ENGINEERING PHYSICS AND MATHEMATICS
Application of meshless local radial point
interpolation (MLRPI) on a one-dimensional
inverse heat conduction problem
Elyas Shivanian
*, Hamid Reza Khodabandehlo
Department of Mathematics, Imam Khomeini International University, Qazvin 34149-16818, Iran Received 25 September 2014; revised 17 June 2015; accepted 19 July 2015
Available online 8 September 2015
KEYWORDS
Local weak formulation; Meshless local radial point interpolation;
MLRPI method; Radial basis function; Inverse heat conduction problems
Abstract In this paper, the meshless local radial point interpolation (MLRPI) method is applied to one-dimensional inverse heat conduction problems. The meshless LRPIM is one of the truly mesh-less methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimen-sions, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. Numerical experi-ments are carried out and compared with implicit Finite difference method.
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1. Introduction
Development of constructive methods for the numerical solu-tion of mathematical problems is a main branch of mathemat-ics. For the numerical solutions of PDEs, finite differences, finite elements, finite volume and spectral methods are traditional well-developed main numerical methods. In the both mathematics and engineering community, meshless meth-ods have attracted much attention in recent years. Extensive
developments have been made in several varieties of meshless methods and applied to many applications in science and engineering.
The main shortcoming of mesh-based methods such as the finite element method (FEM), the finite volume method (FVM) and the boundary element method (BEM) is that these numerical methods rely on meshes or elements. In the last two decades, in order to overcome the mentioned difficulties some techniques the so-called meshless methods have been proposed
[1]. This method is used to establish system of algebraic equa-tions for the whole domain of the problem without the use of predefined mesh for the domain discretization so that set of nodes scattered within the domain of the problem as well as sets of nodes on the boundaries of the domain to represent (but not to discretize) the domain of the problem and its boundaries are used. These sets of scattered nodes are usually called field nodes. There are three types of meshless methods: * Corresponding author. Tel.: +98 9126825371.
E-mail address:[email protected](E. Shivanian). Peer review under responsibility of Ain Shams University.
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meshless methods based on weakforms such as the element free Galerkin (EFG) method[2,3], meshless methods based on col-location techniques (strong forms) such as the meshless collo-cation method based on radial basis functions (RBFs) [4–6]
and meshless methods based on the combination of weak forms and collocation technique. Due to the ill-conditioning of the resultant linear systems in RBF-collocation method, various approaches are proposed to circumvent this problem, Refs.[7–12] being among them. The weak forms are used to derive a set of algebraic equations through a numerical integration process using a set of quadrature domain that may be constructed globally or locally in the domain of the
problem. In the global weak form methods, global background cells are needed for numerical integration in computing the algebraic equations. To avoid the use of global background
Table 1 The L1; L2 and L1 errors calculated by MLRPI for
Example 1with differentDx and Dt at time t ¼ 1:0.
Dt h kEk1 kEk2 kEk1
1 10
1
10 9:612747E 004 2:153922E 003 6:051397E 003 1
20 201 2:511049E 004 8:142094E 004 3:298200E 003 1
40 401 7:489474E 005 3:433787E 004 1:975697E 003 1
80 1
80 2:675784E 005 1:718255E 004 1:394899E 003 1
160 1
160 8:408118E 006 7:668379E 005 8:805627E 004
Table 2 The L1; L2and L1errors calculated by implicit Finite
difference method forExample 1with differentDx and Dt at time t¼ 1:0.
Dt h kEk1 kEk2 kEk1
1 10
1
10 1:917950E 002 5:256130E 002 1:571049E 001 1
20 1
20 5:216780E 003 2:048860E 002 8:885172E 002 1
40 401 1:398108E 003 7:754154E 003 4:809822E 002 1
80 801 3:830978E 004 2:961067E 003 2:606742E 002 1
160 1
160 1:109430E 004 1:178479E 003 1:463599E 002
0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 x u (x,t) Exact solution Numerical solution
Figure 1 Numerical solutions and exact solution at time t¼ 1:0 forExample 1. The solid line corresponds to the exact solution, the stared line corresponds to numerical solution of the MLRPI withDt ¼ 1 40and h¼ 1 40. 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 x u (x,t) Exact solution Numerical solution
Figure 2 Numerical solutions and exact solution at time t¼ 1:0 forExample 1. The solid line corresponds to the exact solution, the stared line corresponds to numerical solution of the implicit Finite difference method withDt ¼1
40and h¼ 1 40.
