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In this article, my focus is the derivation, analysis and implementation of a new modified one-step implicit hybrid block method with five off-step points. The derived method is to solve directly initial value problems of fourth order ordinary differential equations. The approach for the derivation of the method is to interpolate the approximate power series solution to the problem and to collocate its fourth derivative at the grid and off-grid points to generate systems of linear equations for the determination of the unknown parameters. The derived method is tested for consistency, zero stability, convergence and absolute stability. Accuracy and usability of the method are determined with some test problems and the results obtained are found to be better in accuracy than some existing methods.

In sciences and engineering, mathematical models are developed to understand as well as to interpret physical phenomena, many of such phenomena, when modeled, often result into higher order ordinary differential equations of the form:

An old conventional way to solve (1) is the method of first reducing (1) to system of first order differential equation of the form:

and to solve the resulting system of equations by any of the existing methods of solving first order ordinary differential equations. Literatures abounded in this old conventional method of solving problems of type (1) numerically are [

We take our basis function to be a power series of the form:

The fourth derivative of (3) gives

By putting (4) into (1) we have the differential system:

where

By putting these system of equations in the matrix form and then solved to obtain values of parameters

The coefficient of

where

We evaluate (9) at

The first derivative of

Similarly, the second derivative of

The third derivative of

It is noted that the general fourth order odes involve the first, second and third derivatives. The derivatives can be obtained by imposing that:

By using (14) and evaluating (11), (12) and (13) at

By combining the schemes (10), the first, second, third derivatives schemes (15) together and write them in block form, using the definition of implicit block method in [

This equation is solved and we obtained values for

In this section, we carry out the analysis of the basic properties of the new method.

The linear operator of the block (17) is defined as:

By expanding

The block (17) and associated linear operator are said to have order p if

The term

Hence the block (17) has order 7 with error constant:

By rewriting the main method (10c) in the form:

Expanding (24) in Taylor series in the form:

Since

The block (17) is said to be Zero stable if the roots

where

Hence our method is Zero stable.

From main method (10c), the first and second characteristics polynomials of the method are given by:

and

the method (10c) is consistent since it satisfies the following conditions:

1. The order of the method is

2. For the method

3.

4. it follows from here that

5. Note that:

For the principal root r = 1: it is observed that the last condition above is satisfied, hence the main method is consistent.

The necessary and sufficient condition for a numerical method to be convergent is for it to be consistent and Zero stable. Thus since it has been successfully shown from the above condition, it could be seen that our method is convergent.

We consider the stability polynomial written in general form:

where

Adopting the boundary locus method whose equation is given by:

By inserting the values of

From here, it could be seen that the region of absolute stability of the method is given by

To test the accuracy, workability and suitability of the method, I adopted our method to solving some initial value problems of fourth order ordinary differential equations.

Test Problem 1

I consider special fourth order problem:

Whose exact solution is:

My method was used to solve the problem and result compared with [

Test Problem 2

I consider a linear fourth order problem

Whose exact solution is given by:

My method was used to solve the problem and result compared with [

I make use of the following Notations in the table of results:

XVAL: Value of the independent variable where numerical value is taken.

ERC: Exact result at XVAL.

NRC: Our Numerical result at XVAL.

ERR: Error of our result at XVAL.

In this paper, I propose an accurate five off-step points modified implicit block algorithm for the numerical solution of initial value problems of fourth order ordinary differential equations. For better performance of the method, step size is chosen within the stability interval.

XVAL | ERC | NRC | ERR P = 7 K = 1 | ERR in [ |
---|---|---|---|---|

0.1 | 0.1000000833333340 | 0.10000008333349980 | 1.658E−13 | 7.000E−10 |

0.2 | 0.20000266666666690 | 0.20000266666998294 | 3.316E−12 | 8.999E−10 |

0.3 | 0.300020250000000004 | 0.30002025000718312 | 7.183E−12 | 2.999E−09 |

0.4 | 0.400008533333333333 | 0.40000853339982528 | 6.649E−11 | 5.100E−09 |

0.5 | 0.500260416666666665 | 0.50026041667657280 | 9.906E−11 | 7.799E−09 |

0.6 | 0.600648000000000007 | 0.60064800003216824 | 3.217E−11 | 1.180E−08 |

0.7 | 0.701400583333333344 | 0.70140058343576487 | 2.432E−10 | 1.240E−08 |

0.8 | 0.802730666666666670 | 0.80273066698686870 | 3.202E−10 | 1.410E−08 |

0.9 | 0.904920750000000005 | 0.90492075025408587 | 2.540E−10 | 1.880E−08 |

1.0 | 1.00833333333333300 | 1.00833333359573400 | 2.024E−10 | 2.600E−08 |

XVAL | ERC | NRC | ERR P = 7 K = 1 | ERR in [ |
---|---|---|---|---|

0.103150 | 0.001300799589367158 | 0.001300799589367196 | 0.38142683E−18 | 0.49873299E−15 |

0.206250 | 0.002531773700195635 | 0.002531773700195672 | 0.37184370E−17 | 0.67654215E 15 |

0.306250 | 0.003652478978884993 | 0.003652478978887675 | 0.26822346E−16 | 0.31350790E−14 |

0.406250 | 0.004695953223180484 | 0.004695953223180513 | 0.29384802E−16 | 0.94360283E−13 |

0.506250 | 0.005657642360803446 | 0.005657642360803864 | 0.41813224E−15 | 0.22116856E−13 |

0.603125 | 0.006507754608034524 | 0.00650775460803811 | 0.38734880E−15 | 0.43379362E 13 |

0.703125 | 0.007298314767638522 | 0.007298314767638809 | 0.28714827E−15 | 0.77870869E−13 |

0.803125 | 0.007998520222728983 | 0.007998520222737657 | 0.86740034E−14 | 0.12863494E 12 |

0.903125 | 0.008607246703302495 | 0.008607246703309575 | 0.70802448E−14 | 0.19927115E−12 |

1.003125 | 0.009124283967030094 | 0.009124283967034006 | 0.35121472E−14 | 0.29323245E 12 |

The order of my method is of order 7 higher than that of [

The results of my new method when also compared with the block method proposed by [

Monday Kolawole Duromola, (2016) An Accurate Five Off-Step Points Implicit Block Method for Direct Solution of Fourth Order Differential Equations. Open Access Library Journal,03,1-14. doi: 10.4236/oalib.1102667