Compactly supported radial basis functions for solving certain high order partial differential equations in 3D

Compactly supported radial basis functions for solving certain high order partial differential equations in 3D Wen Li a, Ming Li a,, C.S. Chen a,b, Xiaofeng Liu a a College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China b Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, USA Abstract Compactly supported radial basis functions have been selected as basis functions for the derivation of closed-form particular solution in the process of solving certain high order partial differential equations in 3D. The method of particular solutions and the method of fundamental solutions are employed to numerically evaluate the particular solution and homogeneous solution of a 3D problem. Two numerical examples are given to demonstrate the proposed method is applicable for handling large-scale problems. Keywords: compactly supported radial basis functions, the method of particular solutions, the method of fundamental solutions, particular solution, homogeneous solution, plate vibration equation 1. Introduction Classical and numerical solutions of plate vibrations have been an important research subject in science and engineering [11, 13, 14]. A survey for the plate vibration given by Leissa [12] is perhaps the most comprehensive source to date. Various numerical methods have been developed for solving eigenvalues and eigenmodes [5, 30] for plate vibration problems. For solving free vibration problem of plate structure, boundary meshless methods such as the method of fundamental solutions (MFS) [7, 9], boundary knot method (BKM) [6], etc. are very popular because the fundamental solution of the given homogeneous differential partial equation is available. However, when the Corresponding author Email address: liming04@gmail.com (Ming Li) Preprint submitted to Elsevier February 3, 2014

external force is applied to the plate in which an inhomogeneous term will appear in the right hand side of the differential equation, all these boundary methods lose their attractiveness due to the fact that domain discretization and integration are required. To improve the computational efficiency, much effort has been made to dealing with the issue of meshing the domain or boundary. One of the notable approaches is the dual reciprocity method (DRM) [15] which is to transfer domain integral to boundary in the implementation of boundary element methods. The success of the DRM depends on its ability to accurately approximate the inhomogeneous term. The theory of radial basis functions (RBFs) was later introduced by Golberg and Chen [9] to replace the ad-hoc basis function 1 + r which was exclusively used in the DRM at that time. A critical step in the DRM is the derivation of the particular solution associate with the RBFs and various differential operators. Being able to evaluate the particular solution, the above mentioned boundary meshless methods can be easily extended to solving inhomogeneous equations and time-dependent problems [9]. However, the evaluation of the particular solution is by no mean trivial. Before the RBFs have been introduced to the DRM, the associate particular solution is only available for Laplace and bi-harmonic operators. This was primarily due to the difficult of deriving the closed-form particular solution associate with other differential operators. The RBFs provide a rich and flexible class of basis functions that can be chosen and implemented so that the derivation of the closed-form particular solution is more likely to obtain for various types of differential operators. An important advancement in this area is the discovery of the closed-form particular solutions for Helmholtz-type operators using RBFs [4]. The new discovery opens a new research opportunity for approximating the particular solutions for a wide class of differential operators using RBFs. As a result, a number of novel numerical techniques using RBFs have been developed. We refer readers to [9, 21, 23] for further details. Despite the effectiveness of the RBFs in approximating the particular solutions, it is known that most of the RBFs are globally defined basis functions. Due to this reason, the resultant matrix using RBFs is dense and highly ill-conditioned. This poses serious stability problems and high computational cost, in particular for large-scale problems 2

in 3D where a large number of interpolation points is required. All of these issues led us to the search for locally based RBFs. In mid-1990s, the compactly supported radial basis functions (CS-RBFs) were constructed by Wu [27], Wendland [24], and later by Buhmann [1]. Due to the nature of local support, the implementation of the CS-RBFs leads to a sparse matrix in the approximation of a multivariate function or scatter data. The first application of the CS-RBFs in the context of evaluating the particular solution was established by Chen et al. [2] for solving Poisson s equation. Later, the CS-RBFs were extended to Helmholtz-type operators in 3D [10]. Ironically, the closedform particular solution for Helmholtz-type operators in 2D using CS-RBFs is still not available. CS-RBFs were also applied for solving partial differential equations in the context of RBF collocation method [8, 17, 18, 25, 26]. A review paper of the CS-RBFs for solving partial differential equations is given by Chen et al. [3]. The DRM is mainly coupled with the boundary element method to transform the domain integral to the boundary. To alleviate the domain or boundary integral, the meshless methods using RBFs or CS-RBFs have been developed through the use of the method of particular solutions (MPS) [9] which is a two-step approach for solving partial differential equations. Using the MPS, the solution of the given differential equation is split into a homogeneous solution and a particular solution. Notice that the particular solution can be obtained without the restriction of a boundary or boundary conditions, and therefore is not unique. Once a particular solution is available, the original differential equation can be reduced to a homogeneous solution subjected to the modification of the boundary conditions. Many different boundary meshless methods can be used for solving the reduced homogeneous solution. In this paper we will choose the method of fundamental solutions (MFS) for this purpose. The solution process of the MFS is rather straightforward. Our main focus on this paper is the derivation of the particular solution associate with the plate vibration problems using the CS-RBFs in 3D. Over the past few years, the particular solutions for some plate problems have been obtained [19, 20, 22]. In [29] the particular solution for plate vibration problems using polyharmonic splines has been derived in 2D and 3D. In this paper, we will present 3

