The Method of Particular Solutions for Solving Certain Partial Differential Equations

Transcription

1 The Method of Particular Solutions for Solving Certain Partial Differential Equations C.S. Chen, C.M. Fan, P.H. Wen Abstract A standard approach for solving linear partial differential equations is to split the solution into a homogeneous solution and a particular solution. Motivated by the method of fundamental solutions for solving homogeneous equations, we propose the similar approach using the method of particular solutions for solving linear inhomogeneous differential equations without the need of finding the homogeneous solution. This results to a much simpler numerical scheme with the similar accuracy as the traditional approach. To demonstrate the simplicity of the new approach, three numerical examples have been given with excellent results. Key words: The method of fundamental solutions, particular solution, homogeneous solution, radial basis functions, meshless method, polyharmonic splines. Introduction During the past decade, meshless methods have attracted great attention for numerically solving partial differential equations. Various meshless methods have been developed and successfully employed to solve challenging problems in different areas of science and engineering. Among all the meshless methods, the method of fundamental solutions (MFS) is one of the boundary-type meshless methods [3, 4, 3]. The MFS is highly accurate for solving homogeneous equation if the fundamental solution of the given differential operator is known [9, ]. To extend the MFS to inhomogeneous equations or time-dependent problems [,, 2, 5], the method of particular solutions (MPS) has been introduced Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 3946, USA. Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 2224, Taiwan Department of Engineering, Queen Mary, University of London, London E 4NS, UK Correspondence to: C.S. Chen, cs.chen@usm.edu

2 to evaluate the particular solution of the given differential equation. Since the particular solution is not unique, there is a rich variety of numerical techniques developed for this purpose. Radial basis functions (RBFs), polynomial functions, trigonometric functions, etc. [, 5, 7,, 5] have been employed as the basis functions to approximate the particular solution of the given differential equation. Once the particular solution has been evaluated, the given inhomogeneous equation can be converted to the homogeneous equation and thus the MFS can be used to evaluate the homogeneous solution. The numerical solution of the original differential equation can be recovered by adding the homogeneous solution and particular solution. This is a two-stage numerical scheme. For a given linear elliptic partial differential equation, if the fundamental solution and particular solution are both available, then the problem can be solved easily. In the past, a large class of fundamental solutions have been derived and applied to solve various kinds of problems in science and engineering. On the other hand, the particular solutions are more difficult to derive since the inhomogeneous term is a more general function. In some senses, the fundamental solution is a special case of particular solution. The major distinction is that the fundamental solution is a singular function and the general particular solution is a smooth function. When the inhomogeneous term is replaced by the delta function, the particular solution becomes fundamental solution. Both the fundamental solution and particular solution are derived in the infinite space. The main idea of the MFS is that the fundamental solution satisfies the homogeneous equation inside the domain, and one only needs to enforce the fundamental solution on the boundary conditions to obtain the solution of the given homogeneous problem. Hence, when the fundamental solution is available, the given homogeneous problem can be obtained easily by using the MFS. Motivated by the similar idea, the particular solution can be used to solve the inhomogeneous problems; i.e., since the particular solution satisfies the given inhomogeneous equation through the domain without satisfying the boundary conditions, one only needs to impose the boundary conditions to obtain the solution of the given inhomogeneous problem. We normally use various kinds of numerical scheme to approximate the particular solutions as mentioned earlier. It is the purpose of this paper to propose a numerical scheme using radial basis functions to approximate particular solution and then solve the inhomogeneous problems without the need of obtaining homogeneous solution. This means we don t need to use the MFS or other boundary methods to find the homogeneous solution in the solution process. The structure of this paper is as follows. In Section 2, we give a brief review of the traditional approach using the MPS to evaluate the approximate particular solution and the MFS to find the homogeneous solution. In Section 3, we propose the new approach of using the particular solution without the need of finding homogeneous solution. In Section 4, three numerical examples are given to demonstrate our current proposed approach in comparison with the traditional approach. In Section 5, we draw conclusions and discuss directions of future research. 2

