# Polyharmonic Multiquadric Particular Solutions for Reissner/Mindlin Plate

Authors: Chia-Cheng Tsai
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Analytical particular solutions of the polyharmonic multiquadrics  are derived for both the Reissner and Mindlin thick-plate models in a unified formulation. In the derivation, the three coupled second-order partial differential equations are converted into a product operator of biharmonic and Helmholtz operators using the Hörmander operator decomposition technique. Then a method is introduced to eliminate the Helmholtz operator, which enables the utilization of the polyharmonic multiquadrics. Then, the analytical particular solutions of displacements, shear forces, and bending or twisting moments corresponding to the polyharmonic multiquadrics are all explicitly derived. Numerical examples are carried out to validate these particular solutions. The results obtained by the present method are more accurate than those by the traditional multiquadrics and splines.


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## Body

## 1. Introduction

Boundary-type numerical methods have been emerged as a popular research field because only boundary discretizations are required when they are applied to solve homogeneous problems. These methods include the boundary element method (BEM) [1], method of fundamental solutions (MFS) [2, 3], and Trefftz methods (TM) [4, 5]. When they are applied for solving an inhomogeneous partial differential equation, the dual reciprocity method (DRM) can be used [6, 7].In the early development of DRM, the ad hoc radial function,1+r, was exclusively used. In order to improve the accuracy of the computation, Golberg and Chen [8] and Karur and Ramachandran [9] applied the theory of radial basis functions (RBFs) to the DRM. Among these RBFs, Hardy’s multiquadrics (MQ) [10] and Duchon’s augmented polyharmonic spline (APS) [11] are the most popular ones. For example, Golberg [12, 13], Chen [14], and Karur and Ramachandran [9] demonstrated the superiority of the APS over the ad hoc radial function. Then, Golberg et al. [15] further improved the accuracy of the approximated particular solution by utilizing the MQ. Recently, Tsai [16] generalized the MQ to the polyharmonic multiquadrics (PMQ) and showed further accuracy improvement of the PMQ over the traditional MQ. In this study, the accuracy improvement of the PMQ over the MQ and APS is demonstrated for the Reissner and Mindlin thick-plate models.When the MQ is adopted in a DRM procedure, the applicability depends on the availability of the analytical particular solution of the basis function associated with the partial differential operator of a given problem. Golberg et al. [15] derived the analytical particular solutions for the Laplace equation. Samaan and Rashed [17, 18] and Tsai and Hsu [19] found the analytical particular solutions, respectively, for the two- and three-dimensional elasticity problems, which can be converted to a biharmonic equation using the Galerkin-Papkovich vector [20]. Basically, the applicability of MQ is limited to the harmonic and biharmonic operators [21] before Tsai [16] derived the analytical particular solutions of MQ associated with the polyharmonic operators, which are denoted as the PMQ in the following.When applying the DRM for solving the Mindlin thick-plate model [22], Wen et al. [23] and Tsai and Wu [24] found the analytical particular solutions for the ad hoc RBF and the APS, respectively. In addition, the analytical particular solutions for the Chebyshev polynomials [25] and the APS [26] were derived for the Reissner thick-plate model [27, 28]. Basically, a product operator of biharmonic and Helmholtz operators resulted when applying the Hörmander operator decomposition technique [29] to the coupled PDEs of the Reissner and Mindlin thick-plate models as demonstrated in the previous studies [24–26]. This seems to infer that the accuracy improvement of the PMQ over the APS is not possible for the thick-plate models.In this study, a method is introduced to eliminate the Helmholtz operator, which makes the application of the PMQ for the thick-plate models possible. Numerical experiments are carried out to demonstrate the superior accuracy of the PMQ over the traditional MQ and the first few orders of the APS.This paper is organized as follows: the unified Reissner/Mindlin plate model is introduced in Section2. Then, a review of the PMQ is given in Section 3. And the particular solutions of the PMQ associated with the Reissner/Mindlin plate model are derived in Section 4. Section 5 gives the formulation of MFS-DRM. Finally, some numerical experiments are carried out to validate the particular solutions in Section 6 and the conclusions are drawn in Section 7.


