1. A new implementation of the finite collocation method
for time dependent PDEs
Fariba Takhtabnoos, Ahmad Shirzadi n
Department of Mathematics, Persian Gulf University, Bushehr, Iran
a r t i c l e i n f o
Article history:
Received 3 August 2014
Received in revised form
30 September 2015
Accepted 9 November 2015
Keywords:
Local meshless methods
Radial basis functions
Finite collocation method
Time dependent PDEs
Burger's equation
a b s t r a c t
This paper is concerned with a new implementation of a variant of the finite collocation (FC) method for
solving the 2D time dependent partial differential equations (PDEs) of parabolic type. The time variable is
eliminated by using an appropriate finite difference (FD) scheme. Then, in the resultant elliptic type
PDEs, a combination of the FC and local RBF method is used for spatial discretization of the field vari-
ables. Unlike the traditional global RBF collocation method, dividing the collocation of the problem in the
global domain into many local regions, the method becomes highly stable. Furthermore, the computa-
tional cost of the method is modest due to using strong form equation, collocation approach and that the
matrix operations require only inversion of matrices of small size. Different approaches are investigated
to impose Neumann's boundary conditions. The test problems consist of three linear convection–diffu-
sion–reaction equations and a 2D nonlinear Burger's equation. An iterative approach is proposed to deal
with the nonlinear term of Burger's equation.
& 2015 Elsevier Ltd. All rights reserved.
1. Introduction
Meshless methods are very attractive and effective for solving
boundary value problems, because they involve simple pre-
processing, arbitrary node distribution and flexibility of placing
nodes at arbitrary locations, straightforward adaptive refinement,
versatility in solving large deformation and also have the high
order continuity and the ability to treat the evolution of non-
smooth solutions, which is very useful to solve PDE problems.
Many variants of these methods have been proposed and applied
for a wide range engineering and sciences. From a point of view,
meshless methods can be classified into two groups, one using
strong form and another using weak form of governing equations.
Another interesting and useful classification of these methods is a
classification based on the trial approximation and testing
approach, see [1]. Trial approximation says what kind of basis
functions are used to approximate the unknown solution u by
uðxÞ %
P
λiΦiðxÞ. To obtain the unknowns λi, the above approx-
imation should be tested to satisfy the governing equations. Test-
ing can be done using strong form of the governing equations and
collocation approach [2–4], global weak form of the equations and
Galerkin approaches [5,6], local weak form of governing equations
such as meshless local Petrov–Galerkin methods [7–18] or a
combination of them [19–24]. For the purpose of this paper, a brief
introduction to the original strong form RBF collocation method is
required. Let us consider a PDE in the form of:
Lu ¼ f in Ω; and u ¼ g on ∂Ω; ð1:1Þ
where ∂Ω denotes the boundary of the bounded domain Ω in Rd
.
Both given functions f and g : Rd
-R are sufficiently smooth. For
any set of N scattered nodal points in the domain and on the
boundary represented by Ξ ¼ fξkg
N
k ¼ 1, the unknown solution u is
approximated via
uðxÞ % uNðxÞ ¼
X
ξi A Ξ
λiΦiðxÞ; ð1:2Þ
where ΦiðxÞ, i ¼ 1; 2; …; N, are called shape functions constructed
on the set of nodal points Ξ and λi is the unknown coefficient at
node i to be determined. To solve for the N unknowns λ1; …; λN,
the approximation (1.2) is tested to satisfy in the strong form of
the governing equation and boundary condition, Eq. (1.1) at a set of
N scattered nodal points in the domain and on the boundary called
test centers which is generally different from that of RBF centers as
follows:
ðLuÞðxiÞ ¼ f ðxiÞ )
XN
j ¼ 1
LΦjðxiÞ
À Á
λj ¼ f ðxiÞ: ð1:3Þ
This yields a system of N linear equations for the N unknown
coefficients, AΛ ¼ b. More test equations than RBF centers results
in overdetermined system of equations called unsymmetric
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/enganabound
Engineering Analysis with Boundary Elements
http://dx.doi.org/10.1016/j.enganabound.2015.11.007
0955-7997/& 2015 Elsevier Ltd. All rights reserved.
n
Corresponding author.
E-mail addresses: f.takhtabnoos@sutech.ac.ir (F. Takhtabnoos),
shirzadi.a@gmail.com, shirzadi@pgu.ac.ir (A. Shirzadi).
Engineering Analysis with Boundary Elements 63 (2016) 114–124

2. meshless methods first studied in [1]. Although this method is
very simple to be implemented and is flexible with regard to the
enforcement of arbitrary boundary conditions, the problem is the
ill-conditioning of the coefficient matrix A; when the collocation
point xi approaches another collocation point xj, the ith row of the
coefficient matrix A approaches to its jth row and ill-conditioning
happens. For some numerical experiments on the condition
number of the interpolation matrices for RBF see [25]. It is
remarkable that for the Laplace equation with N¼100 number of
nodes in the unit square domain and multiquadric RBFs as basis,
condition number of the final matrix A is about 1015
. Several
adaptive approaches have been proposed to overcome the pro-
blem of ill-conditioning, see for example [26–30]. Meshless local
RBF method first proposed in [31] suggests a local strategy which
result in well-conditioned and banded coefficient matrices. By this
method, to approximate the unknown solution u at an arbitrary
point x, we consider a local region Ωx around this point and derive
the collocation in this subdomain. Finally all equations derived by
local collocation are assembled to obtain the final system of
algebraic equations. Besides this distinct characteristic of locali-
zation, in the context of meshless typed approximation strategies,
other major advantages of the local strategy include: (i) the exis-
tence of the shape functions is guaranteed provided that all the
nodal points within an influence domain are distinct; (ii) the
constructed shape functions strictly satisfy the Kronecker delta
condition; (iii) the computational cost is modest and the matrix
operations require only inversion of matrices of small size which is
equal to the number of nodes inside the influence domain. This
method is briefly explained in the next section. For useful appli-
cation of local RBF collocation method the interested readers are
referred to [32,33]. The finite collocation (FC) method proposed in
[34] is another meshless local strategy for which, besides collo-
cation of unknown solution in local subdomains, the PDE gov-
erning operator is also enforced in the local collocation systems.