Table 3 The L1; L2 and L1 errors calculated by MLRPI for
Example 2with differentDx and Dt at time t ¼ 1:0.
Dt h kEk1 kEk2 kEk1
1
10 101 1:690047E 003 3:771650E 003 1:058519E 002 1
20 1
20 4:270053E 004 1:405035E 003 5:716648E 003 1
40 1
40 1:283556E 004 5:965173E 004 3:449968E 003 1
80 801 4:708274E 005 3:035730E 004 2:472142E 003 1
160 1601 1:483971E 005 1:364903E 004 1:572810E 003
Table 4 The L1; L2and L1errors calculated by implicit Finite
difference method forExample 2with differentDx and Dt at time t¼ 1:0.
Dt h kEk1 kEk2 kEk1
1 10
1
10 3:602432E 002 1:031185E 001 3:091081E 001 1
20 1
20 1:127784E 002 4:552576E 002 1:980004E 001 1
40 401 3:703452E 003 2:026905E 002 1:256497E 001 1
80 801 1:328027E 003 9:657850E 003 8:413821E 002 1
160 1
160 5:274177E 004 5:110765E 003 6:186284E 002
0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 x u (x,t) Exact solution Numerical solution
Figure 3 Numerical solutions and exact solution at time t¼ 1:0 forExample 2. The solid line corresponds to the exact solution, the stared line corresponds to numerical solution of the MLRPI withDt ¼ 1
40and h¼ 1 40.
cells, a so-called local weak form is used to develop the mesh-less local Petrov–Galerkin (MLPG) method[13–23]. When a local weak form is used for a field node, the numerical integra-tions are carried out over a local quadrature domain defined for the node, which can also be the local domain where the test (weight) function is defined. The local domain usually has a regular and simple shape for an internal node (such as sphere and rectangular), and the integration is done numerically within the local domain. Hence the domain and boundary inte-grals in the weak form methods can easily be evaluated over the regularly shaped sub-domains (spheres in 3D or circles in 2D) and their boundaries. In the literature, several meshless weak form methods have been reported such as diffuse element
method (DEM) [24], smooth particle hydrodynamic (SPH)
[25,26], the reproducing kernel particle method (RKPM)[27], boundary node method (BNM) [28], partition of unity finite element method (PUFEM) [29], finite sphere method (FSM)
[30], boundary point interpolation method (BPIM) [31] and boundary radial point interpolation method (BRPIM) [32]. Liu applied the concept of MLPG and developed meshless local radial point interpolation (MLRPI) method [33–35]. Inverse problems arise in many heat transfer situations when experimental difficulties are encountered in measuring or producing the appropriate boundary conditions. It is some-times necessary to determine, the surface temperature of a body from a measured temperature history at a given fixed location inside the body, when the surface itself is inaccessible for measurements. Due to the missing boundary conditions and therefore maximum principle, the solution of the under consideration problem does not depend continuously on the data, and therefore it is known as an ill-posed problem in the sense as described by Hadamard [36] and therefore, its numerical solution requires special care. This ill-posedness is motivated by the fact that in general, in applications the data will be measured quantities and therefore always contaminated by errors. In this paper, we concentrate on the numerical solution of inverse heat conduction problems using the meshless local radial point interpolation (MLRPI) method. The mathematical formulation of the inverse heat conduction problems (IHCP) considered in this study can be described by the following boundary value problem. The governing heat conduction equation in a slab geometry, namely