## 2. Reissner/Mindlin Plate Model

In the following, indicesi,j,k are in the range {1,2,3} and indices α,β,γ are in the range {1,2}. As described in Figure 1, we consider a plate of uniform thickness h with its middle plane being a domain Ω with boundary Γ in the x1-x2 plane and thickness coordinate x3. The plate is subjected to a normal loading with intensity q(x1,x2). Then, the equations of equilibrium are given by
(1)∂Qα∂xα+q=0,∂Υαβ∂xβ-Qα=0,
where Qα is the transverse shear force and Υαβ is the bending or twisting moment.Figure 1
Configuration of a thick plate.In the Reissner [27, 28] and Mindlin [22] thick-plate theory, uα represents an average slope angle and u3 is the lateral deflection of the plate in the middle surface. According to the theories, the constitutive equations are given by
(2)Qα=D1-νλ22uα+∂u3∂xα,Υαβ=D1-ν2∂uα∂xβ+∂uβ∂xα+2ν1-ν∂uγ∂xγδαβ+εν(1-ν)λ2qδαβ
with
(3)λ=πh,ε=0
for the Mindlin plate model and
(4)λ=10h,ε=1
for the Reissner plate model. Combining (1)~(2) results in the following governing equation:
(5)L~u1u2u3=-E~q
with(6)L~=D(1-ν)2(∇2-λ2)+D(1+v)2∂2∂x12D(1+v)2∂2∂x1∂x2-Dλ2(1-ν)2∂∂x1D(1+v)2∂2∂x1∂x2D(1-ν)2(∇2-λ2)+D(1+v)2∂2∂x22-Dλ2(1-ν)2∂∂x2Dλ2(1-ν)2∂∂x1-Dλ2(1-ν)2∂∂x2Dλ2(1-ν)2∇2,E~=εν(1-ν)λ2∂∂x1εν(1-ν)λ2∂∂x21,where D and v are Young’s modulus and Poisson ratio, respectively.Equation (5) is a coupled system of three second-order PDEs for the three unknown functions uj, and, therefore, three boundary conditions are required as
(7)B1·u=H1x1,x2,B2·u=H2x1,x2,B3·u=H3x1,x2,
where Hix1,x2 are given boundary data.