This approach is also more explained in the next section. The
proposed local approaches simulate the steady problems. Due to
importance of time dependent equations in science and engi-
neering, in the current work, we propose a meshless local strategy
for the numerical solution of general initial-boundary value time-
dependent problem of the following type:
∂uðx; tÞ
∂t
¼ L½uðx; tÞŠþf ðx; tÞ; in Ω;
uðx; 0Þ ¼ gðxÞ;
B½uðx; tÞŠ ¼ hðx; tÞ; on ∂Ω; ð1:4Þ
where the operators L and B are linear partial differential operators
on the domain Ω and on the counter ∂Ω, describing the governing
equation and boundary conditions respectively. The operator B
corresponds to the unit operator when Dirichlet conditions are
prescribed, the normal derivative for Neumann conditions, and a
combination of the two in the case of Robin or mixed conditions.
Note that Eq. (1.4) contains diffusion, convection–diffusion and
convection–diffusion–reaction equations. A meshless method for
the numerical computation of the solution of steady Burger's-type
equations based on the global RBF collocation method in combi-
nation with the Newton method is proposed in [35]. The organi-
zation of this paper is as follows: Section 2 briefly describes the
local RBF and FC strategies for discretizing steady equations. Dis-
cretization of the above-mentioned equation is described in
Section 3. Test problems are considered in Section 4. Our conclu-
sions and suggestions for further research all are summarized in
the final section.
2. Meshless local RBF approximations for steady problems
In this section, we briefly review two variants of local meshless
collocation methods using RBF approximations for steady pro-
blems; namely meshless local RBF method introduced by Lee et al.
[31] and meshless FC method proposed by Stevens et al. [34]. These
methods are briefly described in the following two subsections.
2.1. Local RBF approximations for steady problems
To illustrate the local RBF method, we consider the following
steady PDE
LðuðxÞÞ ¼ f ðxÞ; in Ω;
BuðxÞ ¼ hðxÞ; on ∂Ω; ð2:1Þ
where L is a linear partial differential operator and B is the
boundary operator. In the local RBF method, we consider a set of N
scattered nodal points in the domain and on the boundary
represented by Ξ ¼ fξkg
N
k ¼ 1. Then, to approximate the unknown
solution u at an arbitrary point xk; k ¼ 1; …; N, we consider a local
region Ωx around this point, called the domain of influence of
point x, covering a number of n nodal points as shown in Fig. 1. In
the next section this cover will be called also an stencil. Then we
have:
uðxÞ ¼
Xn
i ¼ 1
λiϕiðxÞ; ð2:2Þ
where ϕiðxÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
JxÀxi J2
þc2
q
, n is the number of nodal points
fallen within the influence domain Ωx of x. The c term is known as
the “shape parameter”, and describes the relative width of the RBF
functions about their centers. To obtain the values of the coeffi-
cients λi, we can first evaluate Eq. (2.2) at all nodal points
xi; i ¼ 1; 2; …; n, in each cover. Then the following system of linear
equations will be obtained:
U ¼ ΦΛ; ð2:3Þ
in which
U ¼ ½uðx1Þ; uðx2Þ; …; uðxnÞŠT
and Λ ¼ ½λ1; λ2; …; λnŠT
:
It can be proved that the matrix Φ is non-singular such that ðΦÞÀ 1
can always be computed provided that ca0 and all the nodal
points are distinct points. Upon computing ðΦÞÀ 1
; the vector Λ
can be obtained by writing
Λ ¼ ðΦÞÀ 1
U:
The approximated function, uðxÞ, can now be expressed in terms of
Fig. 1. Influence domain Ωx of a node x embracing 7 neighboring nodes ðn ¼ 7Þ.
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124 115

3. the nodal values at each cover. That is
uðxÞ ¼ JðxÞT
ðΦÞÀ1
U ¼ ΨðxÞT
U ¼
Xn
i ¼ 1
ψiðxÞuiðxÞ: ð2:4Þ
where JðxÞ ¼ ½ϕ1ðxÞ; ϕ2ðxÞ; …; ϕnðxÞŠT
and ΨðxÞ ¼ ðΦÞÀ 1
J ¼ ½ψ1;
ψ2; …; ψnŠT
. The functions ψi; i ¼ 1; 2; …; n, are called the shape
functions for the local RBF interpolation. Note that the shape
functions may be obtained by solving the linear system
ðΦÞT
ΨðxÞT
¼ JðxÞT
; ð2:5Þ
and therefore, it is not need to explicitly compute the matrix
inverse ðΦÞÀ 1
. Once the shape functions of the local RBF are
defined, since the matrix ðΦÞÀ 1
dose not contain any term of x, it
is a trivial task to compute their partial derivatives:
∂Ψ
∂x
¼
∂ψ1
∂x
;
∂ψ2
∂x
; …;
∂ψn
∂x
!T
¼ ðΦÞÀ 1∂JðxÞ
∂x
;
where
∂J
∂x
¼
∂ϕ1
∂x
;
∂ϕ2
∂x
; …;
∂ϕn
∂x
!T
:
The derivative with respect to y (first or higher derivatives) can be
computed in a similar way. Finally, corresponding each node a
local equation will be obtained and all these equations should be
assembled in a global final system. When assembling the local
equations in the final global system, ith row of the global matrix is
the n-vector ½0; …; μðψ1Þ; 0; …; μðψ2Þ; 0; …; μðψnÞ; 0; …; 0Š where μð
ψiÞ is equal to either LðψiÞ or BðψiÞ, depending on which operator
acts on ψi. Note that the number of columns, which μðψkÞ is
located, is the global number of node k in ith stencil. In other
words, this row is the extension of vector ½μðψ1Þ; μðψ2Þ; …; μðψnÞŠ
by patching zeros into entries associated with the nonselected
nodes in the Ωxi
. For more details on local RBF meshless methods
see [31].
2.2. Finite collocation method for steady problems
This method operates on a set of scattered nodes that are
placed within the solution domain and on the solution boundary,
forming a small RBF collocation system around each internal node.
