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The mode of definition of the error at polynomial Richardson’s extrapolation is described. Along with the table of extrapolations the new magnitudes reflecting expediency and efficiency of extrapolation are entered. On concrete examples it is shown that application of Richardson’s extrapolation to a solution of integral equations has appeared rather effective and gives a solution with a high exactitude. Application of formulas of interpolation leads to a solution in the analytical aspect.

Numerical methods of the approached solution are attractive by the universality. In numerical methods three problems are put: obtaining enough exact solutions, accuracy control and algorithmic simplicity of procedures. Richardson’s extrapolation solves these problems. The first work of this theme was [

The purpose of this paper is to construct procedure of an estimation of an error at Richardson’s extrapolation and to apply it to a numerical solution of integral equations of the second kind of Volterra and Fredholm.

We shall briefly remind an essence of Richardson’s extrapolation.

There are many problems from different sections of mathematics in which the difference between exact

Magnitude h is usually a grid step.

We shall designate the approached solution as

Extrapolation of the 1-st order is obtained by elimination of the first term of expansion (2.1) by a linear combination of extrapolations of the zero order. Extrapolation of the j-st order is calculated on the recurrence formula

where usually is accepted

In the end a table of extrapolations is obtained

If magnitude h forms a geometrical progression

that (2.3) receives an aspect

In this case from (3.2) and (2.1) we have

where lack of the top sign at the sum means that the remainder term is included in this sum.

We will name as the index of extrapolation

Extrapolation improves an exactitude

From (3.3) follows

where

Because

By means of the Formula (3.5) table

Obviously, at a diminution h there occurs such moment when terms in expansion (3.3) start to decrease monotonically. We name this mode regular. If extrapolation becomes on a regular mode, the first term should bring the basic contribution in expansion (3.5). Then the relation should be observed

And on the contrary: if the relation (3.7) is observed, we have a regular mode. Magnitude

Let’s enter magnitude

and we shall define its meaning.

From (3.8), we have

Substituting (3.9) in (3.2) we receive

From (3.4), (3.6), (3.9) and (3.10) , we have

Therefore,

By Formulas (3.6), (3.7), (3.11) and assumption

Starting from row (3.5) and definition (3.8), it is possible to establish that there is the expansion for

where the first item has a basic meaning in a regular mode.

Then from (3.13) we receive the relation

on all table irrespective of an index j.

Let’s consider integral equation of Volterra of the second kind.

where

This equation has an exact solution

We shall designate the approximate solution of the Equation (4.1) at the calculation of integrals on a trapezoidal rule with extrapolation by

Procedure of a solution of integral equation of Volterra of the second kind with application of the trapezoidal rule is known (for example [

and (3.2) receives the form

It is sometimes more convenient to use “direct” formulas

which come out by sequential application of the Formula (4.4).

Let’s consider (4.1) on a piece x = 0 ÷ 2.5 with step

By means of the Formula (3.5) table

By means of the Formula (3.7) table

By means of the Formula (3.8) table

The analysis of tables leads to the important conclusions. From

i | ||||||
---|---|---|---|---|---|---|

1 | 50 | 59.1571 | 67.002589 | 66.87888352 | 66.879216515 | 66.8792162898715 |

2 | 100 | 65.0412 | 66.886615 | 66.87921131 | 66.879216290 | |

3 | 200 | 66.4252 | 66.879674 | 66.87921621 | ||

4 | 400 | 66.7660 | 66.879244 | |||

5 | 800 | 66.8509 | 66.8792162899017 |

i | ||||
---|---|---|---|---|

1 | −5.8841 | 0.1160 | −0.0003278 | |

2 | −1.3840 | 0.0069411 | ||

3 | −0.34081 | 0.0004292 | ||

4 | −0.08488 |

i | |||
---|---|---|---|

1 | 4.2514 | 16.708 | 66.889 |

2 | 4.0611 | 16.171 | |

3 | 4.0152 |

i | |||
---|---|---|---|

1 | −0.062845 | −0.044273 | −0.045144 |

2 | −0.015275 | −0.010703 | |

3 | −0.0037927 |

that

The true error is

We have received the precision solution in isolated points. It is interesting to find good interpolation function and to receive a solution in the analytical form. Let’s make uniform set from

designate

The Equation (4.1) for area

The Equation (4.7) defines the solution at

The equation is considered

which has the exact solution

Procedure of a solution of integral equation of Fredholm of the second kind with application of the trapezoidal

x = 2.488 | x = 2.45 | x = 2.35 | x = 1.95 | |
---|---|---|---|---|

83.53814 | 117.853257 | 68.052110 | ?28.347394668 | |

117.853249796849 | 68.05210963589 | ?28.3473946668611 | ||

83.53832 | 117.853249796861 | 68.05210963599 | ?28.3473946668607 |

rule is also known (for example [

From (3.12) we have the estimation of the error of

The remark. Trapezoidal rule is the scheme of 2-nd order of exactitude and s = 2 in expansions (2.1) and (3.3). For a calculation of interpolations of zero order it is possible to use Simpson’s formula (the formula of 4-th order of exactitude). For Simpson’s formula expansion on even degrees is kept, but in (2.1) and (3.3) the first term vanishes. Formally it means: in (2.1) and (3.3) we do replacement

and

If to begin calculations from Simpson’s formula then in the formula (3.2) it is necessary to replace

and “direct” formulas receive the kind

The important note: by means of the formula (3.2), it is easy to establish that Simpson’s formula is extrapolation of 1-st order of a trapezoidal rule and there is no real necessity to use Simpson’s formula. Let’s remind that application of any schemes of a high exactitude demands high smoothness of functions.

Trapezoidal rule with Richardson’s extrapolation is the effective method of a solution of integral equations of the second kind, and with application of interpolation we receive a solution in an analytical aspect. In this case the table of extrapolations enables to make a good estimation of an error of solution. The received solution possesses a high exactitude and can be the standard for other methods of a solution of the equation.

i | ||||||
---|---|---|---|---|---|---|

1 | 2 | 3.8472 | 2.612042 | 2.7205405 | 2.7182699 | 2.718281844 |

2 | 4 | 2.9208 | 2.713759 | 2.7183054 | 2.7182818 | |

3 | 8 | 2.7655 | 2.718021 | 2.7182821 | ||

4 | 16 | 2.7299 | 2.718265 | |||

5 | 32 | 2.7211 | 2.718281828 |

i | ||||
---|---|---|---|---|

1 | 0.9263 | −0.10172 | 0.002235 | −0.000012 |

2 | 0.1553 | −0.00426 | 0.000023 | |

3 | 0.0356 | −0.00024 | ||

4 | 0.0087 |

i | |||
---|---|---|---|

1 | 5.965 | 23.87 | 96.14 |

2 | 4.369 | 17.42 | |

3 | 4.084 |

i | |||
---|---|---|---|

1 | −0.4912 | −0.4916 | −0.5022 |

2 | −0.0897 | −0.0891 | |

3 | −0.0211 |

Igor Petrovich Dobrovolsky, (2015) The Estimation of the Error at Richardson’s Extrapolation and the Numerical Solution of Integral Equations of the Second Kind. Open Access Library Journal,02,1-7. doi: 10.4236/oalib.1102051