3 2 The Traditional Two-Stage Scheme In this section we re-examine the numerical procedure for evaluating the particular solution and homogeneous solution. The motivation of our current proposed method is based on the idea that the fundamental solution can be viewed as a special type of particular solution. Consider the following differential equation Lu(x, y) = f(x, y), (x, y) Ω, () Bu(x, y) = g(x, y), (x, y) Ω, (2) where L is a linear differential operator, B a boundary operator, f(x, y), g(x, y) are known functions, and Ω is an open bounded domain in R d, d = 2, 3 with boundary Ω. Let u p be a particular solution of the governing equation, then it satisfies Lu p = f(x, y), (3) but does not necessarily satisfy the boundary condition. If u p in (3) can be obtained, then the original equation in () (2) can be reduced to the following homogeneous equation through the variable substitution u h = u u p ; i.e., Lu h (x, y) =, (x, y) Ω, (4) Bu h (x, y) = g(x, y) Bu p (x, y), (x, y) Ω. (5) The homogeneous equation (4) (5) can be easily solved using boundary methods such as the MFS [6, 9, ]. The above numerical scheme for solving partial differential equations is pretty standard provided that the particular solution and fundamental solution are both available. The final solution of () (2) can be obtained by adding the particular solution and homogeneous solution; i.e., u = u p + u h. 2. Approximate Particular Solutions One of the most popular schemes to numerically approximate the particular solution u p in (3) is using RBFs. A complete review can be found in Reference []. Typically, this is done by approximating f by a finite series of RBFs {ϕ j } n through interpolation f(x, y) ˆf(x, y) = a j ϕ j (r), (6) where r = (x x j ) 2 + (y y j ) 2 is the Euclidean distance, {(x i, y i )} n is an unisolvent set of interpolation points, and the coefficients {a j } n are to be determined; i.e. by solving a j ϕ j (r k ) = f(x k, y k ), k n. (7) 3

4 An approximate particular solution û p to (3) is given by û p (x, y) = a j Ψ j (r), (8) where {Ψ j } n is obtained by analytically solving LΨ j (r) = ϕ j (r). (9) An accurate approximation of u p depends on how well f is approximated. Consequently, the appropriate choice of basis function {ϕ j } n is of considerable interest. In general, MQ ( r 2 + c 2 ), Conical RBFs (r 2m ), Polyharmonic splines (r 2m ln r) are the most popular RBFs. For a conditional positive definite function such as Polyharmonic splines, additional polynomial terms in (6) are required to ensure the resultant matrix is solvable [8]; i.e., t ˆf(x, y) = a j r 2m ln r + b i p i, along with the constraints i= a j p i (x j, y j ) =, i t, () where {p i } t i= is a basis for P Q, the set of d-variate polynomials of degree Q, and t = ( ) Q+d d is the dimension of PQ. For polyharmonic splines, Q = m +. Then, û p (x, y) = a j Ψ [m] j (r) + t b i χ i, () i= where LΨ [m] j (r) = rj 2m ln r j, (2) Lχ i = p i. (3) A list of closed-form particular solutions for various of L and ϕ is giving in the Appendix. 2.2 Approximation of Homogeneous Solution After the particular solution has been evaluated, the homogeneous solution for (4) and (5) needs to be addressed. The basic idea of the MFS is to approximate the solution 4