the derivation of the particular solution using CS-RBFs for the same problem in 3D. Then, we will further implement the newly derived particular solution for solving plate vibration problems using the MPS and MFS. We will then compare both approaches for the advantages and disadvantages. The rest of paper is organized as follows. In Section 2, a brief review of the CS- RBFs is given. In Section 3, the method of particular solutions is introduced to split the solution into a particular solution and a homogeneous solution. Furthermore, the main solution procedure of obtaining the approximate particular solution is given. In Section 4, the method of fundamental solutions for solving homogeneous equations is introduced. In Section 5, we present the detailed derivation of the closed-form particular solution associate with CS-RBFs. In Section 6, the derived particular solutions in Section 5 have been implemented. Two numerical examples are given to verify that the derived particular solutions are indeed effective. In Section 7, we conclude with some discussion for the direction of future research. 2. CS-RBFs Radial basis functions have been very effective of constructing a multivariate function that interpolates the known data or functions. Let X = {x j } n j=1 be a set of pairwise distinct points in a domain Ω R 3 with associate values {f(x j )} n j=1. For globally supported RBFs φ such as Guassians or multiquadrics, the interpolation matrix A X = (φ( x k x j )) 1 j,k n is non-sparse and often ill-conditioned. For a large number of interpolation points, the resulting condition number can be very large, leading to stability problem and loss of numerical accuracy. Furthermore, the cost of inverting and storing A X is too high. To overcome all these difficulties, compactly supported RBFs (CS-RBFs) have been introduced as a local method. The construction of the CS-RBFs was first established by Wu [27], followed by Wendland [24], and later by Buhmann [1]. In this paper we will focus on the CS-RBFs constructed by Wendland [24]. These functions are piecewise polynomial with minimal degree in term of the given order of smoothness. The interpolation matrix A X is sparse, positive definite and thus suitable for solving large-scale problems. A list of 3D CS-RBFs is given in Table 1. In this table, 4

the cut-off function (r) + is defined to be r if r 0 and to be zero elsewhere. In Table 1, (1 r) 2 + C 0 (1 r) 4 +(4r + 1) C 2 (1 r) 6 +(35r 2 + 18r + 3) C 4 Table 1: Wendland s CS-RBFs in 3D. the radius of the support of the function has been normalized to 1. In the real application, we can re-scale the function in this table with the support of radius α using φ(r/α) for α > 0. The sparseness of the interpolation matrix A X can be suitably adjusted by choosing the scaling factor α. If α is too small, the reproduction quality is poor, while if α is too large, the matrix A X is no longer sparse and it will lost it attractiveness in real applications. Hence, a reasonable choice of the scaling factor α is crucial to compromise between the stability and quality of approximation. For solving partial differential equations, we have the freedom to choose how the interpolation points are distributed inside the solution domain. In general, we choose these interpolation points as uniformly distributed as possible if we do not have the prior knowledge of the solution. In the next section, we will give more details on how to couple the CS-RBFs in Table 1 for the evaluation of particular solutions. 3. The method of particular solutions Let Ω R 3 be a bounded domain with boundary Ω. We denote x = (x, y, z). Consider the following plate vibration problem ( 2 λ 4) u(x) = f(x), x Ω, (1) B 1 u(x) = g(x), x Ω, (2) B 2 u(x) = h(x), x Ω, (3) where B 1 and B 2 are different boundary operators, f, g, and h are known functions. By the MPS [9], the solution of (1) (3) can be split into a particular solution u p and a homogeneous solution u h ; i.e., u = u h + u p. 5