## 3. Polyharmonic Multiquadrics

Before deriving the particular solutions of PMQ associated with the Reissner/Mindlin plate model, we give a short review on the PMQ previously derived by the method of undetermined coefficients together with the Laurent series in [16]. Considering the Lth order PMQ ψL, it is governed by the following hierarchical relation:
(8)∇2ψL=ψL-1,(9)ψ0=c2+r2,(10)∇2iψL=0asr⟶0for0≤i≤L-1.
Here, (10) is introduced so that the PMQ are hierarchically unique and c is a shape parameter to tune the numerical accuracy [10].Then, by using the method of undetermined coefficients, we can find the solution of (8)~(10) as
(11)ψLc2L+1=∑j=0LAL,jR2j1+R2+∑j=0L-1BL,jR2jln⁡1+1+R22+∑j=0L-1CL,jR2j
with
(12)R=rc
for any positive integer L. And, the unknown CL,i in (11) can be solved by using the uniqueness condition (10) as
(13)CL,i=-∑j=0iPL,i,j+∑j=0i-1QL,i,jfor0≤i≤L-1
with
(14)PL,i,j=-1i-j-12i-2j-3!!i-j!2i-jAL,j,QL,i,j=-1i-j+12i-2j-1!!i-j!2i-j+1i-jBL,j.
In (14), the double factorial is defined as
(15)n!!=n·n-2⋯5·3·1nisanoddpositiveintegern·n-2⋯6·4·2nisanevenpositiveinteger1n=0,-1-1n+1/2(-n-2)!!n≤-3isanoddnegativeinteger.
The PMQ defined in (11) can be expanded into the Maclaurin series as
(16)ψLc2L+1=∑i=L∞∑j=0LPL,i,j+∑j=0L-1QL,i,jR2i.
Observing (16), we can conclude that the proposed PMQ are infinitely differentiable since they only consist of even-coefficient polynomials [16, 30]. Then, substituting (16) into (8) and (9) and performing some mathematical manipulations can result in
(17)∑j=0LPL,i+L,j4Li+L!2i!2+∑j=0L-1QL,i+L,j4Li+L!2i!2=-1i-12i-3!!i!2iforallnonnegativeintegeri,
which can be used for solving the unknown coefficients AL,j and BL,j in (11) and (16). In practice, we simply enforce i equal to 0,1,…,2L in (17) which results in 2L+1 linear equations for solving the 2L+1 unknown coefficients AL,j and BL,j of the PMQ. After AL,j and BL,j are obtained, (13) can be used to have CL,j. Then, the PMQ ψL can be computed simply by (11). For the cases when r is very small or equal to zero, one can alternatively use the Maclaurin series (16).For the purpose of a later derivation of the particular solutions associated with the Reissner/Mindlin plate model, we need to find the formulas when the radial differential operatord/rdrM is applied onto (11) and (16). This can be done for arbitrary nonnegative integer M since the proposed PMQ are infinitely differentiable. Then, the required formulas for (11) and (16) can be derived, respectively, as
(18)drdrMψLc2L-2M+1=∑j=0LAL,jVM,jR+∑j=0L-1BL,jWM,jR+∑j=0L-1CL,jUM,jR,drdrMψLc2L-2M+1=∑i=L∞∑j=0LPL,i,j+∑j=0L-1QL,i,jUM,iR
with
(19)UM,jR≡dRdRMR2j=2j!!2j-2M!!R2j-2M,forj≥M≥0,0,forM>j≥0,-1M-2j+2M-2!!-2j-2!!R2j-2M,forj<0,(20)VM,jR≡dRdRMR2j1+R2=∑k=0Mk!M-k!M!UM-k,jRdRdRk1+R2,(21)WM,jR≡dRdRMR2jln⁡1+1+R22=∑k=0Mk!M-k!M!UM-k,jRdRdRkln⁡1+1+R22.
In deriving (18), we have used
(22)c2MdrdrM=dRdRM.
And, in deriving (20) and (21), we have used the product rule of differentiation as follows:
(23)dRdRMfg=∑k=0Mk!M-k!M!dRdRM-kfdRdRkg
with f and g being two arbitrary radial functions. Finally, additional formulas are required to complete (20) and (21), respectively, as follows:
(24)dRdRk1+R2=-1k-12k-3!!1+R22k-1,(25)dRdRkln⁡1+1+R22=1R2-1R21+R2,fork=1,Uk-1,-1R-∑l=0k-1l!k-1-l!k-1!Uk-1-l,-1RdRdRl11+R2,fork>1
with
(26)dRdRl11+R2=-1l2l-1!!1+R22l+1.The proposed formulas can be implemented into a floating-point subroutine for evaluatingd/rdrMψLr with arbitrary nonnegative integers M and L. Therefore, the subroutine can be invoked when implementing numerical methods. In addition, the source codes can be obtained from the author via e-mails.


