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This paper presents numerical solution of elliptic partial differential equations (Poisson's equation) using a combination of logarithmic and multiquadric radial basis function networks. This method uses a special combination between logarithmic and multiquadric radial basis functions with a parameter

Many problems in applied sciences and engineering are reduced to a set of partial differential equations (PDEs). Analytical methods are frequently inadequate for obtaining solution, and usually numerical methods must be resorted. Radial basis function network is a well-known method to interpolate unknown functions and approximate numerical solutions. We have some radial basis functions such as “spline functions,” “Gaussian functions,” “multiquadric functions,” and “logarithmic functions.” All of researchers have tried to increase accuracy of approximate solutions while the stability of their suggested system is stable (the condition number is near to unity as possible). In multiquadric radial basis functions (MQ-RBFs), there are some parameters that influence accuracy of the solution, for instance, the width parameter of a basis function, scattered data points, and so on. In recent years, many researchers have worked on these parameters. They have tested many cases and have obtained different relations for such parameters. Kansa [^{1}–10^{9}. Sharan et al. [

The organization of the present paper is as follows. In Section

The form of a Poisson’s equation is as follows:

In this method, we use the following expression:

In expression (

In this section, we present three experiments, wherein their numerical solutions illustrate some advantages of the new method with high accuracy and show that, in this new way, the system is not ill conditioned.

A method is said to be stable when the obtained solution undergoes small variations as there are slight variations in inputs and parameters and when probable perturbations in parameters that are effective in equations and conditions prevailing them do not introduce, in comparison to the physical reality of the problem, any perturbations in what is returned. We propose here to compare the new method with other numerical methods (i.e., DRBFN and IRBFN methods) by offering experiments and examining the stability of the new method (Tables

Comparison between exact solution and approximate solution of the new method of Experiment

Exact solution | Approximate solution of the new method | Error of the new method | ||
---|---|---|---|---|

0.3333 | 0.2 | 0.4070935392947847 | 0.4070935392947850 | |

0.3333 | 0.4 | 0.4972251717238353 | 0.4972251717238351 | |

0.3333 | 0.6 | 0.6073121961701566 | 0.6073121961701570 | |

0.3333 | 0.8 | 0.7417727914665396 | 0.7417727914665393 | |

0.6667 | 0.2 | 0.8143092188653853 | 0.8143092188653852 | |

0.6667 | 0.4 | 0.9945995259174347 | 0.9945995259174348 | |

0.6667 | 0.6 | 1.214806604220352 | 1.214806604220355 | |

0.6667 | 0.8 | 1.483768137025928 | 1.483768137025920 | |

1.0000 | 0.2 | 1.221402758160170 | 1.221402758160172 | |

1.0000 | 0.4 | 1.491824697641270 | 1.491824697641277 | |

1.0000 | 0.6 | 1.822118800390509 | 1.822118800390500 | |

1.0000 | 0.8 | 2.225540928492468 | 2.225540928492466 | |

1.3333 | 0.2 | 1.628496297454955 | 1.628496297454971 | |

1.3333 | 0.4 | 1.989049869365105 | 1.989049869365101 | |

1.3333 | 0.6 | 2.429430996560666 | 2.429430996560662 | |

1.3333 | 0.8 | 2.967313719959008 | 2.967313719959003 | |

1.6666 | 0.2 | 2.035589836749739 | 2.035589836749740 | |

1.6666 | 0.4 | 2.486275041088941 | 2.486275041088940 | |

1.6666 | 0.6 | 3.036743192730822 | 3.036743192730820 | |

1.6666 | 0.8 | 3.709086511425547 | 3.709086511425548 |

Consider the following two-dimensional Poisson’s equation:

We denote the root-mean-square error by the RMSE from the following relation:

Values of condition number and RMSE for some values of

Condition number | RMSE | |
---|---|---|

1.616697379470667 | ||

1.005685744274181 | ||

1.000056810457828 | ||

1.000000568099880 | ||

1.000000005680999 | ||

1.000000000056810 | ||

1.000000000000585 | ||

1.000000000000003 | ||

1.000000000000000 | ||

1.000000000000000 | ||

1.000000000000000 |

Comparison between exact solution and approximate solution of IRBFN method on the polar coordinate of Experiment

Exact solution | Approximate solution of IRBFN method on the polar coordinate | Error of IRBFN method on the polar coordinate | ||
---|---|---|---|---|