The governing equation for the particular solution is given by ( 2 λ 4) u p (x) = f(x). (4) The particular solution is not required to satisfy any particular set of boundary conditions. Hence, the particular solution is not unique. After a particular solution is obtained, the associate homogeneous solution u h can be obtained by solving the following homogeneous equation with updated boundary conditions: ( 2 λ 4) u h (x) = 0, x Ω, (5) B 1 u h (x) = g(x) B 1 u p, x Ω, (6) B 2 u h (x) = h(x) B 2 u p, x Ω. (7) There are a variety of boundary meshless methods can be used to solve the homogeneous solution. In the next section, the method of fundamental solutions (MFS) will be introduced to solve (5) (7). The solution process of using the MFS is rather straight forward when the fundamental solution of the given differential operator is known. The key issue in this paper is how to obtain a particular solution which is the most challenging part of the solution procedure. In general, the analytical particular solution u p in (4) is available only when f is simple. It is known that the direct evaluation of the particular solution for general differential operators is difficult, if not impossible. In this section we will introduce the basic concept on how to obtain the approximate particular solution. In the past RBFs have been widely adopted for this purpose. In many science and engineering problems where a large number of interpolation nodes are required for interpolation, the CS-RBFs become a method of choice. The initial step of finding the approximate particular solution is to apply CS-RBFs to approximate the forcing term f(x) in (4). Let {x i } n i=1 be a set of uniformly distributed points in Ω and denotes the Euclidean norm. Then we seek to approximate f by ˆf in the following way: f(x) ˆf(x) = n ( ri ) a i φ, x Ω, (8) α i=1 where r i = x x i, φ (r i /α) is a CS-RBF with scaling factor α. By collocation method, 6

{a i } n i=1 is determined by forcing the interpolation conditions f(x j ) = ˆf(x j ), 1 j n. Since the CS-RBF φ is positive definite, the solvability of the following system of equations n ( ) xj x i a i φ = f(x j ), 1 j n (9) α i=1 is guaranteed. Once {a i } n i=1 is determined, an approximate particular solution û p(x) can be expressed as where Φ α is a solution of û p (x) = n a i Φ α (r i ), x Ω, (10) i=1 ( 2 λ 4) ( r Φ α (r) = φ. (11) α) In order to obtain û p efficiently, Φ α (r) in (11) needs to be derived analytically which is the main task of this paper. We observe that (11) is a function of the variable r and can be converted to the ordinary differential equation. As a result, it is possible to find Φ α (r) analytically. The detailed derivation of Φ α (r) will be given in a later section. 4. The method of fundamental solutions As we have mentioned earlier, there are many boundary mesh or meshless methods for solving homogeneous solution (5)-(7) when the fundamental solution of the given differential equation is available. The MFS is known for its simplicity for solving such problems. In this section we will briefly review the MFS, in particular, for solving (5) (7). Let {η i } m i=1 be a set of source points located outside the domain. In the implementation of the MFS, the approximate homogeneous solution can be written as the linear combination of the fundamental solution [30] û h (ξ) = where r i = ξ η i, and m m b 1i G 1 (r i ) + b 2i G 2 (r i ), ξ Ω, (12) i=1 i=1 G 1 (r) = e λr 4πr, G 2(r) = e iλr 4πr 7

are respectively the fundamental solutions of ( λ 2 ) G 1 (r) = 0, and ( + λ 2 ) G 2 (r) = 0. {b 1i } m i=1 and {b 2i} m i=1 are considered as the strength of the source points to be determined. There are several approaches can be applied to find {b 1i } m i=1 and {b 2i} m i=1. For simplicity, the collocation method will be adopted for such purpose. Hence, the same number of boundary collocation points and source points are being used. Notice that û h in (12) has already satisfied the given differential equation (5). As a result, we only need to enforce û h so that it satisfies the boundary conditions (6) (7). Let {ξ j } m j=1 set of boundary collocation points. More explicitly, we have Ω be a m {b 1i B 1 G 1 ( ξ j η i ) + b 2i B 1 G 2 ( ξ j η i )} = g(ξ j ) B 1 u p (ξ j ), j = 1, 2, m, i=1 m {b 1i B 2 G 1 ( ξ j η i ) + b 2i B 2 G 2 ( ξ j η i )} = h(ξ j ) B 2 u p (ξ j ), j = 1, 2, m. i=1 There are more equations than the number of unknowns in (13) (14). Hence, a least squares method is required for solving above system of linear equations. Once {b 1i } m i=1 and {b 2i } m i=1 are determined, the approximate homogeneous solution û h can be evaluated using (12) at any given point in the domain. Despite that the MFS is very simple to set up and easy for program coding, there are a number of computational issues to be aware. One of the most important issues is how to choose the source points {η i } m i=1 (13) (14) outside the domain. In general, when the number of source points become large, we place them closer to the boundary due to the ill-conditioning of the resultant matrix of (13) (14). Otherwise, we choose these source points far from the boundary. There are many algorithms being proposed to deal this challenging issue. We refer readers to References [7, 9, 16] for more details. 8