5 u h by û h which can be expressed as a linear combination of fundamental solutions. Let {(s j, t j )} n b be the source points placed outside the domain of the problem. n b û h (x, y) = α j G(r), (x, y) Ω Ω, (4) where r = (x s j ) 2 + (y t j ) 2. The source points in the MFS can be considered as lying on a fictitious boundary ˆΩ of a region ˆΩ containing Ω. It is an important issue to determine the optimal location of the fictitious boundary. Once the source points have been chosen, the {α j } n b in (4) are generally obtained by collocation. That is, n b points {x k, y k } n b are chosen on Ω and then {α j } n b satisfy the boundary condition in (5) n b α j BG(r k ) = g(x k, y k ) Bu p (x k, y k ), k n b. (5) Because the system of equations resulted from (5) is highly ill-conditioned as the source points move away from the boundary, we have to choose the location of the these source points with care. The optimal choice of source location is still an outstanding research problem. 3 New Proposed Numerical Scheme In this section, we propose a new approach using the particular solution only without the need of finding homogeneous solution for solving () (2). We observe that if we replace f(x, y) in () by Dirac-Delta function δ(x, y), u p in (3) becomes the fundamental solution of differential operator L; i.e., LG = δ(x, y), where G is the fundamental solution of differential operator L. In other words, the fundamental solution is also a particular solution. Since the fundamental solution satisfies the homogeneous equation in the infinite domain, the implementation of the MFS allows us to impose the boundary conditions only on (4) without considering the governing equation inside the domain. Based on the similar idea, it is known that the particular solution satisfies the given differential equation in the infinite domain without the need to satisfy the boundary condition which is similar to the fundamental solution to the given homogeneous equation. However, we notice there is a distinction that the fundamental solution is a singular function, but the particular solution for a given smooth inhomogeneous term is a smooth function. As we shall see, the new approach requires the use of the approximate particular solution û p to obtain the approximate solution of () (2). 5

6 Since û p in (8) is an approximate particular solution, from () we have or which implies Lû p = ˆf(x, y), (x, y) Ω, (6) a j LΨ j (r) = ˆf(x, y), (x, y) Ω, (7) a j ϕ j (r) = ˆf(x, y), (x, y) Ω. (8) If we impose û p in (8) to satisfy the boundary conditions in (2), then û p becomes the approximate solution of the original partial differential equation () (2). To be more specific, we have a j BΨ j (r) = g(x, y), (x, y) Ω. (9) For the numerical implementation, we let {(x k, y k )} n i be the interior points, {(x k, y k )} n i+n b n i + be the boundary points, and n = n i + n b. By collocation method, from (8) (9) we have a j ϕ j (r k ) = f(x k, y k ), k n i, (2) a j BΨ j (r k ) = g(x k, y k ), n i + k n. (2) The above system of equations can be easily solved by standard matrix solver. Once {a j } n is determined, the approximate particular solution in (8) becomes the approximate solution û of () (2); i.e., û(x, y) = a j Ψ j (r). (22) As shown in (2) (2), no homogeneous solution is required in the solution process. The difficulties of locating the fictitious boundary and ill-conditioning of the MFS can be completely avoided. In Figure, we show two set of interpolation points: interior and boundary points. 4 Numerical Results In this section we provide three numerical examples to validate our proposed method. In the following examples, we will examine the effect of using various number of interpolation 6

7 Ω Ω Figure : Internal points ( ), and boundary points ( ) in the computational domain. nodes and order of RBFs on the accuracy of numerical results. Overall, the numerical stability, accuracy, and simplicity of the proposed method are the major issues to be explored in this section. The root-mean-square error (RMSE) and the root-mean-square error of the derivative with respect to x (RMSEx) are used to show the accuracy of the solutions. They are defined as follows: RMSE = n t (û j u j ) 2, (23) n t RMSEx = n t n t ( ûj x u ) 2 j, (24) x where n t is the number of testing nodes located randomly within the domain. û j denotes the numerical solution at the j th node. Since RMSEy is similar to RMSEx, we only show RMSE and RMSEx in the following examples. We use the following formula to choose the location of the source points in the MFS: x s = x b + σ ( x b x c), (25) where x s, x b, and x c denote the source, boundary, and central nodes. The parameter σ determines how far away the source points from the boundary. 7