@u @t¼ @
2 u
@x2þ fðx; tÞ; x 2 X ¼ h0; 1i; x 2 h0; Ti ð1Þ
has to be solved subject to the initial and boundary condition uðx; 0Þ ¼ u0ðxÞ; x 2 ð0; 1Þ;
uð1; tÞ ¼ gðtÞ; t 2 ð0; T; ð2Þ
where u is the temperature,@uðx;tÞ@n is the heat flux, n is the out-ward normal at the boundary of the slab and u0ðxÞ and gðtÞ are prescribed functions. The thermal diffusivity is assumed con-stant and, for simplicity, taken to be unity. In addition, tem-perature readings are provided at an arbitrary space location x¼ d 2 ½0; 1, namely
uðd; tÞ ¼ wðtÞ; t 2 ð0; T; ð3Þ
wherew is a known function. When d ¼ 0 Eqs.(1)–(3)reduce to the direct problem. Based on the formulation in Eqs.(1)–(3)
it is required to determine the temperature
uð0; tÞ ¼ /1ðtÞ; t 2 ð0; T ð4Þ
and the heat flux @uðx; tÞ @n x¼0 ¼ /2ðtÞ; t 2 ð0; T: ð5Þ
See [37–40] for some papers on inverse heat conduction problems.
2. The modified radial point interpolation scheme
In the conventional point interpolation method (PIM) there is a main difficulty that inverse of the polynomial moment matrix (it will be defined later) does not often exist. This condition could always be possible depending on the locations of the nodes in the support domain and the terms of monomials used in the basis. If an inappropriate polynomial basis is chosen for a given set of nodes, it may yield a badly conditioned or even singular moment matrix[1]. In order to avoid the singularity problem in the polynomial point interpolation method (PIM), the radial basis function (RBF) is used to develop the radial point interpolation method (RPIM) for meshless weak form techniques [34,41,42]. The combination of RPIM and polynomial PIM is described as follows: consider a continuous function uðxÞ defined in a domain X, which is represented by a set of field nodes. The uðxÞ at a point of interest x is approxi-mated in the form of
uðxÞ ¼X n i¼1 RiðxÞaiþ Xm j¼1 pjðxÞbj¼ RTðxÞa þ PTðxÞb; ð6Þ where RiðxÞ is a radial basis function (RBF), n is the number of RBFs, pjðxÞ is monomial in the space coordinate x and m is the number of polynomial basis functions. The pjðxÞ in Eq.(6)is, in general, chosen in a top-down approach from the Pascal tri-angle, so that the basis is complete to a desired order and a complete basis is usually preferred. For 1D problems, we use
PTðxÞ ¼ 1; x; x 2; x3; . . . ; xm: ð7Þ
For 2D problems, we have
PTðxÞ ¼ PTðx; yÞ ¼ 1; x; y; xy; x 2; y2; . . . ; xm; ym: ð8Þ 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 x u (x,t) Exact solution Numerical solution
Figure 4 Numerical solutions and exact solution at time t¼ 1:0 forExample 2. The solid line corresponds to the exact solution, the stared line corresponds to numerical solution of the implicit Finite difference method withDt ¼ 1
40and h¼ 1 40.