5. Derivation of particular solutions As we have shown earlier, the approximate particular solution û p in (10) is heavily depended on the derivation of Φ α in (11) which can be explicitly written as follows ( ( 2 λ 4) 1 r ) n ( r p, 0 r α, Φ α (r) = α α) (15) 0, r > α, where p (r/α) is an appropriate polynomial of degree, say k 0, so that φ(r/α) is a C 2k CS-RBF. Since in 3D = 1 ( d r 2 d ), (16) r 2 dr dr (15) is nothing but an ordinary differential equation (ODE) with variable r. How to solve this ODE analytically is essential in obtaining the approximate particular solution. To achieve this goal, we first make a change of variable as follows Φ α (r) = w α(r), r > 0, (17) r Φ j d j ( wα ) α(0) = lim, j = 0, 1, 2. (18) r 0 + dr j r From (16), we can reformulate (15) into an ordinary differential equation with order 4; i.e., d 4 w α dr 4 λ 4 w α = ( r 1 r ) n ( r p, α α) 0 r α, 0, r > α. (19) The general solution of (19) can be obtained as follows Ae λr + Be λr + C cos λr + id sin λr + q α (r), 0 r α, w α (r) = Ee λr + F e λr + G cos λr + ih sin λr, r > α, (20) where q α (r) is a polynomial particular solution of the first equation in (19). For convenience, q α (r) can be obtained easily using symbolic solver such as Mathematica or Maple. Furthermore, there are eight coefficients to be determined in (20). They need to be chosen with care so that Φ α (r) is fourth-order differentiable, in particular at r = 0 and r = α. 9

Theorem 1. Let w α be a solution of (19) with w α (0) = 0. Then Φ α (r) defined by (17) is fourth-order continuously differentiable at r = 0 with Φ α (0) = w α(0), Φ α(0) = w α(0), Φ 2 α(0) = w α (0), 3 Φ α (0) = 0, Φ (4) α (0) = 1 ( λ 4 w 5 α(0) + p(0) ). Furthermore, Φ α (r) satisfies (15) as r 0 +. Proof. Notice that Φ α (0) = lim r 0 + w α (r)/r. Since w α (0) = 0, using L Hospital s rule, it is clear that Φ α is continuous at 0 with Φ α (0) = w α(0). Moreover, for 0 < r < α, Φ α(r) = rw α(r) w α (r) r 2. (21) Since lim r 0 +(rw α(r) w α (r)) = 0, using L Hospital s rule, we have Φ α(0) = lim r 0 + Φ α(r) = lim r 0 + w α(r) 2 = w α(0). 2 From (21), we have Φ α(r) = r2 w α(r) 2rw α(r) + 2w α (r) r 3, 0 < r < α. (22) Since w α (0) = 0, using L Hospital s rule, we have Φ α(0) = lim r 0 + Φ α(r) = lim r 0 + w α (r) 3 = w α (0). 3 From (22), we have Φ α (r) = 1 r 4 ( r 3 w α (r) 3r 2 w α(r) + 6rw α(r) 6w α (r) ), 0 < r < α. (23) Let ϕ α (r) = ( 1 r α) n p ( r α). Since wα (0) = 0, using L Hospital s rule and (19), we have Φ α (0) = lim r 0 + Φ α (r) = lim w α (4) r 0 + (r) 4 = lim r 0 + λ 4 w α + rϕ α (r) 4 = 0. Similarly, for 0 < r < α, we have Φ (4) α (r) = 1 r 5 ( r 4 w (4) α (r) 4r 3 w α (r) + 12r 2 w α(r) 24rw α(r) + 24w α (r) ) = 1 r 5 ( λ 4 r 4 w α (r) + r 5 ϕ α (r) 4r 3 w α (r) + 12r 2 w α(r) 24rw α(r) + 24w α (r) ). 10