## 4. Analytical Particular Solution of Plate Model

Analytical particular solutions are required when the DRM is applied for solving a Reissner/Mindlin plate problem under an arbitrary loading. In this section, the analytical particular solutions of the PMQ associated with the Reissner/Mindlin plate model will be derived, which are governed by(27)L~u1u2u3=-E~ψLr,
where
(28)r=x12+x22.Formally, the particular solutionsujx1,x2, governed by (27), should be obtainable by using the Hörmander operator decomposition technology [29], which begins with the definition of the adjoint operator L~adj as
(29)L~adjL~=L~L~adj=det⁡L~I~,
where det⁡(L~) is the determinant of L~ and I~ is the three-by-three identity matrix. Here, both det⁡(L~) and L~adj can be obtained by linear algebraic manipulations, respectively, defined as(30)det⁡L~=D3λ21-ν24∇2-λ2∇2∇2,(31)L~adj=2∇4-1+ν∇2+1-νλ2∂2∂x12-1+ν∇2+1-νλ2∂2∂x1∂x21-ν∂∂x1∇2-λ2-1+ν∇2+1-νλ2∂2∂x1∂x22∇4-1+ν∇2+1-νλ2∂2∂x221-ν∂∂x2∇2-λ2-1-ν∂∂x1∇2-λ2-1-ν∂∂x2∇2-λ2∇2-λ22∇2-1-νλ2λ2.Then, the particular solutions can be found by assuming
(32)u1x1,x2u2x1,x2u3x1,x2=L~adjE~Fr,
where Fr is an unknown function to be determined. In (32), the required operator L~adjE~ can be obtained as
(33)L~adjE~=D21-ν4∂∂x1∇2-λ2εν∇2+1-νλ2D21-ν4∂∂x2∇2-λ2εν∇2+1-νλ2D21-ν4∇2-λ22-εν∇2-1-νλ2.After substituting (32) into (27) and using (29) and (30), the resulted equation indicates that the following particular solution is required:
(34)-D3λ21-ν24(∇2-λ2)∇2∇2F=ψL.
However, the analytical particular solution of the above equation is not known due to the existence of the Helmholtz operator.In this study, we find that the Helmholtz operator can be eliminated by rewriting (33) as
(35)L~adjE~=D3λ21-ν24∇2-λ2E~1,
where the operator E~1 is defined by
(36)E~1=1Dλ21-ν∂∂x1εν∇2+1-νλ2∂∂x2εν∇2+1-νλ22-εν∇2-1-νλ2.
Then, (29) and (35) can be combined to have
(37)L~L~adjE~=-D3λ21-ν24∇2-λ2∇2∇2E~⟹L~D3λ21-ν24∇2-λ2E~1=-D3λ21-ν24∇2-λ2∇2∇2E~⟹L~E~1=-∇2∇2E~.
The above equations have suggested us to modify the Hörmander operator decomposition technology by assuming
(38)u1x1,x2u2x1,x2u3x1,x2=E~1F1r,
where F1r is an unknown function to be determined. Substituting (38) into (27) and using (37) result in
(39)∇2∇2F1r=ψL.
By using the PMQ definitions (8) and (9), we can have the particular solution of (39) as
(40)F1r=ψL+2.
Substituting (40) into (38) and using (36) can result in the desired analytical particular solutions of displacements as
(41)uαx1,x2=-xαDλ2(1-ν)·ενdrdrψL+1+(1-ν)λ2drdrψL+2,(42)u3x1,x2=-(2-εν)ψL+1+(1-ν)λ2ψL+2Dλ2(1-ν),
where we have used
(43)∇2F1=∇2ψL+2=ψL+1.
In addition, the required formulas of dψL+1/rdr and dψL+2/rdr have been provided in the previous section. Then, the corresponding shear forces and bending/twisting moments can be obtained by using the definitions
(44)Qαx1,x2=D1-νλ22uαM+∂u3M∂xα,(45)Υαβx1,x2=D1-ν2∂uαM∂xβ+∂uβM∂xα+2ν1-ν∂uγM∂xγδαβ+δαβενr2+c2λ21-ν.
Equations (41), (42), (44), and (45) are sufficient to obtain the analytical particular solutions of shear forces and bending/twisting moments, respectively, as follows:
(46)Qαx1,x2=-xαdrdrψL+1,(47)Υαβx1,x2=-δαβεν2-νr2+c2+ν1-νλ2F2rλ21-ν+1λ2ενdrdrψL+1εν2-νr2+c2+ν1-νλ2F2rλ21-ν+(1-ν)λ2drdrψL+2-xαxβλ2ενdrdr2ψL+1+(1-ν)λ2drdr2ψL+2,
where the additional terms d/rdr2ψL+1 and d/rdr2ψL+2 have been given in the previous section.This completes the derivation of the analytical particular solutions of the PMQ associated with the Reissner/Mindlin plate model.