When m¼ 0, only RBFs are used. Otherwise, the RBF is aug-mented with m polynomial basis functions. Coefficients aiand
bj are unknown which should be determined. There are a
number of types of RBFs, and the characteristics of RBFs have been widely investigated[5,43,44]. In the current work, we have chosen the thin plate spline (TPS) as radial basis functions in Eq.(6). This RBF is defined as follows:
RðxÞ ¼ r2mlnðrÞ; m ¼ 1; 2; 3; . . . ð9Þ
Since RðxÞ in Eq.(9)belongs to C2m1(all continuous function to the order 2m 1, so higher-order thin plate splines must be used for higher-order partial differential operators. For the second-order partial differential Eq.(1), m¼ 2 is used for thin plate splines (i.e., second-order thin plate splines). In the radial basis function RiðxÞ, the variable is only the distance between the point of interest x and a node at xi, i.e., r¼ jx xij for 1-D and r¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xiÞ 2þ ðy y iÞ 2 q
for 2-D. In order to determine aiand bjin Eq.(6), a support domain is formed for the point of interest at x, and n field nodes are included in the support domain. Coefficients ai and bj in Eq. (6)can be determined by enforcing Eq.(6)to be satisfied at these n nodes surround-ing the point of interest x. This leads to the system of n linear equations, one for each node. The matrix form of these equations can be expressed as
Us¼ Rnaþ Pmb; ð10Þ
where the vector of function values Us is Us¼ uf 1; u2; u3; . . . ; ung
T; ð11Þ
the RBFs moment matrix is
Rn¼ R1ðr1Þ R2ðr1Þ . . . Rnðr1Þ R1ðr2Þ R2ðr2Þ . . . Rnðr2Þ ... ... ... ... R1ðrnÞ R2ðrnÞ . . . RnðrnÞ 2 6 6 6 6 4 3 7 7 7 7 5 nn ð12Þ
and the polynomial moment matrix is
Pm¼ 1 x1 . . . xm1 1 x2 . . . xm2 ... ... .. . ... 1 xn . . . xmn 2 6 6 6 6 4 3 7 7 7 7 5 nm : ð13Þ
Also, the vector of unknown coefficients for RBFs is
aT¼ af 1; a2; a3; . . . ; ang ð14Þ
and the vector of unknown coefficients for polynomial is
bT¼ bf 1; b2; b3; . . . ; bmg: ð15Þ
We notify that, in Eq.(12), rkin RiðrkÞ is defined as
rk¼ jxk xij: ð16Þ
We mention that there are mþ n variables in Eq.(10). The additional m equations can be added using the following m constraint conditions:
Xn i¼1
pjðxiÞai¼ PTma¼ 0; j ¼ 1; 2; . . . ; m: ð17Þ Combining Eqs.(10) and (17)yields the following system of equations in the matrix form:
~Us¼ Us 0 ¼ Rn Pm PTm 0 a b ¼ G~a; ð18Þ where ~Us¼ uf 1u2. . . un0 0. . . 0g; ~aT ¼ af 1a2. . . anb1. . . bmg: ð19Þ
Because the matrix Rnis symmetric, the matrix G will also be symmetric. Solving Eq.(18), we obtain
~a ¼ a b
¼ G1~U
s: ð20Þ
Eq.(6)can be rewritten as
uðxÞ ¼ RTðxÞa þ PTðxÞb ¼ R TðxÞ; PTðxÞ a b
: ð21Þ
Now, using Eq.(20)we obtain uðxÞ ¼ R TðxÞ; PTðxÞ
G1~Us¼ ~UTðxÞ ~Us; ð22Þ
where ~UTðxÞ can be rewritten as ~UTðxÞ ¼ R TðxÞ; PTðxÞ
G1
¼ /1ðxÞ/2ðxÞ . . . /nðxÞ/nþ1ðxÞ . . . /nþmðxÞ
: ð23Þ
The first n functions of the above vector function are called the RPIM shape functions corresponding to the nodal dis-placements and We show by the vector ~UTðxÞ so that it is
~UTðxÞ ¼ /
1ðxÞ /2ðxÞ . . . /nðxÞ
f g; ð24Þ
then Eq.(22)is converted to the following one: uðxÞ ¼ ~UTðxÞU
s¼ Xn
i¼1
/iðxÞui: ð25Þ
The derivatives of uðxÞ are easily obtained as @uðxÞ @x ¼ Xn i¼1 @/iðxÞ @x ui; @2uðxÞ @x2 ¼ Xn i¼1 @2/ iðxÞ @x2 ui: ð26Þ
Note that R1n usually exists for arbitrary scattered nodes and therefore the augmented matrix G is theoretically non-singular[45,46]. In addition, the order of polynomial used in Eq.(6)is relatively low. We add that the RPIM shape func-tions have the Kronecker delta function property, that is
/iðxjÞ ¼
1; i ¼ j; j ¼ 1; 2; . . . ; n; 0; i – j; j ¼ 1; 2; . . . ; n:
ð27Þ This is because the RPIM shape functions are created to pass through nodal values.