Using L Hospital s rule, we have Φ (4) 1 ( α (0) = lim Φ(4) r 0 + α (r) = lim λ 4 r 4 w r 0 + 5r α(r) + r 4 ϕ 4 α (r) + r 5 ϕ α(r) ) λ 4 w = lim α(r) + ϕ α (r) + rϕ α(r) r 0 + 5 = λ4 w α(0) + p(0). 5 We note that our goal is to find the eight undetermined coefficients in (20). The above theorem gives us assurance that w α (r) is fourth-order continuously differentiable at r = 0 under the condition w α (0) = 0. From (20), w α (0) = 0 implies A + B + C + q α (0) = 0. (24) Next, we need to find the conditions to make sure w α (r) is fourth-order differentiable at r = α by matching the continuity of w α, w α, w α,and w α at r = α in (20). Consequently, we have the following equations (E A)e λα + (F B)e λα + (G C) cos (λα) + i(h D) sin(λα) = q α (α), (25) (A E)e λα + (F B)e λα + (C G) sin (λα) + i(h D) cos(λα) = q α(α) λ, (26) (E A)e λα + (F B)e λα + (C G) cos (λα) + i(d H) sin(λα) = q α(α), λ 2 (27) (A E)e λα + (F B)e λα + (G C) sin (λα) + i(d H) cos(λα) = q α (α). (28) λ 3 From (25), we notice that the right hand side q α (α) is a real number while the left hand side is a complex number. This implies that H = D. Subtracting (25) from (27), we have Similarly, subtracting (26) from (28), we have 2(C G) cos (λα) = q α(α) q λ 2 α (α). (29) 2(G C) sin (λα) = q α (α) q α(α) λ 3 λ. (30) From (29) and (30), it is apparent that the high order of smoothness of Φ α can only occur if q α has additional properties. For instance, third order smoothness requires ( ) ( ) q α(α) q α (α) q λ 2 α (α) sin (λα) + q α(α) cos (λα) = 0, (31) λ 3 λ 11

and fourth order smoothness requires the following additional property: (E A)e λα + (F B)e λα + (G C) cos (λα) + i(h D) sin(λα) = q(4) α (α). (32) λ 4 From (25) and (32), we have q α (4) (α) = q λ 4 α (α). (33) To obtain sufficient smoothness of Φ α (r) at r = 0 and α, the above conditions are not enough since we have eight coefficients to be determined. The additional conditions can be found by considering the singularity terms of 1 Φ α r r and Φ α. In the following examples, we will explicitly obtain these unknown coefficients. Case 1. In this case, we consider φ(r/α) = (1 r/α) 2 +. The general solution of d 4 w α r ( 1 r 2 λ 4 w dr 4 α = α), 0 r α, 0, r > α, is given by Ae λr + Be λr + C cos λr + id sin λr + q α (r), 0 r α, w α (r) = Ee λr + F e λr + G cos λr + ih sin λr, r > α, (34) (35) where From (24), we have q α (r) = 2r2 αλ r 4 λ r3 4 α 2 λ. 4 A + B + C = 0. (36) From (25) (27), it follows that D = H, (37) (E A)e λα + (F B)e λα + (G C) cos (λα) = 0, (38) (A E)e λα + (F B)e λα + (C G) sin (λα) = 0, (39) (E A)e λα + (F B)e λα + (C G) cos (λα) = 2 αλ 6. (40) According to (31), w α is continuous at r = α only if ( ) ( 2 6 sin (λα) + αλ 6 α 2 λ 7 ) cos (λα) = 0, 12

