^{1}

^{1}

^{1}

^{1}

The current numerical technique for solving a system of higher-order ordinary differential equations (ODEs) is to reduce it to a system of first-order equations then solving it using first-order ODE methods. Here, we propose a method to solve higher-order ODEs directly. The formulae will be derived in terms of backward difference in a constant stepsize formulation. The method developed will be validated by solving some higher-order ODEs directly with constant stepsize. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The result presented confirmed our hypothesis.

Differential equations constantly arise in various branches of science and engineering. Many of these problems are in the form of higher-order ordinary differential equations (ODEs). A few examples where these problems can be found are, in the motion of projectiles, the bending of a thin clamped beam and population growth.

The popular practice for solving a system of higher-order ODEs is by reducing it to a system of first-order equations and then solving with first-order methods. These methods worked, so that methods for solving higher-order ODEs have been disregarded as robust codes. However, the work by Krogh [

Related works for solving higher-order ODEs can be found in Collatz [

The advantage of such a code is that the integration or differentiation constants are calculated only once at the start of the first step of integration, whereas other formulations calculate the constants at every step.

In this paper, we will focus only on nonstiff ODEs of the form

Without loss of generality, we will be considering the scalar equation in (

This paper will be organized as follows. First, the integration coefficients of the explicit constant stepsize backward difference formulation of the DI method will be derived. Then, the coefficients of the implicit method are formulated and their relationship with the explicit coefficients is shown. We start the derivation with the coefficients of the first-order system, which is given in Henrici [

The code developed will be using the PECE mode. The predictor and corrector will have the following form:

predictor:

Integrating (

Integrate (

Next, the case of the third-order ODE where

or

Integrating (

or

The generating function

Substituting (

Calculating the integration coefficients directly is time consuming when large numbers of integration are involved. A more efficient way of computing the coefficients is by obtaining a recursive relationship between the coefficients. With this recursive relationship, we are able to obtain the implicit integration coefficient with minimal time consumption. The relationship between the explicit and implicit coefficients is expressed below.

For first-order coefficients,

It can be written as

or

which leads to a recursive relationship

Tables

The explicit integration coefficients for

0 | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

1 | 1/2 | 5/12 | 3/8 | 251/720 | 95/288 | 19087/60480 | |

1/2 | 1/6 | 1/8 | 19/180 | 3/32 | 863/10080 | 275/3456 | |

1/6 | 1/24 | 7/240 | 17/720 | 41/2016 | 731/40320 | 8563/518400 |

The implicit integration coefficients for

0 | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

1 | −1/2 | −1/12 | −1/24 | −19/720 | −3/160 | −813/60480 | |

1/2 | −1/3 | −1/24 | −7/360 | −17/1440 | −41/5040 | −731/120960 | |

1/6 | −1/8 | −1/80 | −1/180 | −11/3360 | −89/40320 | −5849/3628800 |

For error calculations, we will be using the three error tests, namely, absolute error, relative error, and mixed error tests. The error formula is given by,

In (

When

The following notations hold MAX ABS: maximum error when using absolute error test, MAX MIX: maximum error when using mixed error test, MAX REL: maximum error when using relative error test,

For the choice of problems to be tested, we choose four linear problems consisting of a second- and a third-order problem. The third problem is a mix system of second- and first-order equations and the fourth problem is a system of three second-order equations. Our reason for choosing the linear problems is that if the formulae are correct, then they should solve linear problems. The choice of system of equations is to raise the degree of difficulty of solving the problems. The rest of the problems are nonlinear, which occur in physical situations. The choices of the physical problems are those with exact solutions known. We give our comments on the numerical results right after the numerical Tables