3. The time discretization approximation
In the current work, we employ a time-stepping scheme to approximate the time derivative. For this purpose, the follow-ing finite difference approximation for the time derivative operator is used: @uðx; tÞ @t ffi 1 Dt ukþ1ðxÞ ukðxÞ : ð28Þ
Also we employ the following approximation by using the Crank–Nicolson technique:
uðx; tÞ ffi1 2 u kþ1ðxÞ þ ukðxÞ ; @2 uðx; tÞ @x2 ffi 1 2 @2 ukþ1ðx; tÞ @x2 þ @ 2 ukðx; tÞ @x2 ; ð29Þ where ukðxÞ ¼ uðx; kDtÞ.
Using the above discussion, Eq.(1)can be written as: 1 Dt uðkþ1ÞðxÞ uðkÞðxÞ 1 2 @2 uðkþ1ÞðxÞ @x2 þ @ 2 uðkÞðxÞ @x2 ¼1 2 f ðkþ1ÞðxÞ þ fðkÞðxÞ : ð30Þ Suppose thatk ¼ 1 Dt, then we obtain k u ðkþ1ÞðxÞ uðkÞðxÞ1 2 @2 uðkþ1ÞðxÞ @x2 þ @ 2 uðkÞðxÞ @x2 ¼1 2 f ðkþ1ÞðxÞ þ fðkÞðxÞ ; ð31Þ therefore kuðkþ1Þ1 2 @2uðkþ1ÞðxÞ @x2 ¼ ku ðkÞþ1 2 @2uðkÞðxÞ @x2 þ1 2 f ðkþ1ÞðxÞ þ fðkÞðxÞ : ð32Þ
4. The meshless local weak form formulation
Instead of giving the global weak form, the MLRPI method constructs the weak form over local quadrature cell such as Xq, which is a small region taken for each node in the global domainX. The local quadrature cells overlap with each other and cover the whole global domain X. The local quadrature cells could be of any geometric shape and size. In one dimen-sional problems, they are line (interval). The local weak form of Eq.(32)for xi2 Xiq¼ xi rq; xiþ rq can be written as Z Xi q kuðkþ1Þ1 2 @2 uðkþ1ÞðxÞ @x2 mðxÞdx ¼ Z Xi q kuðkÞþ1 2 @2 uðkÞðxÞ @x2 mðxÞdx þ Z Xi q 1 2 f ðkþ1ÞðxÞ þ fðkÞðxÞ mðxÞdx; ð33Þ whereXi
q is the local quadrature domain associated with the point i, andmðxÞ is the Heaviside step function[47,48]:
mðxÞ ¼ 1; x 2 Xq; 0; x R Xq;
ð34Þ as the test function in each local quadrature domain. Hence, we have k Z Xi q uðkþ1ÞmðxÞdx 1 2 Z Xi q @2 uðkþ1ÞðxÞ @x2 mðxÞdx ¼ k Z Xi q uðkÞmðxÞdx þ1 2 Z Xi q @2 uðkÞðxÞ @x2 mðxÞdx þ 1 2 Z Xi q fðkþ1ÞðxÞ þ fðkÞðxÞ mðxÞdx: ð35Þ
Using integration by parts, one has:
Z Xi q @2 uðkÞðxÞ @x2 mðxÞdx ¼ mðxÞ @uðkÞðxÞ @x x¼xiþrq x¼xirq Z Xi q @uðkÞðxÞ @x @mðxÞ @x dx ð36Þ
and using the test function the following local weak equation will be obtained: k Z Xi q uðkþ1Þdx1 2 @uðkþ1ÞðxÞ @x x¼xiþrq x¼xirq ! ¼ k Z Xi q uðkÞdxþ1 2 @uðkÞðxÞ @x x¼xiþrq x¼xirq ! þ1 2 Z Xi q fðkþ1ÞðxÞ þ fðkÞðxÞ dx: ð37Þ
Applying the radial point interpolation method (RPIM) for the unknown functions, the local integral Eq. (37) is trans-formed into a system of algebraic equations with used unknown quantities, as described in the next section.
