/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 9, No. 1, 2016, 64-83

ISSN 1307-5543 – www.ejpam.com

Application of Meshless Methods for Solving an Inverse Heat

Conduction Problem

Malihe Rostamian, Alimardan Shahrezaee∗

Department of Mathematics, Alzahra university, Vanak, Post Code 19834, Tehran, Iran.

Abstract. In this paper, we consider the inverse problem of determining the unknown temperature at

x = 0 and section of initial condition at t = 0 in an inverse heat conduction problem (IHCP). Two new

numerical methods are developed by using the solution of an auxiliary problem and heat polynomials

as basis functions in presence of noisy data. Due to ill-posed IHCP, we use the Tikhonov regularization

technique with the GCV scheme to solve the resulting matrix system of the basis function methods

(BFM). Some numerical examples are presented to illustrate the strength of the methods.

2010 Mathematics Subject Classifications: 35R30, 35K05

Key Words and Phrases: IHCP, Basis function method, Tikhonov regularization method, meshless

method, GCV, Ill-conditioned, Heat polynomials, Auxiliary problem

1. Introduction

Inverse heat conduction problems (IHCPs) arise in many industrial and engineering appli-

cations where heat transfer occurs. In remote sensing, oil exploration, nondestructive evalu-

ation of material and determination of the earth’s interior structure. One of the applications

may be the determination of the surface heat flux histories of reentering heat shield [32]. In

some problems because of the physical situation at the surface that may be unsuitable for at-

taching a sensor, some boundary conditions are unknown. For example, in a shuttle or missile

reentering the earth’s atmosphere from space, the heat flux at the heated surface is needed

[3].

Inverse problems are in nature ’unstable’ because the unknown solutions and parameters

have to be determined from indirect observable data which contain measurement error. The

major difficulty in establishing any numerical algorithm for approximating the solution is the

ill-posedness of the problem and the ill-conditioning of the resulting discretized matrix. So far

many different methods have been applied to solve IHCPs [2, 3, 6–8, 15, 18–20, 22, 24, 26,

28, 29, 31, 36, 37].

∗Corresponding author.

Email addresses: mlh.rostamian@gmail.com (M. Rostamian), ashahrezaee@alzahra.ac.ir (A.

Shahrezaee)

http://www.ejpam.com 64 c© 2016 EJPAM All rights reserved.



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 65

1.1. A Brief Review of Meshless Methods

The finite element method (FEM) and finite difference method (FDM) have been the dom-

inant numerical methods in the solution of partial differential equations (PDEs) for several

decades. Nevertheless, FEM and FDM possess some limitations in the solution of certain en-

gineering problems:

(1) the generation of a well behaved mesh may be extremely time consuming procedure for

complex geometry,

(2) large deformations produce highly distorted meshes,

(3) repeated remeshing with a moving discontinuity problem such as flame propagation is

required,

(4) very slow convergence in high gradient regions is observed, etc.

In order to avoid these problems, an alternative approach, called mesh-less method, has been

developed [17].

Meshless methods for the solution of PDEs can be grouped into two broad categories. One,

as already mentioned, includes methods based on RBF interpolation. The second is based on

the least squares technique. To this second class of methods belong element-free Galerkin

methods [4], local Petrov-Galerkin technique [1], the finite point method [23] and the general

finite difference method [21].

Radial basis functions (RBFs) were first applied to solve the partial differential equations

in 1999 by Kansa [16]. This Kansa’s method is a technique based on direct collocation method.

RBF performs [13, 27, 33] very well in interpolating highly irregular scattered data compared

to many interpolating methods. These functions have been used in the boundary element

method formulation such as the dual reciprocity method DRM [11], the method of fundamen-

tal solution (MFS), the analog equation method (AEM) and the boundary knot method (BKM).

These methods have been successfully applied to solve several non-linear problems [12].