8 Through the rest of this section, we denote n i the number of the internal interpolation points, n b the number of boundary points. Furthermore, these interpolation points are selected uniformly inside the domain and along the boundary. Example We first consider the Poisson problem with Dirichlet boundary condition: u(x, y) = π 2 (y sin(πx) + x cos(πy)), x, y Ω, (26) u(x, y) = y sin(πx) + x cos(πy), x, y Ω. (27) The parametric equation of the computational domain is given by Ω = {(x, y) x = ρ cos θ, y = ρ sin θ, θ 2π}, (28) where ρ = ( ) cos (3θ) + 2 sin 2 3 (3θ). (29) The analytical solution is given by u(x, y) = y sin(πx) + x cos(πy). (3) y Z x 2.5 X Y Figure 2: Computational domain (left) and analytic solution (right) in the extended domain. The profiles of computational domain, distributions of boundary and interior nodes, and analytical solution are presented in Figure 2. To compute RMSE and RMSEx, we set n t = 3, n b = 4, n i =. We first choose MQ as the interpolation function. The profiles of RMSE and RMSEx with respect to shape parameter, c, are shown in Figure 3. It shows that the numerical results deteriorate when c is larger than.6. This is consistent with other studies using MQ in the literature. The solutions using MQ with different number of nodes are shown in Table where n b = 2, 4, 8, 2, and n i = 6,, 28, 284. In this table, we observe there is little difference in term of accuracy using the MFS and MPS. 8

9 2 RMSE RMSEx Error c Figure 3: Profiles of RMSE and RMSEx verse shape parameters. Table : Comparison of RMSE and RMSEx with and without the MFS. n b n i MPS optimal c RMSE 2.53E 4 3.5E 5.36E E 6 RMSEx 2.7E 3.5E E 5 2.8E 5 MFS (σ = ) optimal c RMSE 2.75E 3 2.2E E 5.53E 6 RMSEx 2.88E E E E 6 MFS (σ = 5) optimal c RMSE.46E E 6 3.6E 6 9.E 6 RMSEx.33E E 5 8.6E E 5 9

10 Despite the fast convergence of MQ, the determination of optimal shape parameter is still a challenge. As a result, we adopt the Conical and the polyharmonic splines to approximate the solution instead of using MQ. The numerical results using these RBFs are shown in Tables 2 and 3. Overall, MQ outperforms the Conical and the polyharmonic splines. However, no free parameter is required using the Conical and the polyharmonic splines. It s worth to notice that the derivative can be approximated quite accurately which is important for solving many practical problems. Table 2: RMSE and RMSEx using the Conical RBFs, r 2m. n b n i m = RMSE 7.8E E 3.845E 3 RMSEx 4.486E 2.729E 2.27E 2 m = 2 RMSE.34E E E 4 RMSEx 9.635E E E 3 m = 3 RMSE 5.45E 4.53E E 5 RMSEx 3.292E E E 4 To show the numerical convergence of the proposed scheme, we consider the above problem on an unit square domain. The nodes are distributed evenly in the domain. We denote dh be the distance between closest nodes. The numerical results using different order of Conical RBFs are shown in Figure 4. From these figures, we observe the good stability and convergence of the proposed method using Conical RBFs for solving Poisson s problem. Example 2 In this example we consider the following inhomogeneous modified Helmholtz equation: ( λ 2 )u(x, y) = f(x, y), (x, y) Ω, (3) u(x, y) = g(x, y), (x, y) Ω, (32) where f(x, y) and g(x, y) are chosen according to the following analytical solution u(x, y) = sin(πx) cosh(y) cos(πx) sinh(y). (33)

11 Table 3: RMSE and RMSEx using the polyharmonic splines, r 2m ln(r). m = m = 2 m = 3 n b n i RMSE 6.53E E E 4 RMSEx 5.499E E E 3 RMSE 2.95E 4.344E E 5 RMSEx 2.293E 3.596E E 4 RMSE.87E E 5.923E 5 RMSEx.48E 3 4.7E 4.998E 4 2 n = n = 2 n = 3 2 RMSE RMSEx n = n = 2 n = dh dh Figure 4: Profiles of RMSE (left) and RMSEx (right) using the Conical RBFs r 2m. The computational domain is defined by the following parametric equation: Ω = {(x, y) x = cos θ, y = sin θ, θ 2π}. (34) The profiles of the distribution of internal and boundary nodes, computational domain, and the analytical solution are shown in Figure 5. We set the number of test point to be n t = 2. For the Helmholtz-type equations, the close-form particular solutions are available only for polyharmonic splines [4]. The numerical results are shown in Tables 4 and 5 using different number of nodes and different order of polyharmonic splines. Three combinations of nodes are used, including n b = 5,, 2 and n i = 5, 3, 6. The numerical results are excellent. In Tables 4 and 5, the numerical errors are not sensitive

