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A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at

Numerous problems such as chemical kinetics, orbital dynamics, circuit and control theory, and Newton’s second law applications involve second-order ordinary differential equations (ODEs). Normally, those equations have no analytical solutions. To approximate the solution of such problems several numerical methods were developed on the hands of many scholars such as [

Block methods for solving ODEs were first proposed by Milne ([

To increase the accuracy of the numerical methods further, researchers such as [

This article is organized as follows: in the coming section we demonstrate the derivation of the method, where we consider three off-step points through the approach of interpolation and collocation. The details of the analysis of the method are discussed in Section

An approximate power series basis function taking the form

The hybrid block method formula (

To illustrate that the root of the first characteristic equation satisfies the prior definition we assume that

The linear operator

A block method is said to be consistent if its order

Consistency property is achieved for the hybrid block method from the above analysis since the order

Consistency and zero stability are sufficient conditions for a linear multistep method to be convergent

The hybrid block method equation (

In this section, the efficiency and the performance of the general three-hybrid one-step implicit hybrid block method (

Comparison of the proposed method with [

| Exact solution | Computed solution for | Error in new method | Error for [ |
---|---|---|---|---|

0.100 | 0.904837418035959520 | 0.904837418035948970 | | |

0.200 | 0.818730753077981820 | 0.818730753077964060 | | |

0.300 | 0.740818220681717770 | 0.740818220681694340 | | |

0.400 | 0.670320046035639330 | 0.670320046035611350 | | |

0.500 | 0.606530659712633420 | 0.606530659712602120 | | |

0.600 | 0.548811636094026500 | 0.548811636093992530 | | |

0.700 | 0.496585303791409530 | 0.496585303791373890 | | |

0.800 | 0.449328964117221620 | 0.449328964117184870 | | |

0.900 | 0.406569659740599170 | 0.406569659740561860 | | |

1.000 | 0.367879441171442330 | 0.367879441171404920 | | |

Comparison of the proposed method with [

| Exact solution | Computed solution for | Error in new method | Error for [ |
---|---|---|---|---|

0.100 | 1.050041729278491400 | 1.050041729278491200 | | |

0.200 | 1.100335347731075600 | 1.100335347731075300 | | |

0.300 | 1.151140435936466800 | 1.151140435936466100 | | |

0.400 | 1.202732554054082100 | 1.202732554054081000 | | |

0.500 | 1.255412811882995200 | 1.255412811882994800 | | |

0.600 | 1.309519604203111900 | 1.309519604203112800 | | |

0.700 | 1.365443754271396400 | 1.365443754271398000 | | |

0.800 | 1.423648930193601900 | 1.423648930193606400 | | |

0.900 | 1.484700278594052000 | 1.484700278594060600 | | |

1.000 | 1.549306144334055000 | 1.549306144334067700 | | |

Comparison of the proposed method with [

| Exact solution | Computed solution for | Error in new method | Error for [ |
---|---|---|---|---|

0.100 | −0.105170918075647710 | −0.105170918075647660 | | |

0.200 | −0.221402758160169850 | −0.221402758160169990 | | |

0.300 | −0.349858807576003180 | −0.349858807576003520 | | |

0.400 | −0.491824697641270570 | −0.491824697641271070 | | |

0.500 | −0.648721270700128640 | −0.648721270700129420 | | |

0.600 | −0.822118800390509550 | −0.822118800390510880 | | |

0.700 | −1.013752707470477500 | −1.013752707470479300 | | |

0.800 | −1.225540928492468800 | −1.225540928492471600 | | |

0.900 | −1.459603111156951200 | −1.459603111156955000 | | |

1.000 | −1.718281828459047300 | −1.718281828459052400 | | |

Exact and approximate solutions for solving

| Exact solution | Approximate solution | Error in developed method | Error in [ |
---|---|---|---|---|

0.200 | 0.980066577841241630 | 0.980066580773730210 | | |

0.400 | 0.921060994002885100 | 0.921060786088783830 | | |

0.600 | 0.825335614909678330 | 0.825334546366523920 | | |

0.800 | 0.696706709347165500 | 0.696703745729634890 | | |

1.000 | 0.540302305868139770 | 0.540296131232123940 | | |

Exact and approximate solutions for solving

| Exact solution | Approximate solution | Error in developed method | Error in [ |
---|---|---|---|---|

0.200 | 0.408407044966673130 | 0.628318530716756810 | | |

0.400 | 1.036725575684631900 | 1.256637061492416500 | | |

0.600 | 1.665044106402590500 | 1.884955593121642800 | | |

0.800 | 2.293362637120548900 | 2.513274128242366100 | | |

1.000 | 2.921681167838507500 | 3.141592671767204200 | | |

Comparison of the proposed method with [

| Exact solution | Approximate solution | Error in developed method | Error in [ |
---|---|---|---|---|

| −0.200326851873144250 | −0.200326851873131260 | | |

| 0.200027330586423440 | 0.200027330586373150 | | |

| 0.198830853474466220 | 0.198830853474288970 | | |

| 0.196842430954904000 | 0.196842430954587670 | | |

| 0.194070581011836470 | 0.194070581011452750 | | |

| 0.190527147620306290 | 0.190527147620002170 | | |

A general three-hybrid one-step block method of order

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank Universiti Utara Malaysia for the financial support of the publication of this article.

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