In this paper two meshless approaches based on the use of the solution of an auxiliary

problem and heat polynomials as radial basis functions (RBF) for determining the unknown

temperature at x = 0 and section of initial condition at t = 0 in an inverse heat conduction

problem (IHCP), are devised. Since the basis functions are the general solution of considered

heat equation, the approximation to the temperature distribution only needs to satisfy the

boundary condition and the given measurement data. The proposed methods in this paper

are based on the collocation method. Because of the collocation technique, these methods do

not need to evaluate any integral. These approaches are different from most existing numerical

algorithms in solving dynamical problems where the finite difference quotient will be used to

discretize the time variable. Our methods are also feasible to handle various informal boundary

conditions.

The organization of the paper is as follows: In Section 2, mathematical formulation of the

problem is considered. Heat polynomials are presented in Section 3. In Section 4, numerical

solution based on the use of basis function method is discussed. Some examples are given in

Section 5. Section 6 is adapted to a conclusion of paper.



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 66

2. Mathematical Formulation of the Problem

In this work, we consider the following inverse problem:

Ut(x , t)−α2Ux x(x , t) =0; 0< x < 1, 0< t < tmax , (1)

U(x , 0) =

¨
p(x); 0≤ x < x∗,

f (x); x∗ ≤ x ≤ 1,
(2)

U(0, t) =g(t); 0≤ t ≤ tmax , (3)

U(1, t) =k(t); 0≤ t ≤ tmax , (4)

where 0< x∗ < 1, tmax is final time, α is the thermal diffusivity, p and k are known piecewise-

continuous functions in their domain, while g and f are unknown functions. In order to

determine f and g let us consider additional temperature measurements and heat flux given

at a point x = x∗, 0< x∗ < 1, as overspecified data:

U(x∗, t) =q(t); 0≤ t ≤ tmax , (5)

Ux(x
∗, t) =h(t); 0≤ t ≤ tmax . (6)

Some methods for determination of heat flux at x∗ are discussed in [9, 10].

Problem (1)-(6) may be divided into two separate problems as shown in Figure 1.

←− ←− →
←− ←− →
←− ←− Known q(t) Known k(t) →
←− Unknown g(t) ←− Known h(t) →
←− ←− →
↓ Known p(x) ↓ Unknown f (x) →

x = 0 x = x∗ x = 1

Figure 1: Inverse heat conduction problem (1)-(6).

The first problem is:

Ut(x , t)−α2Ux x(x , t) =0; 0< x < x∗, 0< t < tmax , (7)

U(x , 0) =p(x); 0≤ x ≤ x∗, (8)

U(0, t) =g(t); 0≤ t ≤ tmax , (9)

U(x∗, t) =q(t); 0≤ t ≤ tmax , (10)

Ux(x
∗, t) =h(t); 0≤ t ≤ tmax , (11)

and the second problem is:

Ut(x , t)−α2Ux x(x , t) =0; 0< x∗ < x < 1, 0< t < tmax , (12)

U(x , 0) = f (x); 0< x∗ ≤ x ≤ 1, (13)



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 67

U(1, t) =k(t); 0≤ t ≤ tmax , (14)

U(x∗, t) =q(t); 0≤ t ≤ tmax , (15)

Ux(x
∗, t) =h(t); 0≤ t ≤ tmax , (16)

where g and f are unknowns.

Remark 1. For the problem (1)-(6), we can consider a conducting material that have 2 different

layers and be described by the interval [0,1] with the layers [0, x∗] and [x∗, 1] as following:

U(x , t) =

¨
U1(x , t); 0≤ x < x∗,

U2(x , t); x∗ ≤ x ≤ 1.

In this case, we additionally require the continuity of the temperature and heat flux across two

layers; i.e:

U1(x
∗, t) =U2(x

∗, t),

∂ U1

∂ x
(x∗, t) =

∂ U2

∂ x
(x∗, t).