## 5. MFS-DRM Formulation

Now, we are in a position to review the application of the MFS-DRM procedure [24, 26] for solving the well-posed thick-plate problem governed by (5) and (7). First of all, we need the principle of superposition as
(48)uj=ujh+ujp,
where the particular solution ujp satisfies
(49)L~u1pu2pu3p=-E~q
without specifying any boundary condition. In addition, the homogeneous solution ujh satisfies
(50)L~u1hu2hu3h=000
with modified boundary conditions
(51)B1·uh=H1x1,x2-B1·upB2·uh=H2x1,x2-B2·upB3·uh=H3x1,x2-B3·up,
where uh=(u1h,u2h,u3h) and up=(u1p,u2p,u3p).In order to apply the DRM for solving the particular solutionujp, the loading intensity needs to be first approximated by the PMQ as follows:
(52)qx1,x2≅∑n=1NanψLrn,
where rn is the distance between x1,x2 and the nth DRM field point, ξn1,ξn2, as depicted in Figure 2. In order to solve the unknown coefficient an, (52) should be collocated on the N DRM field points as
(53)qξl1,ξl2=∑n=1NanψLrln
for l=1,2,…,N and rln is the distance between the lth and nth DRM field points. Then, the corresponding particular solution can be approximated by
(54)ujp≅∑n=1Nanujrn
with the particular solutions ujr given by (41) and (42). Also, the particular solutions of shear force and bending or twisting moment can be approximated, respectively, as
(55)Qαp≅∑n=1NanQαrn,(56)Υαβp≅∑n=1NαnΥαβrn.
In (55) and (56), the particular solutions Qα and Υαβ are given in (46) and (47), respectively.Figure 2
Field points of the DRM.After the particular solutions are approximated by the DRM, the modified boundary conditions (51) become well defined. Therefore, it is ready to use the MFS for solving the homogeneous solution. Formally, the homogeneous solution can be approximated by
(57)ujh≅∑k=13∑m=1Mbmkujk*x1,x2;sm1,sm2,
where ujk*(x1,x2;s1,s2) are the fundamental solutions defined by(58)L~u11*u12*u13*u21*u22*u23*u31*u31*u33*=-δ(x1-s1,x2-s2)000δ(x1-s1,x2-s2)000δ(x1-s1,x2-s2)and sm1,sm2 are M source points outside the plate domain as described in Figure 3. In (57), the 3M unknowns bmk can be solved by collocating the modified boundary conditions (51) on M boundary points. More details of the MFS and the explicit fundamental solutions can be found in [24, 26]. After both the particular and homogeneous solutions are solved, the principle of superposition (48) can be applied for obtaining the desired solution.Figure 3
Boundary and source points of the MFS.