12 2.5 Z y x X Y Figure 5: The profiles of the domain (left) and the analytical solution in the extended domain (right). with respect to λ values. This is important since Helmholtz-type equation with higher λ is normally difficult to solve. Furthermore, we notice that, by using the proposed scheme, the excellent solution can be achieved using small number of nodes. From the table, the proposed method can achieve the similar solution of the MFS without taking care the location of source in the MFS. Table 4: RMSE and RMSEx obtained by the MFS and MPS using r 6 ln r, different number of boundary nodes, n b, and interpolation nodes, n i. n b = 5 n b = n b = 2 n i = 5 n i = 3 n i = 6 MPS MFS MPS MFS MPS MFS (σ = 5) (σ = 5) (σ = 5) λ = RMSE.63E 4.69E E 5 2.5E 4.48E 5.29E 5 RMSEx 8.5E 4 2.E 4 2.E E E 5 2.7E 5 λ = RMSE.37E E 5 3.7E 5.4E 5 7.E 6 5.9E 6 RMSEx.58E 3.82E E 4 2.8E 4.29E E 5 λ = RMSE.5E E E 5.E 4.2E E 5 RMSEx 2.43E 3 2.6E E E E 4.6E 3 2

13 Table 5: RMSE and RMSEx obtained by r 2m ln(r) using (n b =, n i = 3). λ = λ = λ = m = m = 2 m = 3 RMSE.E 3.55E E 5 RMSEx 9.5E 3.36E 3 2.E 4 RMSE.44E 3 2.6E 4 3.7E 5 RMSEx.93E 2 3.3E E 4 RMSE.8E 3 2.6E E 5 RMSEx 3.E E E 4 To further investigate the numerical convergence of the proposed method, we consider (3) (32) with λ = in an unit square domain using different number of nodes. The nodes are distributed evenly in the unit square. The mesh distance dh of these nodes are defined the same as in the previous example. The RMSE and RMSEx for m =, 2, 3 are presented in Figure 6. The stability and convergence of the proposed method for Helmholtz-type equation are excellent. From Figure 6, it is clear that the better solution can be obtained by increasing the order of polyharmonic splines or the number of interpolation nodes. Similar to the Conical RBFs, there is no free parameter to be determined. RMSE n = n = 2 n = 3 RMSEx n = n = 2 n = dh dh Figure 6: Profiles of RMSE (left) and RMSEx (right) using different order of polyharmonic splines, r 2m ln(r). 3

14 Example 3 We consider the following inhomogeneous bi-harmonic equation: 2 u(x, y) = f(x, y), x, y Ω, (35) u(x, y) = sin(y 2 + x) cos(y x 2 ), x, y Ω, (36) u(x, y) n = ( (sin(y 2 + x) cos(y x 2 ))) n, x, y Ω, (37) where n is the normal vector along the boundary, and f(x, y) is given according to the analytical solution u(x, y) = sin(y 2 + x) cos(y x 2 ). (38) The computational domain is defined by the following parametric equation: Ω = {(x, y) x = ρ cos θ, y = ρ sin θ, θ 2π}, (39) where ρ = + cos 2 (4θ). (4) 2 2 y Z x 2 2 Y 2 2 X 2 Figure 7: Computational domain (left) and profile of solution (right) in the extended domain. The computational domain, distribution of nodes and profile of solution are shown in Figure 7. We let n t = 47, n b = 8, 2, 2, and n i = 22, 37, 52. We choose the Conical RBFs for m = 2, 3, 4 as the basis function. Since two different boundary conditions are imposed simultaneously for the bi-harmonic equation, we use subroutine LSQRR from the IMSL library to obtain the linear least-square solution. The RMSE and RMSEx obtained by the Conical RBFs are shown in Table 6. Among our test, we notice that the errors are very small by using such small number of nodes. As shown in Table 6, the numerical results are greatly improved when the order of Conical RBFs increases. This is consistent with the previous examples. Furthermore, we compared the solutions by the MPS and the MFS in Table 7 when r 7 is used as the basis function for interpolation. We observe that there is little difference in numerical results using 4