0.3333 | 0.2 | 0.4070935392947847 | 0.407093539278427 | |

0.3333 | 0.4 | 0.4972251717238353 | 0.497225171776793 | |

0.3333 | 0.6 | 0.6073121961701566 | 0.607919812163175 | |

0.3333 | 0.8 | 0.7417727914665396 | 0.741772791449410 | |

0.6667 | 0.2 | 0.8143092188653853 | 0.814309218844426 | |

0.6667 | 0.4 | 0.9945995259174347 | 0.994599525931543 | |

0.6667 | 0.6 | 1.214806604220352 | 1.214806604225190 | |

0.6667 | 0.8 | 1.483768137025928 | 1.483768136992937 | |

1.0000 | 0.2 | 1.221402758160170 | 1.221402758117891 | |

1.0000 | 0.4 | 1.491824697641270 | 1.491824697612953 | |

1.0000 | 0.6 | 1.822118800390509 | 1.822118800390206 | |

1.0000 | 0.8 | 2.225540928492468 | 2.225540928445109 | |

1.3333 | 0.2 | 1.628496297454955 | 1.628496297401382 | |

1.3333 | 0.4 | 1.989049869365105 | 1.989049869316768 | |

1.3333 | 0.6 | 2.429430996560666 | 2.429430996510109 | |

1.3333 | 0.8 | 2.967313719959008 | 2.967313719902712 | |

1.6666 | 0.2 | 2.035589836749739 | 2.035711976964034 | |

1.6666 | 0.4 | 2.486275041088941 | 2.486424223498127 | |

1.6666 | 0.6 | 3.036743192730822 | 3.036925404550813 | |

1.6666 | 0.8 | 3.709086511425547 | 3.709309065459067 |

Values of condition number and RMSE for some values of

Condition number | RMSE | |
---|---|---|

0.005 | 1.471481785169896 | |

0.01 | 2.037248475218950 | |

0.5 | 4.678768168755817 | |

1.0 | 7.977675155476113 | |

1.5 | 10.67180878932467 | |

2.0 | 16.72051422331341 | |

2.5 | 17.54142375428882 | |

3.0 | 22.68488788104916 | |

3.5 | 94.83497808866327 | |

4.0 | 687.3099630448350 | |

4.5 | 2251.857925768735 |

Consider the following two-dimensional Poisson’s equation:

Comparison between exact solution and approximate solution of the new method of Experiment

Exact solution | Approximate solution of the new method | ||
---|---|---|---|

0.0 | 0.0 | 1.000000000000000 | 1.000000001520331 |

0.0 | .25 | 2.117000016612675 | 2.116999962177082 |

0.0 | .5 | 4.481689070338065 | 4.481689131011527 |

0.0 | .75 | 9.487735836358526 | 9.487735908103340 |

0.0 | 1 | 20.08553692318767 | 20.08553702346013 |

1 | 0.0 | 7.389056098930650 | 7.389056124466515 |

1 | .25 | 15.64263188418817 | 15.64263190711622 |

1 | .5 | 33.11545195869231 | 33.11545243577679 |

1 | .75 | 70.10541234668786 | 70.10541228133020 |

1 | 1 | 148.4131591025766 | 148.4131603446419 |

.2 | 0.0 | 1.491824697641270 | 1.491824709011273 |

.2 | 1 | 29.96410004739701 | 29.96410038594428 |

.4 | 0.0 | 2.225540928492468 | 2.225540978673106 |

.4 | 1 | 44.70118449330082 | 44.70118489448246 |

.6 | 0.0 | 3.320116922736547 | 3.320116658933115 |

.6 | 1 | 66.68633104092514 | 66.68633190025831 |

.8 | 0.0 | 4.953032424395115 | 4.953032677031149 |

.8 | 1 | 99.48431564193381 | 99.48431532815737 |

.05 | .05 | 1.284025416687741 | 1.284025760755220 |

.13 | .26 | 2.829217014351560 | 2.829217082663191 |

.46 | .16 | 4.055199966844675 | 4.055199961753327 |

.31 | .42 | 6.553504862191149 | 6.553504821832245 |

.07 | .58 | 6.553504862191149 | 6.553504889746486 |

.12 | .73 | 11.35888208000146 | 11.35888287993308 |

.42 | .91 | 35.51659315162847 | 35.51659237035928 |

.51 | .57 | 15.33288701990720 | 15.33288705221003 |

.68 | .82 | 45.60420832084874 | 45.60420881755932 |

.84 | .37 | 16.28101980178843 | 16.28101996810631 |

.97 | .68 | 53.51703422749116 | 53.51703410219755 |

.17 | .93 | 22.87397954244081 | 22.87397949274820 |

Values of condition number and RMSE for some values of

Condition number | RMSE | |
---|---|---|

15.34291952699858 | ||

4.566804572945806 | ||

2.134108059285889 | ||

1.082012712119965 | ||

1.010904486607965 | ||

1.003900496513122 | ||

1.001720933454105 | ||

1.001551378516199 | ||

1.001534906804984 | ||

1.001537639564036 | ||

1.002332833767664 |

Location of scattered data points (12 of these points are interior points, and 18 points are boundary points) of Experiment