## 6. Numerical Results

In order to validate the derived particular solutions, three numerical cases are considered. Typically,D=λ=2 and ν=0.3 are selected in these numerical studies. Furthermore, the MFS is always arranged to be sufficient and accurate such that the numerical errors are majorly controlled by the DRM. The root mean square errors (RMSEs) of the numerical solutions are defined as
(59)∑i=13∑l=1L-uNUMi,l-uANAi,l23L-,
where uNUMi,l is the numerical solution obtained by the MFS-DRM at the lth point of the L- considered positions and uANAi,l is the corresponding analytical solution of the problem. Here, L- will be chosen to be sufficiently large to ensure the utilization of the RMSEs.In our numerical experiments, we consider the following Reissner/Mindlin plate problem:(60)L~u1u2u3=-DE~sinx1x2+sinh⁡x2+x2sinh⁡x2.
This problem has been considered for both the Mindlin [24] and the Reissner [26] cases and their analytical particular solutions can be unified as
(61)u1=cos⁡x148x2λ2ν-1+εν-61+x22λ2ν-11+x22+1+2x22+x2ενcosh⁡x2+9+x29+2x23+x2λ2ν-19+x29+2x23+x2λ2+121+x2ενsinh⁡x2·48λ21-ν-1,u2=sinx148λ2-ε+λ2ν+3+x23-2x23+x2λ2·λ2ν-1+121+x2ενcosh⁡x2+3-λ2ν-1λ2+22x22+x2-1ενsinh⁡x2·48λ2ν-1-1,u3=sinx1482+λ21-ν-εvx2+6λ2ν-11+x22λ2+εv-21+2x22+x2cosh⁡x2-12εv-21+x2λ2+λ2ν-19+x29+2x23+x22+λ21-ν-εv·sinh⁡x2·48λ21-ν-1.Example 1.
First, we consider a Reissner plate defined in-1,12 subjected to clamped boundary conditions, which are set according to the analytical solutions in (61).The particular solutions of the present problem are solved by the DRM based on the APS [26], Chebyshev polynomials [25], and PMQ. Table 1 gives the RMSEs for the solutions obtained by the DRMs with different bases and N. Basically, the solutions obtained by the Chebyshev polynomials are the most accurate and the application is straightforward if the arbitrary loadings are given by functions. However, when the loadings are measured on some scattered points, a two-stage method [31] is required which is relatively more difficult compared to the direct application of either APS or PMQ. In the table, the superior accuracy of the MQ over the first few orders of the APS can be observed. In addition, the accuracy improvement for the numerical solutions obtained by the high-order PMQ over the traditional MQ can also be observed. Overall, the best accuracy of the high-order PMQ is close to that of the high-order APS while the former RBF does not need augmented polynomials. In practice, it can be cumbersome to implement the particular solutions of the high-order augmented polynomials and thus the superiority of the PMQ over the APS becomes more relevant.Table 1
The RMSEs for Example1 (Div for divergence).
N = 6 × 6
N = 8 × 8
N = 10 × 10
N = 12 × 12
Chebyshev
2.10E − 06
8.62E − 09
2.18E − 09
3.81E − 10
MQ
6.10E − 06
4.30E − 07
4.30E − 07
2.74E − 08
PMQ (L = 2)
6.04E − 06
2.17E − 07
2.05E − 08
5.94E − 09
PMQ (L = 4)
4.51E − 06
1.20E − 07
5.12E − 09
1.66E − 09
APS (order = 1)
2.22E − 04
9.38E − 05
4.96E − 05
2.85E − 05
APS (order = 3)
3.71E − 05
9.08E − 06
3.02E − 06
1.49E − 06
APS (order = 5)
4.88E − 06
7.35E − 07
1.90E − 07
5.90E − 08
APS (order = 7)
Div
8.14E − 08
1.27E − 08
3.20E − 09
APS (order = 9)
Div
Div
2.44E − 09
4.20E − 10
APS (order = 11)
Div
Div
Div
3.81E − 10For both MQ and PMQ, the optimal shape parametersc are searched as demonstrated in Figure 4. Basically, the numerical accuracy improves with increasing c value. However, as c is continuously made larger, the interpolation matrix becomes ill-conditioned [32]. Then, a typical error contour is given in Figure 5 for the solution obtained by the MFS-DRM based on the MQ. In the figure, the larger errors are on the boundary as expected.Optimal shape parameter searches of Example1 for (a) N=8×8 and (b) N=10×10.
(a)
(b)Figure 5
Error contour for Example1 with N=10×10 and c=3.Finally, the RMSE comparison for different numbers of nodes is given in Figure6, in which higher resolution gives better accuracy. Overall, these results should have validated the polyharmonic multiquadric particular solutions of uir.Figure 6
RMSE comparisons of Example1 for different numbers of nodes.Example 2.
Then, we consider a similar Mindlin plate problem by changing the boundary conditions to the free boundary condition on one side. Table2 addresses the RMSEs for the numerical results in this example. The results are very similar except that the errors are slightly larger due to the numerical approximations on the free boundary condition.Table 2
The RMSEs for Example2.
N = 6 × 6
N = 8 × 8
N = 10 × 10
N = 12 × 12
Chebyshev
5.36E − 06
7.98E − 08
4.43E − 08
1.07E − 08
MQ
5.51E − 05
1.03E − 05
1.42E − 06
6.88E − 07
PMQ (L = 2)
4.97E − 05
5.07E − 06
5.51E − 07
1.15E − 07
PMQ (L = 4)
3.12E − 05
3.30E − 06
1.64E − 07
4.21E − 08
APS (order = 3)
1.85E − 03
4.77E − 04
1.64E − 04
6.88E − 05
APS (order = 5)
1.11E − 04
1.13E − 05
2.27E − 06
6.49E − 07
APS (order = 7)
Div
4.20E − 06
7.06E − 07
1.63E − 07
APS (order = 9)
Div
Div
5.66E − 08
1.20E − 08
APS (order = 11)
Div
Div
Div
1.08E − 08In addition, the optimal shape parameter searches and the error comparison for different numbers of nodes are given in Figures7 and 8, respectively. The results are similar to the previous example. Therefore, the polyharmonic multiquadric particular solutions of Qαr and Υαβ(r) are also validated.Optimal shape parameter searches of Example2 for (a) N=8×8 and (b) N=10×10.
(a)
(b)Figure 8
RMSE comparisons of Example2 for different numbers of nodes.


## 7. Discussions

In this paper, analytical particular solutions of the polyharmonic multiquadrics associated with Reissner and Mindlin thick plates in bending are derived. In the derivations, the coupled system of the three second-order governing equations is transformed into a single sixth-order product operator of biharmonic and Helmholtz operators. Then, a method is introduced to eliminate the Helmholtz operator, which enables the use of the polyharmonic multiquadric. Numerical results are performed to validate the derived particular solutions and to demonstrate the superiority of the polyharmonic multiquadrics over the traditional multiquadrics and the augmented polynomial splines.



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