15 Table 6: RMSE and RMSEx using r 2m with different number of boundary and interpolation nodes. m = 2 m = 3 m = 4 n b n i RMSE 3.57E E E 4 RMSEx.25E.746E 2 6.7E 3 RMSE 2.97E E E 4 RMSEx.342E E 3.573E 3 RMSE 8.367E E E 5 RMSEx 5.482E E E 4 Table 7: Comparison of RMSE and RMSEx obtained by the MPS and MFS using r 7. MPS MFS (σ = 5) n b n i RMSE 8.37E E 4 5.7E 5 RMSEx 5.49E E 3 4.4E 4 RMSE 2.72E E 4.32E 4 RMSEx 8.83E E 4.52E 4 the MFS and MPS. As such, the MPS gains advantage since no fictitious boundary is required in the solution process. Hence, the difficulty of choosing source points can be avoided. For convergent test, we solved the same problem in a square domain. The Dirichlet and Neumann boundary conditions along the boundary are derived from the analytical solution. The nodes are evenly distributed and the dh is defined the same as in the previous examples. The results for this convergent test is shown in Figure 8. The 5

16 numbers of nodes that we used are from 3 3 to From these figures, we observe that the proposed method converged stably. RMSE n = 2 n = 3 n = 4 RMSEx n = 2 n = 3 n = dh dh Figure 8: Profiles of RMSE (left) and RMSEx (right) using different order of Conical RBFs, r 2m. 5 Concluding Remarks The derivation of close-form approximate particular solutions has been the focus of our recent research for extending the boundary methods such as the MFS or Trefftz method to solve the inhomogeneous problems. In the past, the availability of fundamental solution and particular solution are equally important for solving such inhomogeneous problems. In this paper, we propose that only the particular solution is required without the need of fundamental solution for solving above mentioned problems. This is due to the fact that the MFS converges much more rapidly than the RBFs. The MFS is most effective for solving the homogenous equations, but not for the inhomogeneous equations if RBFs are employed for the evaluation of particular solutions. However, if Chebyshev polynomial which converges exponentially is used for the evaluation of particular solution, the MFS will be effective for solving the inhomogeneous equation. For the purpose of illustrating the fundamental concept of our new proposed approach, we only deal with certain elementary differential operators. In recent years, the derivation of approximate particular solution has gone beyond the Helmholtz-type differential operators [2, 6]. The similar approach can be extended to solving more general differential equations with variable coefficients and non-linear problems. Further theoretical analysis is required to mathematically justify Ψ in (8) as the basis function for the numerical solution of the given partial differential equation. These research topics will be the focus of our future investigation. Appendix 6

17 For L =, ϕ Ψ ( r2 + c 2 4c 2 + r 2) ( r c 2 c3 3 ln c + ) r 2 + c 2 r 2m r 2m+2 ln r ln r 4(m + ) r2m+2 2 4(m + ) 3 r 2m r 2m+ (2m + ) 2 A particular solution of Ψ = x m y n, m, n, is given by [ n+2 2 ] ( ) k+ m!n!x m+2k y n 2k+2, for m n, (m + 2k)!(n 2k + 2)! k= Ψ(x, y) = [ m+2 2 ] ( ) k+ m!n!x m 2k+2 y n+2k for m < n, (m 2k + 2)!(n + 2k)! k= where [s] is the largest integer that is less than or equal to s. For L = λ 2, ϕ r 2 ln r r 4 ln r r 6 ln r Ψ 4 λ (K (λr) + ln r) r2 ln r 4 4 λ 2 λ, r > ( ( )) 4 4 λ γ + ln 4 λ 4 2 λ, r = 4 64 ( ) λ (K (λr) + ln r) r2 ln r 6 6 λ 2 λ + 2 r2 8r2 λ 96 4 λ, r > ( ( )) 6 64 λ γ + ln 96 λ 6 2 λ, r = ( ) (K λ 8 (λr) + ln r) r2 ln r 576 λ 2 λ + 36r2 + r 4 4 λ ( ) 2 2r2 4 λ 4 λ + 2 r λ, r > ( ( )) λ γ + ln 4224 λ 8 2 λ, r = 8 7