Unlike other meshless local RBF formulations for which the solu-
tion is driven by collocation of unknown solution in a local region,
in the FC method the solution of the PDE is driven by collocation of
unknown solution and PDE governing and boundary operators
within the local systems, resulting in a sparse global collocation
system. So, in the FC method the PDE governing operator is also
enforced in the local collocation systems. To go more in detail,
again consider a set of N ¼ N1 þN2 nodal points for which N1 of
them are in Ω and the remaining N2 are located on the boundary.
Around each xl AΩ; l ¼ 1; …; N1, we choose n neighboring points
contained in the local region Ωx, called a stencil centered at xl and
xl is called a centerpoint. In this paper, solution center is a nodal
point for which the collocation of unknown solution value is done.
Similarly, PDE center and boundary center are nodal points for
which the collocation of PDE governing operator and PDE
boundary operator are done respectively. The nodal point around
which a local system is formed is called the local system cen-
terpoint. Around each strictly interior domain node, a stencil is
generated; Fig. 2 shows 3 Â 3 and 5 Â 5 stencil that are used in FC
method. Therefore a series of N1 RBF local collocation systems are
formed as follows:
ΦðlÞ
ΛðlÞ
¼ d
ðlÞ
; l ¼ 1; 2; …; N1; ð2:6Þ
here ΦðlÞ
represents the collocation matrix for system l, and is
composed as follows. Each local system collocates the unknown
solution value around the periphery of its local domain (solution
centers). At nodes interior to the stencil, including the centerpoint
itself, the governing PDE operator is imposed (PDE centers). Note
that if the stencil intersects the domain boundary, it replaces the
solution centers with collocation of appropriate domain boundary
condition (boundary centers).
Note that here ΦðlÞ
is different from that constructed by Eq. (2.3).
The vector d
ðlÞ
contains the known boundary and PDE operator
values, and also some unknown values of the solution field at the
solution centers. These solution centers lie on the periphery of the
local system domain, and act as a local Dirichlet boundary condition
for collocation system l. Therefor we can obtain the vector Λ as
ΛðlÞ
¼ ðΦðlÞ
ÞÀ1
d
ðlÞ
: ð2:7Þ
By using Eq. (2.2) at the system centerpoint, xðlÞ
c , and applying (2.7), we
obtain
uðlÞ
ðxðlÞ
c Þ ¼ JðlÞ
ðxðlÞ
c ÞΛðlÞ
¼ JðlÞ
ðxðlÞ
c ÞðΦðlÞ
ÞÀ1
d
ðlÞ
¼ WðlÞ
ðxðlÞ
c Þd
ðlÞ
; ð2:8Þ
where WðlÞ
ðxðlÞ
c Þ ¼ JðlÞ
ðxðlÞ
c ÞðΦðlÞ
ÞÀ 1
is a stencil weights vector. Similar to
Eq. (2.5), the weights vector may be obtained by solving the linear
system
ðΦÞT
½WðlÞ
ðxðlÞ
c ÞŠT
¼ ½JðlÞ
ðxðlÞ
c ÞŠT
:
Therefore, by performing Eq. (2.8) for the centerpoint of each local
system l, a series of N1 simultaneous equations are formed for the N1
unknown values of u at the system centerpoints. So, by assembling the
Fig. 2. 3 Â 3 and 5 Â 5 FC stencil. Diamonds represent collocation of the unknown solution value (solution centers). Stars represent collocation of the PDE governing operator
(PDE centers). Filled circles indicate the system centerpoint; the node around which the stencil is formed, and at which the global assembly is performed. Squares denote
boundary nodes. The stencil which labeled with number 3 is a boundary stencil centered at a boundary point.
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124116

4. weight vectors for each of the local systems, a sparse global system is
formed. By solving this sparse matrix, the value of the solution field
will be obtained at interior nodes.
3. Finite collocation method for time dependent PDEs
For solving the time dependent problems of type (1.4) an FD
scheme is used to discretize the time variable. Then, the variations of
the field variables in the resulting elliptic type problems are dis-
cretized by a combination of the local RBF and FC approaches. The FD
approximation of the time derivatives in the θ-method is given as
follows:
θ _uk þ 1
þð1ÀθÞ _uk
¼
uk þ 1
Àuk
Δt
; 0rθr1:
Considering Eq. (1.4) at time instants kΔt and ðkþ1ÞΔt and then
using the θ-method we have
uk þ1
ðxÞÀuk
ðxÞ
Δt
¼ θL½uk þ1
ðxÞŠþð1ÀθÞL½uk
ðxÞŠþFk
ðxÞ; ð3:1Þ
where uk
ðxÞ ¼ uðx; kΔtÞ and Fk
ðxÞ ¼ θf ðx; ðkþ1ÞΔtÞþð1ÀθÞf ðx;
kΔtÞ. Setting θ ¼ 1
2 in Eq. (3.1) we obtain
uk þ1
ðxÞÀ
Δt
2
ðL½ukþ 1
ðxÞŠ ¼ uk
ðxÞþ
Δt
2
L½uk
ðxÞŠþ
Δt
2
þΔtFk
: ð3:2Þ
Similar to Section 2.2, again we consider a set of N ¼ N1 þN2 nodal
points for which N1 of them are in Ω and the remaining N2 are located
on the boundary. Around each xl AΩ; l ¼ 1; …; N1, we choose n
neighboring points contained in the local region Ωx, called a stencil
centered at xl. For locally approximating the field variables u on the
subdomain Ωx we consider
uk
l ðxÞ ¼
Xn
i ¼ 1
λk
l;iϕiðxÞ; ð3:3Þ
where λk
l;i is a real constant. Since we consider that the solution value
is known at time level k, from Section 2.1 the field variables at time
level k can be discretized by
L½uk
ðxÞŠ ¼ L½ΨðxÞT
ŠUk
;
and the right hand side of Eq. (3.2) is discretized as follows:
uk
ðxÞþ
Δt
2
L½uk
ðxÞŠþΔtFk
¼ ΨðxÞT
þ
Δt
2
L½ΨðxÞT
Š
Uk
þΔtFk
: ð3:4Þ
In a stencil l and at time level kþ1, the field variables are approxi-
mated by:
uk þ1
l ðxÞ ¼
Xn
i ¼ 1
λk þ1
l;i ϕiðxÞ: ð3:5Þ
By substituting Eqs. (3.4) and (3.5) in Eq. (3.2) we have:
Xn
i ¼ 1
λk þ 1
l;i ϕiðxÞÀ
Δt
2
Xn
i ¼ 1
λk þ 1
l;i L½ϕiðxÞŠ ¼ ΨðxÞT
þ
Δt
2
L½ΨðxÞT
Š
Uk
þΔtFk
:
ð3:6Þ
After discretization of the right hand side of Eq. (3.6) and considering
Uk
is known, the full discretization of Eq. (3.6) can be done by using
the FC approach.
