point at each block are obtained by using Euler method as predictor and the initial

Computing approximating

The convergence test:

In order to make the selection of the next step size, we follow again the techniques used in [

In this section, we will discuss the stability of the proposed method derived in the previous section on a linear general second-order problem:

Firstly, the test equation in (

Stability region for direct block one-step method when

In case

Stability region for direct block one-step method when

The stability region is plotted using MATHEMATICA, and the shaded region inside the boundary in Figures

We have tested the performance of the proposed method on four problems. For the first three problems, a comparison is made between the solutions obtained in [

We have

First-order systems:

We have

First-order systems:

We have

First-order systems:

We have

The codes are written in C language and executed on DYNIX/ptx operating system. The numerical results for Problems

Results of total steps and time for Problem

Results of total steps and time for Problem

Results of total steps and time for Problem

Solution of Problem ^{−8}.

In Figures

In Table

The ratios execution times and steps for solving Problems

TOL | PROB 1 | PROB 2 | PROB 3 | |||

RSTEP | RTIME | RSTEP | RTIME | RSTEP | RTIME | |

2.53 | 1.90 | 2.83 | 1.55 | 2.56 | 1.20 | |

2.38 | 2.31 | 2.88 | 1.57 | 2.45 | 1.32 | |

2.89 | 3.05 | 3.11 | 2.84 | 3.19 | 2.30 | |

4.74 | 5.26 | 4.34 | 1.49 | 3.47 | 3.17 |

Comparison between 2P(A) and 2P(B) methods for solving Problem

TOL | Method | TS | FS | MAXE | AVERR | FCN | TIME |
---|---|---|---|---|---|---|---|

10^{−2} | 2P(A) | 48 | 2 | 385 | 1002 | ||

2P(B) | 19 | 0 | 121 | 527 | |||

10^{−4} | 2P(A) | 143 | 2 | 1145 | 2423 | ||

2P(B) | 60 | 0 | 367 | 1047 | |||

10^{−6} | 2P(A) | 546 | 3 | 4369 | 9287 | ||

2P(B) | 189 | 0 | 1133 | 3045 | |||

10^{−8} | 2P(A) | 2819 | 3 | 22553 | 47821 | ||

2P(B) | 595 | 0 | 3571 | 9093 |

Comparison between 2P(A) and 2P(B) methods for solving Problem

TOL | Method | TS | FS | MAXE | AVERR | FCN | TIME |
---|---|---|---|---|---|---|---|

10^{−2} | 2P(A) | 17 | 4 | 137 | 408 | ||

2P(B) | 6 | 0 | 47 | 264 | |||

10^{−4} | 2P(A) | 46 | 3 | 369 | 708 | ||

2P(B) | 16 | 1 | 131 | 452 | |||

10^{−6} | 2P(A) | 171 | 2 | 1369 | 2694 | ||

2P(B) | 55 | 1 | 429 | 950 | |||

10^{−8} | 2P(A) | 750 | 2 | 1741 | 3844 | ||

2P(B) | 173 | 1 | 1343 | 2588 |

Comparison between 2P(A) and 2P(B) methods for solving Problem

TOL | Method | TS | FS | MAXE | AVERR | FCN | TIME |
---|---|---|---|---|---|---|---|

10^{−2} | 2P(A) | 23 | 7 | 185 | 296 | ||

2P(B) | 9 | 0 | 69 | 247 | |||

10^{−4} | 2P(A) | 54 | 2 | 433 | 597 | ||

2P(B) | 22 | 0 | 177 | 451 | |||

10^{−6} | 2P(A) | 201 | 1 | 1609 | 2252 | ||

2P(B) | 63 | 0 | 503 | 980 | |||

10^{−8} | 2P(A) | 666 | 1 | 5329 | 7700 | ||

2P(B) | 192 | 0 | 1537 | 2426 |

In Table

In Tables

In this paper, we have constructed the direct two-point block one-step method which is efficient and suitable for solving general second-order ODEs directly. The block method has shown acceptable solutions and managed to solve the second-order ODE faster compared to the existing method.

Tolerance

Method employed

Total steps taken

Total failure steps

Magnitude of the maximum error of the computed solution

The average error

Total function calls

The execution time taken in microseconds

Implementation of the direct block method in [

Implementation of the direct two-point block one-step method by solving the problem directly

The ratio steps,

The ratio execution times,

The authors would like to thank the Universiti Putra Malaysia for providing financial support through Graduate Research Fellowship (GRF) during the study period.