5. Discretization for MLRPI method
In this section, we consider Eq. (37) to see how to obtain discrete equations. Consider N regularly located points on the boundary and domain of the problem so that the distance between two consecutive nodes in each direction is constant and equal to h. Assuming that uðxi; kDtÞ; i ¼ 1; 2; . . . ; N are known,our aim is to compute uðxi; ðk þ 1ÞDtÞ; i ¼ 1; 2; . . . ; N. So, We have N unknowns and to compute these unknowns We need N equations. As it will be described, corresponding to each node We obtain one equation. For nodes which are located in the interior of the domain, i.e., for xi2 interior X, to obtain the discrete equations from the locally weak forms (37), substituting approximation formulas(25) and (26)into local integral Eq.(37)yields kX N j¼1 Z Xi q /jðxÞdx ! uðkþ1Þj 1 2 XN j¼1 @/jðxÞ @x x¼x iþrq @/jðxÞ @x x¼x irq ! uðkþ1Þj ¼ kX N j¼1 Z Xi q /jðxÞdx ! uðkÞj þ 1 2 XN j¼1 @/jðxÞ @x x¼x iþrq @/jðxÞ @x x¼x irq ! uðkÞj þ1 2 Z Xi q fðkþ1ÞðxÞ þ fðkÞðxÞ dx ð38Þ or equivalently kX N j¼1 Z Xi q /jðxÞdx ! 1 2 XN j¼1 @/jðxÞ @x x¼x iþrq @/jðxÞ @x x¼x irq ! " # uðkþ1Þj ¼ kX N j¼1 Z Xi q /jðxÞdx ! þ1 2 XN j¼1 @/jðxÞ @x x¼x iþrq @/jðxÞ @x x¼x irq ! " # uðkÞj þ1 2 Z Xi q fðkþ1ÞðxÞ þ fðkÞðxÞ dx: ð39Þ
6. Numerical implementation for MLRPI method For xi¼ 1 from boundary condition(2), we set
ukþ1ðxiÞ ¼ gððk þ 1ÞDtÞ: ð40Þ
Also, for xi¼ d using condition(3)we have:
The matrix forms of Eqs. (39) and (40) for all N nodal points in domain and in boundary of the problem are given below: kX N j¼1 Ai;j1 2 XN j¼1 Bi;j " # uðkþ1Þj ¼ kX N j¼1 Ai;jþ1 2 XN j¼1 Bi;j " # uðkÞj þ Eiðk; k þ 1Þ ð42Þ where Ai;j¼ Z Xi q /jðxÞdx; Bi;j¼ @/@xjðxÞ x¼xiþrq @/jðxÞ @x x¼xirq ! ; Eiðk; k þ 1Þ ¼ 1 2 Z Xi q fðkþ1ÞðxÞ þ fðkÞðxÞ dx: ð43Þ
Assuming Ai;j¼ kAi;j12Bi;j, Bi;j¼ kAi;jþ 1 2Bi;j, E k¼ E1ðk; k þ 1Þ; E2ðk; k þ 1Þ; . . . ; ENðk; k þ 1Þ ½ T , and U¼ ðuiÞN1, Eq.(42)yields AUðkþ1Þ¼ BUðkÞþ Ek: ð44Þ
Furthermore, to satisfy Eqs.(40) and (41), for nodes in(40) and (41), we set
Eki ¼ 0; 8j : Bi;j¼ 0; Ai;j¼ 1; i ¼ j; 0; i – j:
ð45Þ At the first time level, when n¼ 0, according to the initial conditions that were introduced in Eq.(2), we apply the fol-lowing assumptions: uð0Þ¼ u0; where u0¼ ½u0ðx1Þ; u0ðx2Þ; . . . ; u0ðxNÞ T . 7. Numerical experiments
In this section the numerical results obtained from application of the meshless local radial point interpolation method (MLRPI) for solving the one-dimensional linear telegraph equation are presented. We show the results obtained for two examples using the meshless method described above. In both examples, the domain integrals are evaluated with 7 points Gaussian quadrature rule. These examples are chosen for comparison with the results of implicit Finite difference method for this equation. In all problems the regular node dis-tribution is used. Also in order to implement the meshless local weak form in all cases considered, the radius of the local quadrature domain rq¼ 0:8h is selected, where h is the distance between the nodes in x direction (h¼ Dx). The size of rqis such that the union of these sub-domains must cover the whole glo-bal domain. The radius of support domain to local radial point interpolation method is rs¼ 4rq. This size is significant enough to have sufficient number of nodes (n) and gives an appropriate approximation. Also, in Eq.(6), we set m¼ 4.