In fact, Eq. (3.6) is collocated at the PDE center (the point
marked with star in Fig. 2). For the nodal point which is located on
the periphery of this stencil and does not belong to the boundary
∂Ω (called solution center xj), the unknown solution values are
collocated as follows:
uk þ1
l ðxjÞ ¼
Xn
i ¼ 1
λk þ 1
l;i ϕiðxjÞ: ð3:7Þ
For a nodal point which is located on the periphery of this stencil
and also belongs to the boundary ∂Ω (called a boundary point xd,
or boundary center denoted by squares in Fig. 2), the corre-
sponding boundary condition in Eq. (1.4) is collocated as follows:
Buk þ 1
l ðxdÞ ¼
Xn
i ¼ 1
λkþ 1
l;i BϕiðxdÞ: ð3:8Þ
where B denotes the corresponding boundary condition in (1.4)
(Dirichlet or Neumann). Combining Eqs. (3.6)–(3.8) result in a
series of N1 local square systems of equations as follows:
ΦlΛk þ1
l ¼ dl; l ¼ 1; 2; …; N1; ð3:9Þ
Now the coefficients Λk þ 1
l can be obtained from
Λk þ 1
l ¼ ðΦlÞÀ1
dl: ð3:10Þ
By using Eq. (3.5) at the system centerpoint, xc, we can write
uk þ 1
l ðxcÞ ¼ JT
ðxcÞΛk þ1
l : ð3:11Þ
Therefore, by performing Eq. (3.11) for the centerpoint of each
local system l, one equation is obtained. So, a series of N1 simul-
taneous equations are formed for the N1 unknown values of u at
the system centerpoints. Now, the boundary nodes can be treated
by each of the following ways:
(a) As proposed in [34], we can solve these N1 number of
equations to obtain N1 solution values at internal nodes. Then, in
the case of Neumann's boundary condition, one should interpolate
to obtain solution values at boundary points.
(b) Corresponding each boundary node we can also consider
one stencil (for example the stencil which labeled with number
3 in Fig. 2) centered at that boundary point and the corresponding
BC is imposed at centerpoint. Considering that we have N2
boundary points, we have an N Â N system of equations for N
unknowns for which N ¼ N1 þN2. For a boundary point xd, the
following equation is imposed at centerpoint:
Buk þ 1
l ðxdÞ ¼
Xn
i ¼ 1
λkþ 1
l;i BϕiðxdÞ:
In the next section, it is shown that this approach has the highest
accuracy among the others.
(c) In the case of regular nodes, corresponding boundary nodes, we
add N2 number of equations to the final system obtained by using FD
discretization for imposing Neumann's boundary condition. For FD
discretization, while one can use forward or backward differences
which are of order O(h), in the current work we use the following FD
schemes which are of order Oðh
3
Þ (h is the difference between two
consecutive nodes)
∂uk
ðx; yÞ
∂x
x ¼ a
¼
1
h
À
11
6
uk
ða; yÞþ3uk
ðaþh; yÞÀ
3
2
uk
ðaþ2h; yÞ
þ
1
3
uk
ðaþ3h; yÞ
;
∂uk
ðx; yÞ
∂x
x ¼ b
¼
1
h
11
6
uk
ðb; yÞÀ3uk
ðbÀh; yÞþ
3
2
uk
ðbÀ2h; yÞ
À
1
3
uk
ðbÀ3h; yÞ
;
∂uk
ðx; yÞ
∂y
y ¼ a
¼
1
h
À
11
6
uk
ðx; aÞþ3uk
ðx; aþhÞÀ
3
2
uk
ðx; aþ2hÞ
þ
1
3
uk
ðx; aþ3hÞ
;
∂uk
ðx; yÞ
∂y
y ¼ b
¼
1
h
11
6
uk
ðx; bÞÀ3uk
ðx; bÀhÞ
þ
3
2
uk
ðx; bÀ2hÞÀ
1
3
uk
ðx; bÀ3hÞ
; ð3:12Þ
where the domain is ða; bÞ2
. So, we can obtain the value of u at all the
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124 117

5. nodes without using the current interpolation matrix systems and
updated data-vector, d; which is the strategy used in [34].
In both approaches (b) and (c) we finally obtain the solution values
in the domain and its boundary simultaneously by solving the final
global system, but Ref. [34] first obtains the solution values on the
internal nodes and then uses the local systems to obtain the solutions
on the boundary. In both cases (b) and (c) the stencils corresponding
to the internal nodes are considered exactly the same as [34].
4. Numerical demonstration
Test problems we consider in this section consist of diffusion,
convection–diffusion–reaction and a 2D nonlinear Burger equa-
tion. Accuracies of the numerical results are measured by infinity
norm error ðJeJ1Þ or root mean square (RMS) error defined by:
JeJ1 ¼ fjuNðziÞÀun
ðziÞj; zi AZg;
and
RMS ¼
X
zi AZ
ðuNðziÞÀun
ðziÞÞ2
jZj
!1=2
;
where uN is the numerical solution, un
is the exact solution and Z is
the number of testing nodes.