18 A particular solution for ( λ 2 )Ψ = x m y n, m, n, is given by For L = 2, [ m 2 ] [ n 2 ] Ψ(x, y) = k= l= (k + l)!m!n!x m 2k y n 2l λ 2k+2l+2 k!l!(m 2k)!(n 2l)!. References ϕ Ψ r 2m r 2m+3 (2m + ) 2 (2m + 3) 2 [] K.E. Atkinson. The numerical evaluation of particular solutions for Poisson s equation. IMA Journal of Numerical Analysis, 5:39 338, 985. [2] C.S. Chen, Sungwook Lee, and C.-S. Huang. Derivation of particular solution using chebyshev polynomial based functions. International Journal of Computational Methods, 4():5 32, 27. [3] W. Chen. Symmetric boundary knot method. Engineering Analysis with Boundary Elements, 26: , 22. [4] W. Chen and M. Tanaka. New insights into boundary-only and domain-type RBF methods. Int. J. Nonlinear Sci. & Numer. Simulation, :45 5, 2. [5] A.H.-D. Cheng. Particular solutions of laplacian, helmholtz-type, and polyharmonic operators involving higher order radial basis functions. Eng. Analy. Boundary Elements, 24:53 538, 2. [6] H.A. Cho, C.S. Chen, and M.A. Golberg. Some comments on mitigating the illconditioning of the method of fundamental solutions. Engineering Analysis with Boundary Elements, 3:45 4, 26. [7] H.A. Cho, M.A. Golberg, A.S. Muleshkov, and X. Li. Trefftz methods for time dependent partial differential equations. Computers, Materials, and Continua, : 38, 24. [8] J. Duchon. Splines minimizing rotation invariant semi-norms in Sobolev spaces: constructive theory of functions of several variables. In W. Schempp and K. Zeller, editors, Lecture Notes in Mathematics 57, pages 85. Springer-Verlag, Berlin,

19 [9] G Fairweather and A. Karageorghis. The method of fundamental solution for elliptic boundary value problems. Advances in Computatonal Mathematics, 9:69 95, 998. [] M.A. Golberg and C.S. Chen. The method of fundamental solutions for potential, Helmholtz and diffusion problems. In M.A. Golberg, editor, Boundary Integral Methods: Numerical and Mathematical Aspects, pages WIT Press, 998. [] M.A. Golberg and C.S. Chen. A mesh free method for solving nonlinear reactiondiffusion equations. Journal of Computer Modeling in Engineering & Science, 2:87 96, 2. [2] M.S. Ingber, C.S. Chen, and J.A. Tanski. A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations. Internat. J. Numer. Methods Engrg., 6:283 22, 24. [3] Z-C. Li, T-T. Lu, H-Y. Hu, and A. H-D. Cheng. Trefftz and Collocation Methods. WIT Press, 28. [4] A.S. Muleshkov, M.A. Golberg, and C.S. Chen. Particular solutions of Helmholtztype operators using higher order polyharmonic splines. Comp. Mech., 23:4 49, 999. [5] S.Y. Reutskiy, C.S. Chen, and H.Y. Tian. A boundary meshless method using chebyshev interpolation and trigonometric basis function for solving heat conduction problems. International Journal Numerical Methods in Engineering, 74:62 644, 28. [6] C.C. Tsai, A. H-D. Cheng, and C.S. Chen. Particular solutions of splines and monomials for polyharmonic and products of helmholtz operators. submitted. 9