which is equivalent to For r α, tan (λα) = 3 λα. Φ α (r) = w α(r) r = A e λr r + B eλr r + C cos (λr) r sin (λr) + id + 2r r αλ 1 4 λ r2 4 α 2 λ. (41) 4 It follows that for r α, and 1 Φ α (r) r r = A (λ e λr + id ( λ r 2 cos (λr) r 2 + e λr r 3 ) + B ) sin (λr) r 3 (λ eλr r 2 ) ( eλr sin (λr) C λ + r 3 r 2 ) cos (λr) r 3 + 2 αλ 4 r 2 α 2 λ 4, (42) Φ α (r) = Aλ 2 e λr r + Bλ 2 eλr r Cλ2 cos (λr) r 2 sin (λr) idλ + 4 r αλ 4 r 6 α 2 λ. (43) 4 Notice that there are singularities in (42) and (43) that need to be removed. By Taylor series expansion, it is known that e λr = 1 λr + 1 2 λ2 r 2 1 6 λ3 r 3 + O(r 4 ), (44) cos(λr) = 1 1 2 λ2 r 2 + 1 24 λ4 r 4 + O(r 6 ), (45) sin(λr) = λr 1 6 λ3 r 3 + O(r 5 ). (46) Applying (44) (46) to (42), we can remove the singularity of 1/r, 1/r 2, and 1/r 3 as follows A ( λ 2 + 12 ) λ2 + B (λ 2 12 ) λ2 C (λ 2 12 ) λ2 + 2 αλ = 0 (47) 4 A (λ λ) + B(λ λ) + id(λ λ) = 0 (48) A B C = 0 (49) We observe that (48) is an identity and (49) is identical to (36). Hence, no useful information is available in these two equations. Next, similar to the above procedure, we can remove the singularity of 1/r in (43); i.e., Aλ 2 + Bλ 2 Cλ 2 + 4 = 0. (50) αλ4 13

However, (50) is identical to (47) and is redundant. Hence, from the de-singularization process, we only obtain one additional condition which is (50). Overall we have six algebraic equations and eight unknowns to be determined. Since the particular solution is not unique, we have the freedom of choosing two of the unknowns as the free parameters. From (36) (40) and (50) and by elementary row operation, we obtain the following augmented matrix 1 0 0 0 0 1 0 0. 0 1 0 0 0 1 0 0. 2 0 0 1 0 0 0 0 0. αλ 6 0 0 0 1 0 0 0 1. 0 0 0 0 0 1 1 0 0. 0 0 0 0 0 0 1 0 A B C D E F G H tan(λα) 1 2αλ 6 e λα 2 αλ 6 1 tan(λα) 2αλ 6 e λα (cosh(λα)+sinh(λα) tan(λα)+2) αλ 6. 1 αλ 6 cos(λα) + 2 αλ 6 The last row of (51) simply denotes the corresponding unknowns to be determined. From the above augmented matrix, we have C = 2 αλ, 6 1 G = αλ 6 cos(λα) + 2 αλ. 6 For convenience, we can choose the two free unknowns to be (51) F = H = 0. Consequently, we have A = tan(λα) 1 2αλ 6 e λα 2 αλ 6, B = 1 tan(λα) 2αλ 6 e λα, D = 0, E = (cosh(λα) + sinh(λα) tan(λα) + 2) αλ 6. 14

Furthermore, we have Φ α (0) = 1 tan(λα) + 2 αλ 5 e λα αλ 1 5 λ, 4 1 Φ α (0) r r = 1 tan(λα) 3αλ 3 e λα + 2 3αλ 3 2 α 2 λ 4, Φ α (0) = 1 tan(λα) αλ 3 e λα + 2 αλ 3 6 α 2 λ 4. Case 2. Let φ(r/α) = ( 1 r α) 4 + ( 4 r α + 1). Then q α (r), 0 r α, in (20) is given by q α (r) = 480 α 3 λ + 1800r 8 α 4 λ r 8 λ 1440r2 + 10r3 4 α 5 λ 8 α 2 λ 20r4 4 α 3 λ + 15r5 4 α 4 λ 4r6 4 α 5 λ. 4 Similar to the last case, we can obtain Φ α (r) with the following coefficients A = 30(18λα 24 tan(λα) 48eλα + λ 2 α 2 8λ 2 α 2 e λα + λ 2 α 2 tan(λα) + 24) α 5 λ 10 e λα, B = 30(18λα 24 tan(λα) + λ2 α 2 + λ 2 α 2 tan(λα) + 24) α 5 λ 10 e λα, C = 240λ2 α 2 1440 α 5 λ 10, D = 0, E = 540(1 e2λα ) α 4 λ 9 e λα 30(e2λα 8e λα + 1 + tan(λα) e 2λα tan(λα)) α 3 λ 8 e λα 720(e2λα 2e λα tan(λα) + e 2λα tan(λα) + 1) α 5 λ 10 e λα, F = 0, G = 240λ2 α 2 1440 α 5 λ 10 60λ2 α 2 1440 α 5 λ 10 cos(λα), H = 0. Moreover, we have Φ α (0) = 1 λ 4 1440eλα + 1440 tan(λα) 1440 + λα ( 1800e λα 1080 ) 1 Φ α (0) r r 60 tan(λα) 240eλα + 60 α 3 λ 7 e λα, α 5 λ 9 e λα = 20(18λα 24 tan(λα) + λ2 α 2 + λ 2 α 2 tan(λα) + 24) α 5 λ 7 e λα + 20(λ3 α 3 4λ 2 α 2 24), α 5 λ 7 α 3 λ + 1080 60 tan(λα) 1440 tan(λα) +. 5 α 4 λ6 α 3 λ 5 α 5 λ 7 Φ α (0) = 60 α 2 λ 4 180 15