MAX ABS | MAX MIX | MAX REL | |
---|---|---|---|

4.73734 + 026 | 1.18857 − 003 | 1.18857 – 003 | |

4.70433 + 023 | 1.16697 − 006 | 1.16697 – 006 | |

4.42360 + 020 | 1.18335 − 009 | 1.18335 − 009 | |

4.05706 + 020 | 1.01668 − 009 | 1.01668 − 009 | |

3.69529 + 021 | 9.26023 − 009 | 9.26024 − 009 |

MAX ABS | MAX MIX | MAX REL | |
---|---|---|---|

1.60072 + 023 | 1.40364 − 003 | 1.40364 − 003 | |

1.52595 + 020 | 1.33620 − 006 | 1.33620 − 006 | |

1.55425 + 017 | 1.36098 − 009 | 1.36098 − 009 | |

1.27710 + 016 | 1.11807 − 010 | 1.11807 − 010 | |

2.20767 + 017 | 1.93311 − 009 | 1.93311 − 009 |

MAX ABS | MAX MIX | MAX REL | |
---|---|---|---|

1.74071 − 002 | 1.68285 − 003 | 6.12179 − 003 | |

2.32581 − 005 | 4.04838 − 006 | 5.34483 − 005 | |

2.38274 − 008 | 4.39132 − 009 | 5.74568 − 007 | |

5.33390 − 009 | 5.47762 − 010 | 9.42528 − 009 | |

4.54132 − 008 | 4.54132 − 008 | 5.25218 − 007 |

MAX ABS | MAX MIX | MAX REL | |
---|---|---|---|

1.40347 + 001 | 1.55464 + 000 | 6.27279 + 002 | |

1.78122 − 002 | 1.75004 − 002 | 1.60254 + 001 | |

2.32649 − 005 | 2.32643 − 005 | 9.97054 − 001 | |

2.44607 − 008 | 2.44600 − 008 | 9.82644 − 001 | |

9.90539 − 010 | 9.90531 − 010 | 9.99724 − 001 |

MAX ABS | MAX MIX | MAX REL | |
---|---|---|---|

2.34537 − 001 | 4.96059 − 004 | 1.00000 + 000 | |

2.85286 − 004 | 2.86491 − 007 | 1.00000 + 000 | |

2.91084 − 007 | 6.19617 − 010 | 1.00083 + 000 | |

3.00856 − 008 | 7.74497 − 013 | 1.80152 + 000 | |

2.43305 − 007 | 1.09024 − 011 | 7.08729 + 000 |

MAX ABS | MAX MIX | MAX REL | |
---|---|---|---|

1.01336 − 004 | 8.29238 − 005 | 5.15346 − 002 | |

8.33352 − 008 | 8.33254 − 008 | 9.08294 − 004 | |

9.71863 − 011 | 9.71317 − 011 | 4.58972 − 006 | |

5.45085 − 010 | 5.45075 − 010 | 3.45420 − 004 | |

4.75544 − 009 | 4.75543 − 009 | 1.91598 − 002 |

MAX ABS | MAX MIX | MAX REL | |
---|---|---|---|

6.95439 − 002 | 5.63803 − 002 | 2.97860 − 001 | |

8.31669 − 005 | 7.12977 − 005 | 4.99583 − 004 | |

8.60027 − 008 | 7.37166 − 008 | 5.16016 − 007 | |

8.66052 − 011 | 7.42330 − 011 | 5.19631 − 010 | |

1.86645 − 012 | 1.55538 − 012 | 9.33222 − 012 |

Source: Krogh [

This is a linear equation used by Krogh [

Source: Omar and Suleiman [

This is a third-order problem with an exponential solution. The difference between Problems

Source: Bronson [

For this problem, all error tests worked well.

Source: Bronson [

This problem does not work for relative error test because of the small value of the solution for certain values of

Source: Russel and Shampine [

This problem is the symmetrical bending of a laterally loaded circular plate.

The numerical results of this problem show the failure to control the error using relative error test. This is because the solution is zero when

Source: Shampine and Gordon [

This problem is Newton’s equations of motion for the two-body problem.

Again, relative error test does not work too well for this problem because

Source: Robert Jr. [

For this problem, all error tests worked well.

All the numerical results show that the errors in the mixed error mode give a reliable error estimate for all the problems given. The absolute error mode failed to give meaningful error results for Problems

The research work done shows that the method developed for solving higher-order ODEs directly using the backward difference is successful. We recommend that, for multistep method, the error control procedure should use the mixed error test. This research suggests the potential of this work developing a robust code for solving higher-order ODEs directly.

This paper has been supported by the UPM Graduate Research Fellowship (GRF) and Science Fund.