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Mathematical Problems in Engineering
Volume 2013, Article ID 613082, 11 pages
http://dx.doi.org/10.1155/2013/613082
Research Article

Analytical Particular Solutions of Multiquadrics Associated with Polyharmonic Operators

Department of Marine Environmental Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan

Received 9 January 2013; Accepted 29 March 2013

Academic Editor: Kue-Hong Chen

Copyright © 2013 Chia-Cheng Tsai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We derive two- and three-dimensional analytical particular solutions of multiquadrics (MQ) associated with the polyharmonic operators, named as the polyharmonic multiquadrics (PMQs). The methods of undetermined coefficients are constructed by observing the first few orders of the PMQs which are obtained by the symbolic software, Mathematica. By expanding the PMQs into the Laurent series, the unknown coefficients of the PMQs can be determined. The homogeneous parts of the PMQs are suitably arranged so that the PMQs are hierarchically unique and infinitely differentiable. Mathematica codes are provided for obtaining the PMQs of arbitrary orders. The derived PMQs are validated by numerical solutions for Poisson’s equation. Numerical results indicate that the solutions obtained by the PMQs are more accurate than those by the MQ.

1. Introduction

A radial basis function (RBF) is a real-valued function whose value depends only on the distance from a certain prescribed center. Traditionally, sums of RBFs are used for approximating given functions or reconstructing scattered data. Among the various RBFs, the Gaussian function, Hardy’s multiquadrics (MQ) [1], Duchon’s augmented polyharmonic spline (APS) [2], and Wendland’s compactly supported RBF (CSRBF) [3] are the most popular ones. In the early 1990s, Kansa [4, 5] made the first attempt to use the MQ for approximating the solutions of partial differential equations (PDEs). The Kansa method is meshless, simple and has been used for a wide range of PDEs.

Alternatively, Nardini and Brebbia developed the dual reciprocity method (DRM) for approximating the particular of the considered PDE by the RBFs, and then solved the complementary problem by the boundary element method (BEM) [6]. In addition to the BEM, there are also other boundary-type numerical methods such as the method of fundamental solutions (MFS) [711] and the Trefftz methods [12, 13].

In an early development of DRM, the ad hoc function, , was exclusively used. In order to improve the accuracy of the computation, researchers applied the RBF theory to the DRM [14, 15]. For example, Golberg [9, 16], Chen [17], and Karur and Ramachandran [15] demonstrated the superiority of the APS over the ad hoc trial function. Then, Golberg et al. [18] further improved the accuracy of the approximated particular solution by utilizing the exponential convergence rate of the MQ. A comprehensive review on the DRM development in this period was given by Golberg and Chen [19].

When the MQ is adopted in a DRM procedure, the applicability depends on the availability of the analytical particular solution associated with the partial differential operator of a given problem. Golberg et al. [18] derived the analytical particular solutions for the Laplace equation. Samaan and Rashed [20, 21] found the analytical particular solutions for the two-dimensional problems of elasticity and elastodynamics, which could be converted to a biharmonic equation using the Galerkin-Papkovich vector [22]. Basically, the applicability of MQ is limited to the harmonic and biharmonic operators as shown in Yao’s thesis [23]. However, Yao’s method cannot be easily generalized since the method she applied to cancel the singularity is not straightforward for polyharmonic operators.

In the appendix of a recent study [24], the present author has found analytical particular solutions of the MQ associated with the fourth- and third-ordered polyharmonic operators of two and three dimensions, respectively. In this study, the theory for deriving these polyharmonic multiquadrics (PMQs) is presented. In other words, we derive the analytical particular solutions of multiquadrics (MQ) associated with the two- and three-dimensional polyharmonic operators. The methods of undetermined coefficients are constructed by observing the first few orders of the PMQs which are obtained by straight integrations via the symbolic software, Mathematica. Furthermore, the Laurent expansions are used to find the unknown coefficients and to make the PMQs infinitely differentiable. Numerical experiments are carried out to validate the derived PMQs and to demonstrate the superior accuracy of the PMQs over the MQ for solving Poisson’s equation.

This paper is organized as follows: the problem is mathematically modeled in Section 2. Then, the two- and three-dimensional PMQs are derived in Sections 3 and 4, respectively. Some numerical experiments are carried out to validate the PMQ in Section 5 and the conclusions are drawn in Section 6.

2. Definition of the Problem

In this study, we derive the two- and three-dimensional polyharmonic multiquadrics (PMQs) and of order , which are governed by with In (1), (2), and the following, stands for () or . If we make the following change of variables equations (1) and (2), respectively, become Here, is defined as (3) while replacing by . Equations (7) and (6) can be deduced to

It is clear that the solutions of (6)-(7) are also the solutions of (8). However, the converse is not always true since the homogeneous solutions of (8) can be arbitrarily chosen. In the next two sections, a specific choice will be made to ensure the solutions of (8) are also the solutions of (6)-(7). Then, the PMQs governed by (1)-(2) can be obtained by using (5).

