#### Abstract

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 . Further, the condition number which arises in the process is discussed, and a comparison is made between them with our earlier studies and previously known ones. It is shown that the system is stable.

#### 1. Introduction

The organization of the present paper is as follows. In Section 2, we give the theories of the new method. In Section 3, we provide some numerical experiments on the two-dimensional Poisson’s equation with the Dirichlet, the Neumann, and curved boundary conditions and we propose to examine the stability of the method and its behavior towards input variations. Finally, in Section 4, some conclusions are presented.

#### 2. The New Method of the Present Study

The form of a Poisson’s equation is as follows: in which is a known function. Also, Poisson’s equation can be in the following two Dirichlet and Neumann boundary conditions: where and are boundaries of the domain. Also, the vector is the outer unit normal to the boundaries and , are two known functions of . In MQ approximation scheme, we approximate the unknown function by an expression. In this study, we consider two-dimensional Poisson’s equation.

In this method, we use the following expression: The derivatives in the two-dimensional Cartesian coordinates are presented in the following:

In expression (2.4), the set of weights is to be found. In the present study, the closed form of approximation function (2.4) is first obtained from a set of training points and the derivative functions are then calculated directly by differentiation of such closed activity. The nonzero parameter protects of having zero values inside the logarithmic function. Also, we decrease the parameter to improve approximate solutions. In fact, this parameter controls the accuracy and the stability of the system. In each experiments, we evaluate the condition number of the system (condition number , that is the coefficient matrix of the system) and try to inspect the affect of the parameter on the stability of our system.

#### 3. Illustrative Numerical Experiments

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.

##### 3.1. Stability of the Solution

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 1 and 3).

Experiment 1. Consider the following two-dimensional Poisson’s equation: with the following Dirichlet boundary conditions on and : The exact solution is: .
We denote the root-mean-square error by the RMSE from the following relation: where is exact solution and is approximate solution at points . We have considered those 20 points that we had used for IRBFN method on the polar coordinates . As we have shown in earlier work , the parameter influences on accuracy, partially. We have shown that increasing the values of this parameter causes instability, and usually can not affect on accuracy adequately (see Table 4). Although, in that method, we could improve our results by focusing on other parameters such as substituting the scattered point places, in this new way, we can improve our solutions by changing the values of parameter without having instability. In contrast of MQ-RBFs, in this special combination, as parameter decreases, the accuracy increases. Also, since, for values greater than , the condition number is not near to unity and we will not have a stable system sufficiently, so we have considered at least . For instance, for this value of , we have , and, when we continue to decrease this parameter over and over, we get a better solution. For instance, in the best position while in IRBFN method on the polar coordinates  in the best position, we had . Also, in this new way, when reaches to about and smaller values, our accuracy and condition number are almost fixed. As we are decreasing parameter , the accuracy is going to be better (see Table 2). Further, when we decrease values quite enough, the accuracy and condition number almost do not change (see Table 2). Note that, for smaller values of , condition number is very near to unity (equals unity in double precision).

Experiment 2. Consider the following two-dimensional Poisson’s equation: with the following Neumann and Dirichlet boundary conditions on and : The exact solution is , where and are, respectively, 2 and 3. This experiment was solved using MQ approximation scheme by Kansa . The author used a total of 30 points, including 12 scattered data points in the interior and 18 along the boundary. The reported results showed that the norm of error is . Later, Mai-Duy and Tran-Cong  used IRBFN method and got a greater accuracy. They reported the norm of error is for this experiment. In this study, we have used those same points (see Figure 1) and have achieved better accuracy in comparison with those two previous works. By using present approach, the norm of error that we have gotten in the best position is (). The results are shown in Tables 5 and 6. There are the same properties and results about this new way that we have explained in Experiment 1 (more accuracy and stability).

Experiment 3. Consider the following two-dimensional Poisson’s equation in the elliptical region: The great diameter of the ellipse is , and small diameter is . The boundary condition is on all of boundary points. The equation of ellipse is Since the above ellipse is a symmetrical region, then we have solved this equation on the first quarter. The Dirichlet and the Neumann boundary conditions are The analytical solution is .
The results have been computed for and . We have used 28 points that 17 of them are boundary points and 11 are interior points (see Figure 2) which were selected at random. In this new way, we have achieved a better accuracy in comparison with IRBFN method in the polar coordinates  (see Tables 7, 8, and 9). When we decrease parameter , RMSE decreases too, and so we have better accurate solution. Also, the condition number closes to unity more and more. For (), it almost equals to unity (see Table 8).
We have shown in , in using of MQ-RBFs on the polar coordinate, when we have used 28 data points and we have been increasing the width parameter , the accuracy of our solution has been increasing a little, though our system has been going to nonstability (see Table 9). In contrast, in this new way, when we are decreasing the parameter , the accuracy of our solution is increasing too. Also, the condition number of our system is decreasing and closes to unity (see Table 8).
Here, we would like to emphasize that this experiment had been also solved by  and , and the norm of error is in the DRBFN method , wherein the norm of errors in the DRBFN and the IRBFN methods are and , respectively.

#### 4. Conclusion

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 (the width parameter) all times for preventing inaccuracy of solutions or increasing the value of condition number and having an ill-conditioned system. In this new way that is enough to consider the value of the parameter smaller than . In the aforesaid experiments, the accuracy is better than those before results obtained by [1, 3, 5, 16, 17] and the condition number of the systems is equal to unity. So we have some complete stable systems and more accurate solutions.

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.

#### Acknowledgments

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.