Consider the following two-dimensional Poisson’s equation in the elliptical region:

The results have been computed for

We have shown in [

Here, we would like to emphasize that this experiment had been also solved by [

Comparison between exact solution and approximate solution of the new method of Experiment

Exact solution | Approximate solution of the new method | Error of the new method | ||
---|---|---|---|---|

0.0 | 0.5 | 38.87195121951220 | 38.87195121952383 | |

0.0 | 1.5 | 37.65243902439024 | 37.65243902440016 | |

0.0 | 3.5 | 31.55487804878049 | 31.55487804879181 | |

0.0 | 5.5 | 20.57926829268293 | 20.57926829269648 | |

0.0 | 7.5 | 4.72560975609756 | 4.72560975611862 | |

2.0 | 0.0 | 37.46341463414634 | 37.46341463417008 | |

4.0 | 0.0 | 32.78048780487805 | 32.78048780491504 | |

6.0 | 0.0 | 24.97560975609756 | 24.97560975615097 | |

8.0 | 0.0 | 14.04878048780488 | 14.04878048786474 | |

10.0 | 0.0 | 0.00000000000000 | 0.00000000005427 | |

2.0 | 1.6 | 35.90243902439024 | 35.90243902440453 | |

4.0 | 1.6 | 31.21951219512195 | 31.21951219515048 | |

6.0 | 1.6 | 23.41463414634146 | 23.41463414638886 | |

8.0 | 1.6 | 12.48780487804878 | 12.48780487810293 | |

2.0 | 4.0 | 27.70731707317073 | 27.70731707318550 | |

4.0 | 4.0 | 23.02439024390244 | 23.02439024392700 | |

6.0 | 4.0 | 15.21951219512195 | 15.21951219516124 | |

8.0 | 4.0 | 4.292682926829268 | 4.29268292687660 | |

2.0 | 5.6 | 18.34146341463414 | 18.34146341465220 | |

4.0 | 5.6 | 13.65853658536585 | 13.65853658539350 | |

6.0 | 5.6 | 5.85365853658536 | 5.85365853662464 | |

9.798 | 1.6 | −0.00031375609756 | −0.00031375604674 | |

8.660 | 4.0 | 0.00171707317073 | 0.00171707321643 | |

7.141 | 5.6 | 0.00238790243902 | 0.00238790247911 | |

2.0 | 7.838 | 0.00350975609756 | 0.00350975612416 | |

4.0 | 7.332 | 0.00108292682927 | 0.00108292685914 | |

6.0 | 6.4 | 0.00000000000000 | 0.00000000003483 | |

8.0 | 4.8 | 0.00000000000000 | 0.00000000004390 |

Values of condition number and RMSE for some values of

Condition number | RMSE | |
---|---|---|

2.817474760824544 | ||

1.590396300485768 | ||

1.253627364959679 | ||

1.062100283211657 | ||

1.050234416869035 | ||

1.030320686741992 | ||

1.015443720207797 | ||

1.005553497698158 | ||

1.003855850797432 | ||

1.002467354513745 | ||

1.000616708637105 |

Values of condition number and RMSE for some values of

Condition number | RMSE | |
---|---|---|

0.001 | 1.433292009119422 | |

0.01 | 1.833381468640938 | |

0.5 | 1.641721037366621 | |

1.0 | 2.368533981239234 | |

1.5 | 6.742734977317889 | |

2.0 | 7.268116615890860 | |

2.5 | 19.12284221789496 | |

3.0 | 56.72352660443333 | |

3.5 | 83.76935856474893 | |

4.0 | 101.1762326758897 | |

4.5 | 164.9216307945755 |

Location of scattered data points (11 points are interior points, and 17 points are boundary points) of Experiment

In the present paper, we have introduced a new way for numerical solution of Poisson’s partial differential equation by a special combination between logarithmic and MQ-RBFs. We have showed that by this new method it does not need to control the parameter

It should be noted that the computations associated with the experiments discussed above were performed by using Maple 13 on a PC, CPU 2.4 GHz.

This research paper has been financially supported by the office of vice chancellor for research of Islamic Azad University, Bushehr Branch, for the first author. The authors are very grateful to both reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper. Also, authors acknowledge the editor professor Roberto Natalini, for managing the review process for this paper.