6. Numerical results To verify the particular solution we derived in the last section, two numerical examples have been presented in this section. MATLAB on a Intel Core i5 in Window 7 Home Basic 64 bit. The computation were performed using In the following examples, we choose CS-RBF φ(r/α) = (1 r/α) 2 + as the basis function to approximate the inhomogeneous term of the given differential equation in 3D. We denote N the number of interpolation points inside the domain and M the number of boundary collocation points on the surface of the domain. In the implementation of the MFS, we choose the same number of source points on the surface of a fictitious sphere located outside the domain. For the purpose of identifying the neighboring points of each local support efficiently, we employ the kd-tree algorithm which is available in the Matlab Central File Exchange in the internet. To measure the accuracy, we choose N t test points {x j } Nt j=1 uniformly distributed inside the domain. The error will be evaluated based on the root-mean-square error (RMSE) which is defined as follows RMSE = 1 N t (û(x j ) u(x j )) 2 (52) N t j=1 where û(x j ) and u(x j ) are the numerical and analytical solution at the test point x j respectively. accuracy of the solution. We will study the influence of the scaling factor α with respect to the Example 1. Let Ω be an unit sphere with center at origin. In this example we consider the same problem as in [29] with λ 4 = 1000 : ( 2 λ 4) u(x, y, z) = f(x, y, z), (x, y, z) Ω, u(x, y, z) = 1 ( x 5 + y 5 + z 5), (x, y, z) Ω, 120 u(x, y, z) = g(x, y, z), (x, y, z) Ω, where f(x, y, z) and g(x, y, z) are defined based on the analytical solution u(x, y, z) = 1 ( x 5 + y 5 + z 5), (x, y, z) Ω Ω. 120 16

Figure 1: The uniformly distributed points on the unit sphere. To evaluate the approximate particular solution, we choose N = 5557 uniformly distributed interpolation points inside the sphere. For the MFS, we choose M = 400 boundary points on the surface of the unit sphere and the same number of source points on the surface of a sphere with radius 2. The uniformly distributed boundary points on the unit sphere are shown in Figure 1. To evaluate the RMSE error, we choose N t = 485 uniformly distributed internal points inside the domain. In Table 2, for various scaling factor α, we show the RMSE error, the number of nonzero entry of the sparse matrix A φ, percentage of sparseness of the A φ, and the CPU time. α RMSE(f) RMSE(u) NZ NZ(%) CPU (seconds) 0.2 1.47E 01 5.22E 01 165, 043 0.53 3.07 0.3 9.11E 02 3.25E 02 680, 923 2.20 5.77 0.4 5.85E 02 1.98E 02 1, 572, 329 5.09 5.82 0.45 4.95E 02 7.47E 03 2, 030, 465 6.57 5.91 0.46 4.83E 02 5.82E 03 2, 141, 657 6.93 6.11 0.47 4.75E 02 4.59E 03 2, 405, 369 7.78 6.28 0.48 4.68E 02 3.90E 03 2, 521, 129 8.16 6.54 0.49 4.58E 02 3.86E 03 2, 776, 825 8.99 6.72 Table 2: RMSE and sparseness for various scaling factor α. 17