4.1. Test problem 1
For the first test problem we consider the following convection–
diffusion–reaction equation with variable coefficients on the domain
Ω ¼ ½0; π
2Š Â ½0; π
2Š which is also considered in [36]. The boundary
conditions are considered to be of both Neumann and Dirichlet type:
taking the initial value
uðx; y; 0Þ ¼ 1þ sin ðc1xÞ sin ðc2yÞ;
where the exact solution is given by:
uðx; y; tÞ ¼ 1þeÀt
sin ðc1xÞ sin ðc2yÞ;
provided that c1 and c2 are odd integers; the numerical results in this
example are obtained with c1 ¼ c2 ¼ 1. To see the convergence of the
proposed method with respect to the time step Δt, Table 1 is pre-
sented. The results of this table are RMS errors obtained at time
instant t¼1, with N ¼ 112
number of nodal points and different time
step Δt. Increasing accuracy can be seen by decreasing the time step
Δt until Δt ¼ 0:05. After that cancelation error affects the accuracy
suggesting Δt ¼ 0:05 as best value of the time step. Also, in order to
see the spatial convergence of the proposed method, Table 2 is
presented. The results of this table are obtained at time instant t¼1
with Δt ¼ 0:05 and different number of nodal points N. By going
through each column of Table 2 one can see increasing accuracy with
increasing number of nodal points. Throughout this paper, Cond
stands for the condition number of global matrix.
To compare the results obtained with three approaches of boundary
treatments mentioned at the end of Section 3 see Fig. 3. This figure
presents the RMS error versus the number of the nodal points N at time
instant t¼0.5 obtained with two kinds of stencils and three approaches
of Section 3. Comparisons are made with two kinds of stencils, namely
3Â 3 and 5Â 5 stencils. No remarkable differences can be seen when
using stencils with small size while the differences are more clear
when using 5Â 5 stencils. The results reveal that using boundary
stencils centered at boundary points to impose BC gives the highest
accuracy. Considering the results of this test problem, we use 5Â 5
stencil, approach b and Δt ¼ 0:05 in all of the following test problems,
otherwise it will be stated. Also, we investigate the effect of varying the
time variable t on RMS error. Fig. 4 summarizes these results.
4.2. Test problem 2. Rotating cone
As second test problem, we consider a diffusion equation with
source
∂u
∂t
¼ Δuþf ;
over a unit square ðx; yÞA½0; 1Š Â ½0; 1Š. The Dirichlet condition
along the whole domain boundary, initial condition and the source
term f are extracted from the given exact solution
uðx; y; tÞ ¼ 0:8 expðÀ80½ðxÀrðtÞÞ2
þðyÀsðtÞÞ2
ŠÞ;
where rðtÞ ¼ 1=4ð2þ sin ðπtÞÞ and sðtÞ ¼ 1=4ð2þ cos ðπtÞÞ. We inves-
tigate first the effect of varying the time instant. Table 3 shows the
RMS error at different time instants. As we expected the error is not
decreasing. The initial and numerical solutions with representation of
error profiles at time instants t¼0.5, t¼0.75 and t¼1.0 are given in
Fig. 5 which is obtained with N ¼ 612
number of nodal points.
4.3. Test problem 3
As third test problem we consider a convection–diffusion
equation
∂u
∂t
¼ aΔuÀb Á ∇u;
Table 1
Numerical results obtained at time instant t¼1, with N ¼ 112
number of nodal
points and different time steps Δt for Ex. 1.
Δt 3 Â 3 stencil 5 Â 5 stencil
0.2 5:4236 Â 10À3
2:8885 Â 10À 3
0.1 1:3021 Â 10À3
8:1188 Â 10À 4
0.05 1:4102 Â 10À3
2:6747 Â 10À 4
0.02 1:4102 Â 10À3
2:4510 Â 10À 2
0.01 3:4102 Â 10À3
6:002 Â 10À 1
∂u
∂t
¼
1
c2
1 þc2
2
ΔuÀ
1
c1
sin ðc1xÞux À
1
c2
sin ðc2yÞuy þð cos ðc1xÞþ cos ðc2yÞÞuÀ cos ðc1xÞÀ cos ðc2yÞ ðx; yÞA 0;
π
2
2
;
uðx; 0; tÞ ¼ 1 0rxr
π
2
;
uy x;
π
2
; t
¼ 0 0rxr
π
2
;
uð0; y; tÞ ¼ 1 0ryr
π
2
;
ux
π
2
; y; t
¼ 0 0ryr
π
2
;
8
:
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124118

6. over the square ðx; yÞA½0; 2Š Â ½0; 2Š. The Dirichlet boundary con-
dition along the whole domain boundary and the initial condition
are extracted from the given exact solution
uðx; y; tÞ ¼
1
1þ4t
exp
ÀðxÀb1tÀ0:5Þ2
að1þ4tÞ
À
ðyÀb2tÀ0:5Þ2
að1þ4tÞ
!
;
where a¼0.01 and b ¼ ðb1; b2Þ ¼ ð0:8; 0:8Þ which is also considered
in [37]. As we see in Fig. 6 the initial condition is a Gaussian pulse
with unit height centered at x ¼ 0:5; y ¼ 0:5. In Table 3 we show the
RMS errors at some time instants t¼0.5, t¼1.0, t¼1.5 and t¼2 with
N ¼ 612
number of nodal points. Fig. 7 represents the approximate
solutions and corresponding contours at time instants t¼0.5, t¼1.0,
t¼1.5 and t¼2 with representation of error profile obtained with
N ¼ 612
number of nodal points. Considering Figs. 6 and 7, the
convection in the direction of the vector b can be seen clearly.