To solve the inverse problems (7)-(11) and (12)-(16), let us consider the following auxiliary

problem

Ut(x , t)−α2Ux x(x , t) =0; 0< x < 1, 0< t < tmax , (17)

U(x∗, t) =q(t); 0≤ t ≤ tmax , (18)

Ux(x
∗, t) =h(t); 0≤ t ≤ tmax , (19)

where q and h are known functions. Following the same method as in [30], we assume that a

solution of (17)-(19) is represented as a power series [5],

U(x , t) =

∞∑

j=0

c j(t)(x − x∗) j ,

where the coefficient c j(t) are to be determined.

By substituting U into Ut −α
2Ux x = 0, we get:

c j+2 =
c
′

j
(t)

α2( j + 1)( j + 2)
, ( j = 0,1,2, . . .). (20)

Consequently, we obtain the following formal expression for U(x , t):

U(x , t) =

∞∑

j=0

[
d jq(t)

d t j

(x − x∗)2 j

α2 j(2 j)!
+

d jh(t)

d t j

(x − x∗)2 j+1

α2 j(2 j + 1)!
]. (21)



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 68

The series in (21), however, may not be uniformly convergent. But, according to [5], if q and

h are infinitely differentiable functions that are of the Holmgren class [5], i.e. they are infinitely

differentiable functions defined on [0, tmax] that satisfy:

|q( j)(t)| ≤ C1(2 j)!x∗
(−2 j)

, |h( j)(t)| ≤ C1(2 j)!x∗
(−2 j)

, ∀ j ≥ 0,∀t ∈ (0, tmax),

where C1 > 0 is some constant. We should recall from [5] that the power series in (21) is uniformly

convergent and U in (21) is a solution to (17)-(19).

The solution (21) exists and is unique but not always stable [5].

3. Heat Polynomials

We consider a solution U of (1) in the form [5]:

U(x , t) = φ(x)ψ(t). (22)

Substituting (22) into (1) results:

U(x , t) = eλx+λ2α2 t . (23)

By expanding equation (23) into a Taylor’s series with respect to λ, we obtain:

U(x , t) =

∞∑

n=0

χn(x , t)
λn

n!
, (24)

where χn are the heat polynomials which hold in equation (1).

To compute χn, setting

eλx =

∞∑

n=0

anλ
n,

and

eλ
2αt =

∞∑

n=0

bnλ
n,

where

an =
xn

n!
, n= 0,1,2, . . . ,

and

bn =

(
α2d td

d! ; n= 2d,

0; n= 2d + 1, d = 0,1, . . . ,

it follows that:

eλx+λ2α2 t =

∞∑

n=0

cnλ
n, (25)



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 69

where

cn =

n∑

j=0

b jan− j =

[ n
2 ]∑

d=0

α2d td

d!

xn−2d

(n− 2d)!
. (26)

Here, [ n
2 ] denotes the largest integer less than or equal to n

2 . From equations (24), (25) and

(26), it is clear that:

χn(x , t) = n!

[ n
2 ]∑

d=0

α2d td

d!

xn−2d

(n− 2d)!
. (27)

4. Numerical Procedures

Let∆= {(x j , t j), j = 1, . . . , n+m+l} be a set of scattered nodes such that∆=∆1∪∆2∪∆3,

where

∆1 ={(x j , t j), 0≤ x j ≤ 1, t j = 0, j = 1, . . . , n},

∆2 ={(x j , t j), x j = x∗, 0≤ t j ≤ 1, j = n+ 1, . . . , n+m},

∆3 ={(x j , t j), x j = x∗, 0≤ t j ≤ 1, j = n+m+ 1, . . . , n+m+ l},

and suppose that Γ = {(x j , t j), j = 1, . . . , n+m+ l} be also a set of scattered nodes such that

Γ = Γ1 ∪∆2 ∪∆3, where

Γ1 = {(x j , t j), x j = 1, 0≤ t j ≤ 1, j = 1, . . . , n}.