From Table 2, we can see that with the increasing of scaling factor α, the accuracy improves, but in a slow pace. We believe the main reason is due to the use of low order CS-RBF φ(r/α) = (1 r/α) 2 + which is only C 0. The corresponding Φ α we derived earlier is C 4. The accuracy of the final solution is largely depended on the accuracy of approximating the inhomogeneous term f. In Table 2, we can see RMSE of f and u are correlated. Next let us compare our results with Reference [29] where polyharmonic splines up to order four (r 2n ln r, n = 1, 2, 3, 4) have been used as the basis functions. The best results obtained in [29] using 260 boundary points and 360 interior points are 1.14E 3, 1.07E 3, 3.11E 4, 2.15E 4 for polyharmonic splines of order one through four respectively. Note that polyharmonic splines of order four is C 4 which is relatively smooth. Despite of using globally supported RBFs with high smooth basis functions, the final accuracy is not much better than the results presented in Table 2. We would like to emphasize that the purposes of using global and local support of RBFs are different. The global method has higher accuracy but only applicable for small or median scale of problems due to the full size matrix is involved in the solution process. The CS-RBFs is aimed at handling large-scale problems but suffer the loss of accuracy. We do not recommend to use large α in the real applications. A much higher price of CPU time has to be paid with little improvement of accuracy. It defeats the purpose of using CS-RBFs. Example 2. In this example we consider a domain with more complicate geometry. In addition, we consider both Dirichlet and Neumann boundary conditions as follows: ( 2 λ 4) u(x, y, z) = f(x, y, z), (x, y, z) Ω, u(x, y, z) = 1 λ 4 ex+y+z, u(x, y, z) = g(x, y, z), n (x, y, z) Ω, (x, y, z) Ω, where λ = 5, f(x, y, z) and g(x, y, z) are given based on the analytical solution u(x, y, z) = 1 λ 4 ex+y+z, (x, y, z) Ω Ω. The domain Ω, as shown in Figure 2, is a two joined spheres which is defined as follows: { { ( Ω = (x, y, z) : min x 3 ) 2 (, x + 3 ) } } 2 + y 2 + z 2 < 1. 4 4 18

Figure 2: The computational domain of a two joined spheres. 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 2 1 0 1 2 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 Figure 3: The boundary collocation points ( ) and source points ( ). To evaluate the approximate particular solutions, we choose N = 7807 uniformly distributed interior points. To approximate homogeneous solution using the MFS, we choose M = 300 uniformly distributed boundary points on the surface of the domain and the same number of source points on the surface of a fictitious sphere of radius 2. The distribution of boundary and source points is shown in Figure 3. To estimate the RMSE, we choose N t = 736 uniformly distributed internal testing points. The results are shown in Table 3. We observe that we can obtain reasonable accuracy for 19

α = 0.43 in which case the density of the nonzero entries of the interpolation matrix A α is only 3.18%. Similar to the last example, RMSE of f and u are correlated. The slow convergent rate of CS-RBFs somehow affects the final accuracy of u. Since f is a known function, the RMSE of f can be used as an indicator to estimate the error of solution u. It is known that the stability of the globally supported RBFs is often an issue due to the ill-conditioning problem of the resultant matrix. In Table 3, the solution converges steadily proportional to the scaling factor α. α RMSE (f) RMSE(u) NZ NZ(%) CPU (seconds) 0.35 7.27E 02 2.94E 01 1, 209, 433 1.98 4.03 0.4 6.01E 02 3.56E 02 1, 650, 641 2.71 4.26 0.41 5.77E 02 2.23E 02 1, 740, 017 2.85 4.43 0.42 5.48E 02 1.26E 02 1, 896, 617 3.11 4.54 0.43 5.18E 02 5.62E 03 1, 940, 073 3.18 4.61 0.44 4.89E 02 4.29E 04 1, 952, 001 3.20 4.71 0.45 4.62E 02 3.47E 03 2, 299, 913 3.77 4.80 Table 3: RMSE and sparseness for various scaling factor α. 7. Conclusions In this paper we derive the closed-form particular solution in associate with the differential operator 2 λ 4 using CS-RBFs. For the numerical implementation, the method of particular solutions has been applied to split the solution into particular solution and homogeneous solution. Using the MFS to obtain the homogeneous solution and the MPS to evaluate the particular solution, our numerical procedure is simple and effective. In particular, using CS-RBFs, we have achieved the challenge of solving a fourth order of differential equation in 3D. In addition, the solution is pretty stable with respect to the scaling factor α. Despite our success in solving 3D problem, the closedform particular solution of the 2D version of plate vibration problem is not available. To the best of our knowledge, CS-RBFs have not been applied for solving the Cauchy 20

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