4.4. Nonlinear test problem: Burger's equation
The final test problem simulates the 2D unsteady nonlinear
Burger's equation
∂u
∂t
þuux þuuy ¼ αΔu: ð4:13Þ
in the region Ω ¼ ½0; 1Š Â ½0; 1Š. Initial and Dirichlet boundary
conditions are extracted from the exact solution [38]:
uðx; y; tÞ ¼
1
1þeðx þy À tÞ=2α;
where α ¼ 1
R and R is Reynold's number. Similar to the time dis-
cretization process given in Section 3, considering Eq. (4.13) at
time instants kΔt and ðkþ1ÞΔt and then using the θ-method we
have:
uk þ 1
ðxÞþ
Δt
2
uk þ 1
x ðxÞþuk þ1
y ðxÞ
uk þ 1
ðxÞÀα
Δt
2
Δuk þ 1
ðxÞ
¼ uk
ðxÞÀ
Δt
2
uk
xðxÞþuk
yðxÞ
uk
ðxÞþα
Δt
2
Δuk
ðxÞ:
To handle the nonlinear term uk þ 1
x ðxÞþuk þ1
y ðxÞ
uk þ 1
ðxÞ, we
Table 2
Numerical results at time instant t¼1 with Δt ¼ 0:05 and different number of nodal points for Ex. 1.
N 3 Â 3 stencil 5 Â 5 stencil
RMS JeJ1 Cond RMS JeJ1 Cond
82
2:7108 Â 10À 3
5:0120 Â 10À 3 20.3916 9:8268 Â 10À4
1:6154 Â 10À 3 20.4025
92
2:1130 Â 10À 3
4:0213 Â 10À 3 20.3019 6:0466 Â 10À4
9:8808 Â 10À 4 20.2894
102
1:7149 Â 10À 3
3:1710 Â 10À 3 20.2997 3:9291 Â 10À4
6:4257 Â 10À 4 20.2621
112
1:4102 Â 10À 3
2:6002 Â 10À 3 20.3926 2:6747 Â 10À4
4:3603 Â 10À 4 20.3081
122
1:1021 Â 10À 3
2:1001 Â 10À 3 20.9025 1:8958 Â 10À4
3:0568 Â 10À 4 20.4206
132
9:4704 Â 10À 4
1:8124 Â 10À 3 23.5068 1:3927 Â 10À4
2:2115 Â 10À 4 20.5974
142
8:0820 Â 10À 4
1:5410 Â 10À 3 26.6132 1:0567 Â 10À4
1:6895 Â 10À 4 20.8400
152
6:9804 Â 10À 4
1:3021 Â 10À 3 30.0070 8:2605 Â 10À5
1:4422 Â 10À 4 21.1502
162
6:0921 Â 10À 4
1:2013 Â 10À 3 33.6679 6:6866 Â 10À5
1:4572 Â 10À 4 21.5254
172
5:3655 Â 10À 4
1:0120 Â 10À 3 37.5893 1:0483 Â 10À4
4:3462 Â 10À 4 21.9488
0 200 400 600 800 1000
10
6
10
5
10
4
10
3
10
2
N
RMS
a,5× 5 stencil
b,5× 5 stencil
c,5× 5 stencil
a,3× 3 stencil
b,3× 3 stencil
c,3× 3 stencil
Fig. 3. RMS error versus N for Ex. 1 obtained with 3 Â 3 and 5 Â 5 stencils and
different approaches to impose boundary conditions.
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x 10
t
RMS
Fig. 4. RMS error versus t for Ex. 1.
Table 3
Numerical results obtained at different time instants with N ¼ 612
number of nodal
points for Exs. 2 and 3.
t Ex. 2 Ex. 3
RMS error RMS error
0.5 5:4236 Â 10À 6
8:6014 Â 10À 6
1.0 1:3021 Â 10À 5
2:3021 Â 10À 5
1.5 3:4102 Â 10À 5
6:4102 Â 10À 5
2.0 9:8827 Â 10À 5
8:8827 Â 10À 5
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124 119

7. 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
x
t=0
y
u(x,y,0)
0
0.2
0.4
0.6
0.8
1 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
y
t=0.5
x
u(x,y,0.5)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
x 10
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
x
t=0.75
y
u(x,y,0.75)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
x 10
0
0.2
0.4
0.6
0.8
1 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
y
t=1
x
u(x,y,1)
0
1
2
3
4
5
x 10
Fig. 5. Profiles of approximate solution for Test problem 2 with error representation.
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
x
t=0
y
u(x,y,0)
x
y
t=0
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 6. Initial solution and its contour plot for Ex. 3.
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124120

8. 0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
x
approximate t=0.5
y
u(x,y,0.5)
0
1
2
3
4
5
6
x 10
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
x
approximate t=1
y
u(x,y,1)
0
1
2
3
x 10
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
x
approximate t=1.5
y
u(x,y,1.5)
0
2
4
6
8
10
12
14
16
18
x 10
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
x
approximate t=2
y
u(x,y,2)
0
0.2
0.4
0.6
0.8
1
1.2
x 10
x
y
approximate t=0.5
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.05
0.1
0.15
0.2
0.25
x
y
approximate t=1
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
x
y
approximate t=1.5
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.02
0.04
0.06
0.08
0.1
0.12
x
y
approximate t=2
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fig. 7. Profiles of the approximate solution (left) and corresponding contour plots (right) at different time instants for Ex. 3.
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124 121

9. propose an iterative procedure in each time step by replacing
uk þ1;0
x and uk þ 1;0
y by ux
k
and uy
k
, respectively, at the zeroth itera-
tion. In other words, we solve the following equation:
uk þ1;mþ 1
ðxÞþ
Δt
2
uk þ 1;m
x ðxÞþuk þ1;m
y ðxÞ
uk þ 1;mþ 1
ðxÞ
Àα
Δt
2
Δuk þ 1;mþ1
ðxÞ ¼ uk
ðxÞÀ
Δt
2
uk
xðxÞþuk
yðxÞ
uk
ðxÞþα
Δt
2
Δuk
ðxÞ;
where m stands for the mth iteration. In a stencil, the field
variables are approximated by:
uk þ 1;mþ 1
l
ðxÞ ¼
Xn
i ¼ 1
λk þ 1
l;i ϕiðxÞ: ð4:14Þ
So, for PDE centers the following equation is collocated:
Xn
i ¼ 1
λk þ 1
l;i ϕiðxÞþ
Δt
2
uk þ 1;m
x ðxÞþuk þ1;m
y ðxÞ
Xn
i ¼ 1
λk þ1
l;i ϕiðxÞþα
Δt
2
Xn
i ¼ 1
λk þ 1
l;i ΔϕiðxÞ ¼ uk
ðxÞÀ
Δt
2
uk
xðxÞþuk
yðxÞ
uk
ðxÞþα
Δt
2
Δuk
ðxÞ:
The numerical results obtained with different Δt and number of
iterations are presented in Table 4. The results show that the effect of
iteration is more clear for biggest Δt. This table suggests using one
iteration with the time step Δt ¼ 0:05 at our computations. Approx-
imate solutions with α ¼ 0:05 at time instants t¼0.5, t¼0.75, t¼1 and
t¼1.25 are presented in Fig. 8 with distribution of error profile. Fig. 9
shows the cross section of the approximate solutions at x¼0.4 and
time instant t¼1 with different α. The cross sections of the approx-
imate and exact solutions at x¼0.4 with α ¼ 1 and α ¼ 0:05 at time
instants t ¼ 0:5, 1, 1.5 and 2 are presented in Fig. 10. To see the con-
vergence of the approximate solution by increasing the number of
nodal points, N, Fig. 11 is presented. This figure is obtained by con-
sidering different values of the parameter α.