Then, discretizing of the boundary conditions (8), (10) and (11) in problem (7)-(11) at points

∆ and boundary conditions (14)-(16) in problem (12)-(16) at points Γ may be considered as

following:

U(x j , 0) =p j , j = 1,2, . . . , n, (28)

U(x∗, t j−n) =q j−n, j = n+ 1, n+ 2, . . . , n+m, (29)

Ux(x
∗, t j−n−m) =h j−n−m, j = n+m+ 1, . . . , n+m+ l, (30)

and

U(1, t j) =k j , j = 1,2, . . . , n, (31)

U(x∗, t j−n) =q j−n, j = n+ 1, n+ 2, . . . , n+m, (32)

Ux(x
∗, t j−n−m) =h j−n−m, j = n+m+ 1, . . . , n+m+ l. (33)

An approximate solution of problems (7)-(11) and (12)-(16) can be expressed as the following

form [25]:

U∗(x , t) =

n+m+l∑

j=1

λ jϕ(x − x j , t − t j), (34)

where

ϕ(x , t) = eU(x , t + T ), (35)



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 70

T > tmax is a constant, λ j are unknown constants which remain to be determined separately

in problem (7)-(11) and (12)-(16) and eU is selected by the following two cases:

(BFM1) eU is given by (21) [25].

(BFM2) eU =
∑y

n=1χn(x , t) where χn is given by (27), y is the number of heat polynomials

and may be chosen arbitrary.

Using conditions (28)-(30) and (31)-(33), the values of the λ j can be obtained by solving

the following matrix equation:

Aλ = b, (36)

where

A=




ϕ(xr − x j , tr − t j)

ϕ(x∗ − x j , t i − t j)
∂ ϕ
∂ x (x

∗ − x j , ts − t j)





(n+m+l)×(n+m+l)

, (37)

b is a n+m+ l × 1 vector which in problem (7)-(11) is:

b =




pr

qi

hs





(n+m+l)×(1)

, (38)

and in problem (12)-(16) is

b =




kr

qi

hs





(n+m+l)×(1)

, (39)

where r = 1,2, . . . , n, i = n+ 1, n+ 2, . . . , n+m, s = n+m+ 1, n+m+ 2, . . . , n+m+ l and

j = 1,2, . . . , n+m+ l.

Since the IHCP is ill-posed, matrix A in equation (36) is ill-conditioned. Here, we use

Tikhonov regularization method to solve equation (36) [14, 34]. The Tikhonov regularized

solution λµ for equation (36) is defined to be the solution to the following least square prob-

lem:

min
λ
{‖ Aλ− b ‖2 +µ2 ‖ λ ‖2}, (40)

where ‖ · ‖ denotes the usual Euclidean norm and µ is called the regularization parameter.

We use the GCV method to determine a suitable value of µ.

Denote the regularized solution of equation (36) by λµ
∗
. The approximated solution U∗µ

for problems (7)-(11) and (12)-(16) my be given as:

U∗µ(x , t) =

n+m+l∑

j=1

λ
µ∗

j
ϕ(x − x j , t − t j), (41)

Finally the heat temperature at surface x = 0 and initial condition in x ∈ [0, x∗] are given,

respectively, by:

g(t) =

n+m+l∑

j=1

λ
µ∗

j
ϕ(0− x j , t − t j); 0≤ t ≤ tmax , (42)



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 71

and

f (x) =

n+m+l∑

j=1

λ
µ∗

j
ϕ(x − x j , 0− t j); 0< x∗ ≤ x ≤ 1. (43)

5. Numerical Results and Discussion

In this section, we present and discuss the numerical results by employing BFM1 and BFM2

and compare them with each other. To test the accuracy of the approximate solution, we use

the root mean square error (RMS) defined as [35]:

RMS(ψ(x)) =

√√√√ 1

Nt

Nt∑

i=0

(ψExact(x i)−ψApprox imate(x i))
2,

where Nt is total number of testing points in the domain of function ψ(x).

We apply the noisy data egi = gi +σ× rand(1) where gi is the exact data and rand(1) is a

random number between (0,1) and the magnitude σ indicates the noise level of measurement

data. The results are brought in tables and figures. All the computations are performed on the

PC (pentium(R) 4 CPU 3.20 GHz).