Example 1. We suppose that the exact solution of the first example is given by
uðx; tÞ ¼ 4 expðtÞx2ð1 xÞ2; ðx; tÞ 2 ½0; 1 ½0; 1
and d¼ 0:4. The initial and boundary conditions can be
obtained by using the exact solution, where fðx; tÞ is defined accordingly, i.e.,
fðx; tÞ ¼ 4 expðtÞx2ð1 xÞ2 4 expðtÞð12x2 12x þ 2Þ:
Tables 1 and 2as well asFigs. 1 and 2compares the results of MLRPI with those obtained by implicit Finite difference method. As it is seen, MLRPI is superior to implicit Finite dif-ference scheme.
Example 2. We suppose that the exact solution of this example is given by
uðx; tÞ ¼ 4t3x2ð1 xÞ2; ðx; tÞ 2 ½0; 1 ½0; 1:
As previous example setting d¼ 0:4, the initial and bound-ary conditions can be obtained by using the exact solution, where fðx; tÞ is defined accordingly, i.e.,
fðx; tÞ ¼ 12t2x2ð1 xÞ2 4t3ð12x2 12x þ 2Þ:
Tables 3 and 4as well asFigs. 3 and 4compares the results of MLRPI with those obtained by implicit Finite difference method. As it is seen, MLRPI is superior to implicit Finite dif-ference technique.
8. Conclusions
In this paper meshless local radial point interpolation (MLRPI) method has been applied to solve one-dimensional inverse heat conduction problem. The present method pro-vides a local quadrature domain and a local support domain for each node so that the integration and the interpolation are done on these domains. In this method, the shape functions have been constructed by the radial point interpolation method. A time stepping scheme was employed to approxi-mate the time derivative. The Heaviside step function was used as the test function in the local weak form method in MLRPI. All integrations are regular, therefore the Gaussian quadrature rule was used to calculate the numerical integration for local weak form. In the current work, to demonstrate the accuracy and usefulness of this method, two numerical examples have been presented. As demonstrated by the computational results, it is very easy to implement the proposed method for similar problems.
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Dr. Elyas Shivanian was born in Zanjan pro-vince, Iran on August 26, 1982. He started his master course in applied mathematics in 2005 at Amirkabir University of Technology and has finished MSc. thesis in the field of fuzzy linear programming in 2007. After his Ph.D. in the field of prediction of multiplicity of solu-tions of boundary value problems, at Imam Khomeini International University in 2012, he became an Assistant Professor at the same university. His research interests are analytical and numerical solutions of ODEs, PDEs and IEs. He has several papers on these subjects. He also has published some papers in other fields, for more information see
http://scholar.google.com/citations?user=MFncks8AAAAJ&hl=en.
Mr. Hamid Reza Khodabandehlo was born in Zanjan province, Iran. He started his master course in applied mathematics in 2011 at Imam Khomeini International University and has finished MSc. thesis in the field of numerical solution of partial differential equations.