3. Two-Dimensional PMQ

It is clear that the solution of (8) can be found by straight integrations. For the two-dimensional case, it can be expressed as follows: where the integral operator is defined as . When observing the first few orders of the PMQs obtained by the symbolic software, Mathematica [25], we can find that where the first two series form a particular solution with undetermined coefficients and and the last two series give the homogeneous solutions with integration constants and . To make infinitely differentiable, we obviously need and then (10) becomes

In (12), the coefficients and should be determined such that the PMQ in (12) satisfies (8) and the arbitrary coefficients should be chosen to ensure the solution satisfies the hierarchical relation (6) and (7). Therefore, we expand (12) in the Maclaurin series as where is the double factorial defined as The detailed derivation is given in Appendix A. In order to ensure the solution satisfies the hierarchical relation (6) and (7), we can make the following selection: Then, (13) becomes The PMQ , given in (16) or equivalently in (12) with defined in (15), is the unique infinitely differentiable PMQ with its first even coefficients of the Maclaurin series being zero if the undetermined coefficients and can be found by (8). The proposed condition can be alternatively stated as where the following relation has to be considered: Applying to the PMQ in (16) and using (18) indicate that also satisfies (17) and of the same form in (16). Therefore, it can be understood that the prescribed PMQ also satisfies the hierarchical relation (6) and (7).

In order to determine and , (16) is substituted into (8) which results in where (18) and the following Maclaurin series are used: with being the usual combination formula.

Substituting by in the left-hand side of (19) and comparing the coefficients of , we have or equivalently The identity (22) should be satisfied for every nonnegative integer to ensure the PMQ is the solution of (8). Since the left-hand side of (22) is a -ordered polynomial of the variable , it can be used to uniquely determine the unknown coefficients and if the system is not singular. After and are obtained, (15) can be used to have . Therefore, the two-dimensional PMQ can be computed simply by (12). For the cases when is very small, one can alternatively use (16).

In practice, one can simply set equal to in (22) which results in linear equations for solving the unknown coefficients and of the PMQ of order . Mathematica codes for obtaining the 2D-PMQ are given in Algorithm 1. Alternatively, if a specific low order of 2D-PMQ is considered, one can use the explicit coefficients given in Tables 1, 2, and 3 and the results are plotted in Figure 1. Basically, the PMQs of higher orders are more flat in the near field and grow faster in the far field, as shown in the figure.

tab1
Table 1: .
tab2
Table 2: .
tab3
Table 3: .

alg1
Algorithm 1: Mathematica codes for obtaining 2D-PMQs.

fig1
Figure 1: Plots of the 2D-PMQs in (a) near field and (b) far field.

4. Three-Dimensional PMQ

Similarly, the three-dimensional PMQs can be obtained by the method of undetermined coefficients which begins with Here, the first two series form a particular solution with undetermined coefficients and and the last two series give the homogeneous solutions with integration constants and . Here, it should be noted that the three-dimensional PMQ involving the inverse hyperbolic sine function has been recently derived in the author’s previous study [24] and will be further extended in this study. To determine , , , and , we first expand (23) in terms of the Laurent series as The detailed derivation is given in Appendix B. In order to make infinitely differentiable, it is required to cancel the singular term and all the odd coefficients in (24) [26]. In other words, we need Similar to (17), we additionally consider the following relation: with Then we use (24) and (26) to have the following relation: Substituting (28) into (24) gives The PMQ , given in (29) or equivalently in (23) with and defined in (28) and (25), is the only infinitely differentiable PMQ satisfying (26) if the undetermined coefficients and can be found by using (8). Then, we consider the following relation.

Equation (27) with together with (29) is sufficient to infer that the prescribed PMQ also satisfies the hierarchical relation (6) and (7) as explained in the two-dimensional case. Substituting (29) into (8) and using (20) and (27) result in Substituting by for the left-hand side of (30) and comparing the coefficients of , we have or equivalently Similar to the two-dimensional cases, (32) can be used to uniquely determine the unknown coefficients and since its left-hand side is a -ordered polynomial of the variable . Similarly, Mathematica codes for obtaining the 3D-PMQ are given in Algorithm 2. Also, the first few orders of 3D-PMQ are given in Tables 4, 5, and 6 and plotted in Figure 2. In the figure, we can also observe that the PMQs of higher orders are more flat in the near field and grow faster in the far field.

tab4
Table 4: .
tab5
Table 5: .
tab6
Table 6: .

alg2
Algorithm 2: Mathematica codes for obtaining 3D-PMQs.

fig2
Figure 2: Plots of the 3D-PMQs in (a) near field and (b) far field.