Example 1. Consider the following problem:

Ut =αUx x ; 0< x < 1, 0< t < 1, (44)

U(x , 0) =

¨
1
2(x −

1
2)

2; 0≤ x < 1
2 ,

f (x); 1
2 ≤ x ≤ 1,

(45)

U(1, t) =
1

4
+ 2t; 0≤ t ≤ 1, (46)

U(
1

2
, t) =2t; 0≤ t ≤ 1, (47)

Ux(
1

2
, t) =2t; 0≤ t ≤ 1, (48)

where α= 2 in 0< x < 1
2 and α= 1 in 1

2 < x < 1. The exact solution of this problem is:

U(x , t) =

¨
U1(x , t) = 1

2(x −
1
2)

2 + 2t; 0≤ x < 1
2 ,

U2(x , t) = (x − 1
2)

2 + 2t; 1
2 ≤ x ≤ 1,

g(t) =
1

8
+ 2t,

f (x) =(x −
1

2
)2.

Tables 1 and 2 present the absolute errors |gNumeric − gExact | and | fNumeric − fExact | for various

levels of error (σ = 0%, 1%, 3%), respectively. Figures 2 and 3 show the exact and approximate



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 72

values of g(t) and f (x) for noiseless and noisy data (σ = 0%, 1%, 3%, 5%). From these

Tables and these Figures, the numerical results are quite satisfactory. In Table 3 we present the

errors RMS as a function of the parameter T with noisy data (σ = 1%). It can be seen from

this Table that numerical results are not only stable with respect to parameter T , but also almost

they maintain at the same level of accuracy over a wide rage of values T . Similar conclusions can

be drawn from the results for Example 2 which are shown in Table 7. To see the effectiveness of

the collocation points, we present the RMS(g) and RMS( f ) with varying number of collocation

points in Table 4. It is noted that the increase of the number of nodal points improve the accuracy

of the numerical results. From all tables and figures we conclude that the approximate results of

BFM1 for noiseless data are more accurate than BFM2.

Table 1: The absolute errors of g(t) when n= m= l = 6, T = 5 with noiseless and noisy data.

BFM1 BFM2

t σ = 0% σ = 1% σ = 3% σ = 0% σ = 1% σ = 3%

0.0 1.4× 10−14 9.3× 10−4 2.4× 10−3 1.7× 10−9 1.0× 10−3 2.6× 10−3

0.1 1.4× 10−14 8.9× 10−4 2.3× 10−3 1.9× 10−9 9.0× 10−4 2.3× 10−3

0.2 1.4× 10−14 8.5× 10−4 2.1× 10−3 1.6× 10−9 7.9× 10−4 2.0× 10−3

0.3 1.4× 10−14 8.1× 10−4 2.0× 10−3 1.8× 10−9 7.0× 10−4 1.8× 10−3

0.4 1.4× 10−14 7.7× 10−4 1.9× 10−3 1.5× 10−9 6.3× 10−4 1.6× 10−3

0.5 1.4× 10−14 7.3× 10−4 1.8× 10−3 8.4× 10−10 5.6× 10−4 1.5× 10−3

0.6 1.4× 10−14 6.9× 10−4 1.7× 10−3 1.4× 10−9 5.8× 10−4 1.4× 10−3

0.7 1.3× 10−14 6.5× 10−4 1.6× 10−3 7.5× 10−10 5.8× 10−4 1.4× 10−3

0.8 1.3× 10−14 6.1× 10−4 1.5× 10−3 0.9× 10−11 5.9× 10−4 1.5× 10−3

0.9 1.3× 10−14 5.6× 10−4 1.4× 10−3 2.6× 10−10 6.4× 10−4 1.6× 10−3

1.0 1.3× 10−14 5.2× 10−4 1.3× 10−3 3.8× 10−10 7.1× 10−4 1.8× 10−3

Table 2: The absolute errors of f (x) when n= m= l = 5, T = 5 with noiseless and noisy data.

