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Fourth order differential equations are considered to develop the class of methods for the numerical solution of boundary value problems. In this paper, we have discussed the regions of absolute stability of fourth order boundary value problems. Methods proposed and derived in this paper are applied to solve a fourth-order boundary value problem. Numerical results are given to illustrate the efficiency of our methods and compared with exact solution.

The determination process for the numerical solution of initial value problems in ordinary differential equations can be classified into two categories-single step methods and multistep methods. Single step methods are those in which the approximation for the point _{n} where y_{n} has already been computed. The methods discussed in this paper are essentially based on the idea that the solution is best approximated by polynomials. The motivation for the work carried out in this paper arises from the methods based on numerical differentiation for the first-order differential equations, special multistep methods based on numerical integration for the solution of the special second-order differential equations by Henrici [

The special fourth order differential equation

occurs frequently in many number of problems of science and engineering.

A general linear multistep method of step number k for the numerical solution of equation (1) is given by

where a_{j}, b_{j} are constants and “h” is the step length.

Introducing the polynomials

Equation (2) can be written as

In Equation (4), “E” is the shift operator defined by

Applying (4) to

The roots

Let _{.} Then

Differentiating (7) four times with respect to x, we get

Replacing

where

Taking r = 0 in (8), a class of methods can be attained which are given by

The coefficients

Differences in (10) are expressed in terms of function values.

After simplification, the Equation (10) will turn out into the form

The coefficients

The local truncation error of the formula (11) is given by

Table 1. Coefficients of;.

M | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 1 | 2 |

Table 2. Coefficients of;,.

K | J | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

4 | 1 | −4 | 6 | −4 | 1 | |||||

5 | 3 | −14 | 26 | −24 | 11 | −2 | ||||

6 | ||||||||||

7 | ||||||||||

8 | ||||||||||

9 |

It follows that the k-step method (14) has the order

For the method (13), we have

The regions of absolute stability of the method for k = 4, 5, 6, 7, 8 and 9 are shown in

The region of absolute stability of the method (13) for k = 4, 5 and 6

The region of absolute stability of the method (13) for k = 7, 8 and 9

In this section, we have applied ND methods to solve the differential equation

in the interval

The fourth order numerical differentiation method derived in this paper for k = 6 is

The methods based on numerical integration are found to be closed regions of absolute stability; the methods

. Solution by fifth order ND with h = 0.01

X | Exact Solution | Numerical Solution by fifth order ND | Absolute Error |
---|---|---|---|

0.10 | 9.500016525794E−02 | 9.500016525794E−02 | 7.813194535800E−15 |

0.20 | 1.800052419090E−01 | 1.800052419090E−01 | 5.329070518201E−15 |

0.30 | 2.550394442682E−01 | 2.550394442681E−01 | 5.662137425588E−15 |

0.40 | 3.201646548346E−01 | 3.201646548346E−01 | 3.719247132494E−15 |

0.50 | 3.754975976986E−01 | 3.754975976986E−01 | 3.774758283726E−15 |

0.60 | 4.212257271813E−01 | 4.212257271813E−01 | 2.775557561563E−15 |

0.70 | 4.576217499812E−01 | 4.576217499812E−01 | 2.720046410332E−15 |

0.80 | 4.850567100313E−01 | 4.850567100313E−01 | 8.881784197001E−16 |

0.90 | 5.040115785530E−01 | 5.040115785530E−01 | 1.221245327088E−15 |

1.00 | 5.150873072263E−01 | 5.150873072263E−01 | 2.997602166488E−15 |

1.10 | 5.190133197868E−01 | 5.190133197868E−01 | 1.887379141863E−15 |

1.20 | 5.166544364783E−01 | 5.166544364783E−01 | 4.662936703426E−15 |

1.30 | 5.090162464214E−01 | 5.090162464214E−01 | 5.440092820663E−15 |

1.40 | 4.972489648573E−01 | 4.972489648573E−01 | 5.440092820663E−15 |

1.50 | 4.826498351587E−01 | 4.826498351587E−01 | 8.826273045770E−15 |

1.60 | 4.666641592316E−01 | 4.666641592316E−01 | 9.992007221626E−15 |

1.70 | 4.508850642096E−01 | 4.508850642096E−01 | 1.304512053935E−14 |

1.80 | 4.370521379547E−01 | 4.370521379547E−01 | 1.637578961322E−14 |

1.90 | 4.270490905788E−01 | 4.270490905789E−01 | 1.471045507628E−14 |

2.00 | 4.229006237963E−01 | 4.229006237963E−01 | 1.737499033538E−14 |

. Solution by fifth order ND with h = 0.02

X | Exact Solution | Numerical Solution by fifth order ND | Absolute Error |
---|---|---|---|

0.10 | 9.500016525794E−02 | 9.500016525718E−02 | 7.650130529058E−13 |

0.20 | 1.800052419090E−01 | 1.800052419082E−01 | 7.430722703816E−13 |

0.30 | 2.550394442682E−01 | 2.550394442675E−01 | 6.964984144986E−13 |

0.40 | 3.201646548346E−01 | 3.201646548340E−01 | 6.305511668359E−13 |

0.50 | 3.754975976986E−01 | 3.754975976981E−01 | 5.463962615693E−13 |

0.60 | 4.212257271813E−01 | 4.212257271809E−01 | 4.425348976156E−13 |

0.70 | 4.576217499812E−01 | 4.576217499809E−01 | 3.217981436876E−13 |

0.80 | 4.850567100313E−01 | 4.850567100311E−01 | 1.886824030350E−13 |

0.90 | 5.040115785530E−01 | 5.040115785529E−01 | 4.274358644807E−14 |

1.00 | 5.150873072263E−01 | 5.150873072265E−01 | 1.179056852152E−13 |

1.10 | 5.190133197868E−01 | 5.190133197871E−01 | 2.902122986370E−13 |

1.20 | 5.166544364783E−01 | 5.166544364787E−01 | 4.607425552194E−13 |

1.30 | 5.090162464214E−01 | 5.090162464221E−01 | 6.451505996097E−13 |

1.40 | 4.972489648573E−01 | 4.972489648581E−01 | 8.351652702743E−13 |

1.50 | 4.826498351587E−01 | 4.826498351598E−01 | 1.023958695612E−12 |

1.60 | 4.666641592316E−01 | 4.666641592328E−01 | 1.214361944335E−12 |

1.70 | 4.508850642096E−01 | 4.508850642110E−01 | 1.404265592697E−12 |

1.80 | 4.370521379547E−01 | 4.370521379563E−01 | 1.590561016229E−12 |

1.90 | 4.270490905788E−01 | 4.270490905806E−01 | 1.771249813487E−12 |

2.00 | 4.229006237963E−01 | 4.229006237983E−01 | 1.950384298510E−12 |

based on numerical differentiation are found to be absolutely stable outside some closed boundaries. We have obtained the solution by numerical differentiation methods which are derived in this paper and are more accurate. The absolute errors are very small.