5. Validation

By the standard MFS-DRM, the PMQs derived in this paper have been validated for the cases of fourth and third orders, respectively, in two and three dimensions [24]. The results basically indicate that the solutions of the MQ are more accurate than those of the low-ordered APS. Numerical results for other orders of polyharmonic problems should be similar and thus will not be repeated here.

Alternatively, we apply the PMQs to the standard MFS-DRM for solving Poisson’s equation where is the considered domain, is the given forcing term, and denotes the Laplacian operator. For simplicity, we consider the boundary condition where is the boundary of and is the given boundary data. In the formal MFS-DRM, the solution should be decomposed into two parts where the particular solution satisfies but not necessarily the boundary condition (34). And the homogeneous solution is defined by The solution of (37) can be formally obtained by the MFS [711] and will not be additionally described.

On the other hand, in order to solve the particular solution, we approximate the forcing term by where is the Euclidean distance between the spatial coordinates and prescribed points in and is the PMQ of order . In order to solve the unknown coefficients , (38) should be collocated on the prescribed points as where . After the forcing term is approximated in (38), the particular solution can be correspondingly approximated as

For , the solution procedure is reduced to the formal MFS-DRM based on the MQ [18].

5.1. Two-Dimensional Case

We consider a two-dimensional problem governed by Poisson’s equation (33) with The boundary condition (34) is set according to the exact solution as Since the maximum value of the solution in is equal to , we use the maximum error in this example. And the sources of the MFS are typically located on the boundary of a square with edge length equal to 5.

We have hierarchically performed our tests for . Among the similar results, the maximum errors for are addressed in Table 7. This should have validated the two-dimensional PMQs for . For every individual test, a preliminary search is always performed for finding the optimal shape parameter as demonstrated in Figure 3. It can be observed from Table 7 that the best maximum errors (marked by *) usually occur at and are about an order of magnitude better than those obtained by the MQ and the case of more collocation points (larger ) gives better accuracy. If we observe the PMQ coefficients, we will find that eight is the minimum order of the PMQ with some coefficients over the limit of 64-bit unsigned integer. To be more detailed, is equal to whose denominator is just larger than .

tab7
Table 7: Maximum errors of the two-dimensional Poisson's problem.
fig3
Figure 3: Search for the optimal shape parameter for the 2D case with (a) and (b) .

5.2. Three-Dimensional Case

Then, we consider a three-dimensional problem governed by Poisson’s equation (33) with The boundary condition (34) is set according to the exact solution as Since the maximum value of the solution in the computational domain is equal to , we also use the maximum error in this example. And the sources of the MFS are typically located on the boundary of a cuboid with edge length equal to 5.

Similarly, we also hierarchically perform numerical tests for and address the maximum errors for in Table 8. This should also validate the three-dimensional PMQs for . Figure 4 gives the preliminary search for finding the optimal shape parameter . It can also be observed from Table 8 that the best maximum errors basically occur at and are about an order of magnitude better than those obtained by the MQ. Similarly, we have equal to whose denominator is larger than .

tab8
Table 8: Maximum errors of the three-dimensional Poisson's problem.
fig4
Figure 4: Search for the optimal shape parameter for the 3D case with (a) and (b) .

It seems that the PMQ of higher order gives more accurate results if there is no numerical ill-conditioning. However for the case of and , the error goes worse as shown in Table 8. This result seems to indicate that the PMQ of higher order also requires more collocation points. In practice, one can safely choose and find some improvement over the MQ.

6. Conclusion

The polyharmonic multiquadrics (PMQs) are derived. The methods of undetermined coefficients are constructed and the PMQs are solved by the Laurent expansion. Some additional conditions are introduced to make the derived PMQs hierarchically unique and infinitely differentiable. Numerical experiments were carried out to validate the derived PMQs. Basically, the numerical results obtained by the PMQ showed some improvement over those by the multiquadrics.

The method introduced in this study can be extended to derive the polyharmonic particular solutions of the generalized multiquadrics with which is currently under investigation.

Appendices

A. Maclaurin Series of (12)

Since the Maclaurin series of is given in (20), we begin with the Maclaurin series of as Then, we observe the relation Equations (A.1) and (A.2) can be combined to have Equations (20) and (A.3) are sufficient to find the Maclaurin series of (12).

B. Laurent Series of (23)

First we observe the relation Equations (A.1) and (B.1) can be combined to have Equations (20) and (B.2) are sufficient to find the Maclaurin series of (23).

Acknowledgment

The National Science Council of Taiwan is gratefully acknowledged for providing financial support to carry out the present work under Grant no. NSC 101–2628-E-022-E-MY2.

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