BFM1 BFM2

x σ = 0% σ = 1% σ = 3% σ = 0% σ = 1% σ = 3%

0.50 1.5× 10−15 1.1× 10−3 2.9× 10−3 2.2× 10−12 1.2× 10−3 3.1× 10−3

0.55 1.4× 10−15 1.1× 10−3 3.0× 10−3 2.2× 10−12 1.2× 10−3 3.2× 10−3

0.60 1.5× 10−15 1.2× 10−3 3.1× 10−3 2.2× 10−12 1.3× 10−3 3.4× 10−3

0.65 1.6× 10−15 1.2× 10−3 3.3× 10−3 2.2× 10−12 1.3× 10−3 3.5× 10−3

0.70 1.4× 10−15 1.3× 10−3 3.4× 10−3 2.2× 10−12 1.4× 10−3 3.6× 10−3

0.75 1.4× 10−15 1.4× 10−3 3.6× 10−3 2.2× 10−12 1.4× 10−3 3.8× 10−3

0.80 1.4× 10−15 1.4× 10−3 3.7× 10−3 2.1× 10−12 1.5× 10−3 3.9× 10−3

0.85 1.4× 10−15 1.5× 10−3 3.8× 10−3 2.1× 10−12 1.5× 10−3 4.0× 10−3

0.90 1.4× 10−15 1.5× 10−3 4.0× 10−3 2.0× 10−12 1.6× 10−3 4.1× 10−3

0.95 1.4× 10−15 1.6× 10−3 4.1× 10−3 2.0× 10−12 1.6× 10−3 4.2× 10−3

1.00 1.4× 10−15 1.6× 10−3 4.2× 10−3 1.9× 10−12 1.7× 10−3 4.4× 10−3



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 73

Table 3: The values of RMS(g) and RMS( f ) for various values of T with noisy data (σ = 1%)

and n= m= l = 5.

BFM1 BFM2

T RMS(g) RMS( f ) RMS(g) RMS( f )

1.01 7.6084689× 10−4 0.0013829 7.1926508× 10−4 0.0014809

3.01 7.6060300× 10−4 0.0013832 7.1925926× 10−4 0.0014802

5 7.6089619× 10−4 0.0013833 7.1925262× 10−4 0.0014802

7 7.6095451× 10−4 0.0013828 7.1923964× 10−4 0.0014802

10 7.6081813× 10−4 0.0013829 7.1925280× 10−4 0.0014802

20 7.6078507× 10−4 0.0013834 7.1914739× 10−4 0.0014803

Table 4: The values of RMS(g) and RMS( f ) for various values of m, n, l with noisy data

(σ = 1%) and T = 5.

BFM1 BFM2

n m l RMS(g) RMS( f ) RMS(g) RMS( f )

2 2 2 8.3821721× 10−4 0.0014576 0.0372292 0.1767561

5 5 5 7.6089619× 10−4 0.0013833 7.1925262× 10−4 0.0014802

7 7 7 7.5595261× 10−4 0.0013745 7.1620889× 10−4 0.0014589

10 10 10 7.4725586× 10−4 0.0013689 7.1131845× 10−4 0.0014448

15 15 15 7.4064272× 10−4 0.0013650 7.0804579× 10−4 0.0014359

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.5

1

1.5

2

2.5

t

g(
t)

 

 

Exact
σ=0%

σ=1%

σ=3%

σ=5%

(a) BFM1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.5

1

1.5

2

2.5

t

g(
t)

 

 

Exact
σ=0%

σ=1%

σ=3%

σ=5%

(b) BFM2

Figure 2: The exact and approximate solution of g(t) when n = m = l = 6, T = 5 with

noiseless and noisy data.