Table 4
The results obtained for JeJ1 at time instant t¼1 with different Δt, N ¼ 112
and
different number of iterations for Ex. 4.
m Δt ¼ 0:02 Δt ¼ 0:05 Δt ¼ 0:1
0 3:3196 Â 10À 5
3:4707 Â 10À 5
3:7394 Â 10À 5
1 3:2938 Â 10À 5
3:2926 Â 10À 5
3:3154 Â 10À 5
2 3:3001 Â 10À 5
3:3195 Â 10À 5
3:3863 Â 10À 5
3 3:3001 Â 10À 5
3:3196 Â 10À 5
3:3867 Â 10À 5
4 3:3001 Â 10À 5
3:3196 Â 10À 5
3:3865 Â 10À 5
0
0.2
0.4
0.6
0.8
1 0
0.5
1
0
0.2
0.4
0.6
0.8
1
y
t=0.5
x
u(x,y,0.5)
0
0.5
1
1.5
2
x 10
0
0.2
0.4
0.6
0.8
1 0
0.2
0.4
0.6
0.8
1
0
0.5
1
y
t=0.75
x
u(x,y,0.75)
0
0.5
1
1.5
2
2.5
3
3.5
x 10
0
0.2
0.4
0.6
0.8
1 0
0.5
1
0
0.5
1
y
t=1
x
u(x,y,1)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 10
0
0.2
0.4
0.6
0.8
1 0
0.2
0.4
0.6
0.8
1
0
0.5
1
y
t=1.25
x
u(x,y,1.25)
0
1
2
3
4
5
x 10
Fig. 8. Profiles of approximate solutions at different time instants with representation of error for Ex. 4.
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124122

10. 5. Conclusions
2D time dependent PDEs were considered to be solved with a
local strong form meshless method called the finite collocation
method. The time variable was eliminated by using an appropriate
finite difference (FD) scheme. Then the resulting elliptic type PDEs
were discretized by a meshless local RBF method at time level k
and a finite collocation approach at time level kþ1. Both Dirichlet
and Neumann's type boundary conditions were considered. The
test problems consist of three linear convection–diffusion–reac-
tion equations and a 2D nonlinear Burger's equation while an
iterative approach was proposed to deal with the nonlinear term
of Burger's equation. It is well-known that the final matrix of
global RBF collocation approach is highly ill-conditioned. In the
local collocation methods such as FC method, the final matrix is
well conditioned, but again we have the ill-conditioning of the
local systems. However, since the local systems are of small size,
the ill conditioning of the local system is remarkably better than
its global counterpart. The numerical results showed the effec-
tiveness of the method and we suggest the application of the
method for more complicated problems of physics.
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
u(0.4,y,1)
t=1
α=0.05
α=0.1
α=0.5
α=1.0
Fig. 9. Cross section of approximate solution at x¼0.4 and time instant t¼1 with
different values of α for Ex. 4.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
y
u(0.4,y,t)
app, t=0.5
ex, t=0.5
app, t=1.0
ex, t=1.0
app, t=1.5
ex, t=1.5
app, t=2.0
ex, t=2.0
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
u(0.4,y,t)
app, t=0.5
ex, t=0.5
app, t=1.0
ex, t=1.0
app, t=1.5
ex, t=1.5
app, t=2.0
ex, t=2.0
Fig. 10. Cross section of approximate solution (marker) and exact solution (lines) at x¼0.4 and different time instants with α ¼ 1 (left) and α ¼ 0:05 (right) for Ex. 4.
0 500 1000 1500 2000 2500
10
10
10
10
N
RMS
t=1
α=0.5
α=0.75
α=1.0
α=1.5
α=2
Fig. 11. RMS error versus the number of nodal points, N, obtained with different α
for Ex. 4.
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124 123

11. Acknowledgments
This project was supported by the Research Council of the
Persian Gulf University.
References
[1] Schaback R. Unsymmetric meshless methods for operator equations. Numer
Math 2010;114:629–51.
[2] Hosseini VR, Chen W, Avazzadeh Z. Numerical solution of fractional telegraph
equation by using radial basis functions. Eng Anal Bound Elem 2014;38:31–9.
[3] Sun L, Chen W, Zhang C. A new formulation of regularized meshless method
applied to interior and exterior anisotropic potential problems. Appl Math
Model 2013;37(1213):7452–64.
[4] Yao G, Tsai C, Chen W. The comparison of three meshless methods using radial
basis functions for solving fourth-order partial differential equations. Eng Anal
Bound Elem 2010;34(7):625–31.
[5] Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. Int J Numer
Methods Eng 1994;37(2):229–56.
[6] Lu Y, Belytschko T, Gu L. A new implementation of the element free Galerkin
method. Comput Methods Appl Mech Eng 1994;113(34):397–414.
[7] Atluri S, Zhu T. A new meshless local Petrov–Galerkin (MLPG) approach in
computational mechanics. Comput Mech 1998;22(2):117–27.
[8] Atluri S, Zhu T. A new meshless local Petrov–Galerkin (MLPG) approach to
nonlinear problems in computer modeling and simulation. Comput Modeling
Simul Eng 1998;3:187–96.
[9] Sladek J, Sladek V, Solek P, Zhang C. Fracture analysis in continuously non-
homogeneous magneto-electro-elastic solids under a thermal load by the
MLPG. Int J Solids Struct 2010;47(10):1381–91.
[10] Sladek J, Sladek V, Zhang C. Stress analysis in anisotropic functionally graded
materials by the MLPG method. Eng Anal Bound Elem 2005;29(6):597–609.
[11] Sladek J, Sladek V, Solek P, Pan E. Fracture analysis of cracks in magneto-
electro-elastic solids by the MLPG. Comput Mech 2008;42(5):697–714.
[12] Mirzaei D, Dehghan M. MLPG approximation to the p-Laplace problem.
Comput Mech 2010;46(6):805–12.
[13] Dehghan M, Mirzaei D. Meshless local Petrov–Galerkin (MLPG) method for the
unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall
conductivity. Appl Numer Math 2009;59(5):1043–58.
[14] Mirzaei D, Dehghan M. Meshless local Petrov–Galerkin (MLPG) approximation
to the two dimensional sine-Gordon equation. J Comput Appl Math 2010;233
(10):2737–54.
[15] Ching H-K, Batra RC. Determination of crack tip fields in linear elastostatics by
the meshless local Petrov–Galerkin (MLPG) method. CMES: Comput Model
Eng Sci 2001;2(2):273–89.
[16] Shirzadi A. Meshless local integral equations formulation for the 2d
convection-diffusion equations with a nonlocal boundary condition. CMES:
Comput Model Eng Sci 2012;85(1):45–63.
[17] Ahmad Shirzadi. Numerical simulations of 1D inverse heat conduction pro-
blems using overdetermined RBF-MLPG method. Commun Numer Anal
2013;2013:1–11. http://dx.doi.org/10.5899/2013/cna-00172.
[18] Shirzadi A, Sladek V, Sladek J. A meshless simulations for 2d nonlinear
reaction–diffusion Brusselator system. CMES: Comput Model Eng Sci
2013;95(4):259–82.
[19] Abbasbandy S, Shirzadi A. A meshless method for two-dimensional
diffusion equation with an integral condition. Eng Anal Bound Elem 2010;34
(12):1031–7.
[20] Abbasbandy S, Shirzadi A. MLPG method for two-dimensional diffusion
equation with Neumann's and non-classical boundary conditions. Appl Numer
Math 2011;61:170–80.
[21] Abbasbandy S, Sladek V, Shirzadi A, Sladek J. Numerical simulations for cou-
pled pair of diffusion equations by MLPG method. CMES: Comput Model Eng
Sci 2011;71(1):15–37.
[22] Shirzadi A, Ling L. Convergent overdetermined-RBF-MLPG for solving second
order elliptic PDEs. Adv Appl Math Mech 2013;5(1):78–89.
[23] Shirzadi A, Sladek V, Sladek J. A local integral equation formulation to solve
coupled nonlinear reaction-diffusion equations by using moving least square
approximation. Eng Anal Bound Elem 2013;37(1):8–14.
[24] Shirzadi A, Ling L, Abbasbandy S. Meshless simulations of the two-
dimensional fractional-time convection–diffusion–reaction equations. Eng
Anal Bound Elem 2012;36:1522–7.
[25] Boyd JP, Gildersleeve KW. Numerical experiments on the condition number of
the interpolation matrices for radial basis functions. Appl Numer Math
2011;61(4):443–59.
[26] Libre NA, Emdadi A, Kansa EJ, Shekarchi M, Rahimian M. A multiresolution
prewavelet-based adaptive refinement scheme for RBF approximations of
nearly singular problems. Eng Anal Bound Elem 2009;33(7):901–14.
[27] Libre NA, Emdadi A, Kansa EJ, Shekarchi M, Rahimian M. A fast adaptive
wavelet scheme in RBF collocation for nearly singular potential PDEs. CMES:
Comput Model Eng Sci 2008;38(3):263–84.
[28] Ling L, Hon YC. Improved numerical solver for Kansa's method based on affine
space decomposition. Eng Anal Bound Elem 2005;29(12):1077–85.
[29] Ling L, Kansa EJ. Preconditioning for radial basis functions with domain
decomposition methods. Math Comput Model 2004;40(13):1413–27.
[30] Ling L, Schaback R. An improved subspace selection algorithm for meshless
collocation methods. Int J Numer Methods Eng 2009;80(13):1623–39.
[31] Lee CK, Liu X, Fan SC. Local multiquadric approximation for solving boundary
value problems. Comput Mech 2003;30:396–409.
[32] Li M, Chen W, Chen C. The localized RBFs collocation methods for solving high
dimensional PDEs. Eng Anal Bound Elem 2013;37(10):1300–4.
[33] ul Islam S, Vertnik R, arler B. Local radial basis function collocation method
along with explicit time stepping for hyperbolic partial differential equations.
Appl Numer Math 2013;67:136–51 NUMAN 2010.
[34] Stevens D, Power H, Meng C, Howard D, Cliffe K. An alternative local collo-
cation strategy for high-convergence meshless PDE solutions, using radial
basis functions. J Comput Phys 2013;254:52–75.
[35] Bouhamidi A, Hached M, Jbilou K. A meshless method for the numerical
computation of the solution of steady Burgers-type equations. Appl Numer
Math 2013;74:95–110.
[36] de la Hoz F, Vadillo F. The solution of two-dimensional advection–diffusion
equations via operational matrices. Appl Numer Math 2013;72:172–87.
[37] Dehghan M, Mohebbi A. High-order compact boundary value method for the
solution of unsteady convection–diffusion problems. Math Comput Simul
2008;79(3):683–99.
[38] Goyon O. Multi level schemes for solving unsteady equations. Int J Numer
Methods Fluid 1996;229(22):937–59.
F. Takhtabnoos, A. Shirzadi / Engineering Analysis with Boundary Elements 63 (2016) 114–124124