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 74

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

f(
x)

 

 

Exact
σ=0%

σ=1%

σ=3%

σ=5%

(a) BFM1

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

x

f(
x)

 

 

Exact
σ=0%

σ=1%

σ=3%

σ=5%

(b) BFM2

Figure 3: The exact and approximate solution of f (x) when n = m = l = 5, T = 5 with

noiseless and noisy data.

Example 2. Let us consider the following problem:

Ut =Ux x ; 0< x < 1, 0< t < 1, (49)

U(x , 0) =

(
3(1−x)2−1

6 + 2
π2 cos(π(1− x)); 0≤ x < 1

5 ,

f (x); 1
5 ≤ x ≤ 1,

(50)

U(1, t) =−
1

6
+ t +

2

π2
exp(−π2 t); 0≤ t ≤ 1, (51)

U(
1

5
, t) =

23

150
+ t +

2

π2
cos(

4

5
π)exp(−π2 t); 0≤ t ≤ 1, (52)

Ux(
1

5
, t) =−

4

5
+

2

π
sin(

4

5
π)exp(−π2 t); 0≤ t ≤ 1, (53)

The exact solution of this problem is

U(x , t) =
3(1− x)2 − 1

6
+ t +

2

π2
cos(π(1− x))exp(−π2 t),

g(t) =
1

3
+ t −

2

π2
exp(−π2 t),

f (x) =
3(1− x)2 − 1

6
+

2

π2
cos(π(1− x)).

Tables 5 and 6 present the absolute errors |gNumeric − gExact | and | fNumeric − fExact | for various

levels of error (σ = 0%, 1%, 3%), respectively. Figures 4 and 5 show the exact and approximate

values of g(t) and f (x) for noiseless and noisy data (σ = 0%, 1%, 3%, 5%). From these Tables

and Figures we conclude that the numerical solutions are still in good agreement with the exact

solutions. The stability of the numerical solutions with respect T is studied in Table 7. Furthermore

the results of use of different numbers of the collocation points are shown in Table 8. As can see the

number of nodal points can be effect on the results. Also, from all tables and figures we conclude

that the approximate results of BFM1 for noiseless data are more accurate than BFM2.



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 75

Table 5: The absolute errors of g(t) when n = m = l = 12, T = 1.01 with noiseless and noisy

data.

BFM1 BFM2

t σ = 0% σ = 1% σ = 3% σ = 0% σ = 1% σ = 3%

0.0 1.9× 10−8 1.7× 10−3 3.7× 10−3 1.3× 10−2 1.4× 10−2 1.6× 10−2

0.1 6.7× 10−9 1.4× 10−3 3.1× 10−3 1.9× 10−4 1.5× 10−3 3.1× 10−3

0.2 2.1× 10−9 1.3× 10−3 2.9× 10−3 2.4× 10−3 1.0× 10−3 4.6× 10−4

0.3 5.4× 10−10 1.3× 10−3 2.8× 10−3 4.2× 10−3 2.8× 10−3 1.3× 10−3

0.4 2.7× 10−11 1.3× 10−3 2.8× 10−3 2.9× 10−3 1.5× 10−3 9.6× 10−5

0.5 8.2× 10−11 1.3× 10−3 2.8× 10−3 1.4× 10−3 2.7× 10−3 4.2× 10−3

0.6 4.1× 10−11 1.3× 10−3 2.8× 10−3 5.2× 10−3 6.5× 10−3 8.0× 10−3

0.7 5.6× 10−11 1.3× 10−3 2.8× 10−3 3.7× 10−3 5.1× 10−3 6.5× 10−3

0.8 1.7× 10−10 1.3× 10−3 2.8× 10−3 3.3× 10−3 2.0× 10−3 5.0× 10−4

0.9 3.0× 10−10 1.3× 10−3 2.8× 10−3 7.4× 10−3 6.0× 10−3 4.5× 10−3

1.0 4.1× 10−10 1.3× 10−3 2.8× 10−3 1.5× 10−2 1.6× 10−2 1.8× 10−2

Table 6: The absolute errors of f (x) when n= m= l = 12, T = 1.01 with noiseless and noisy

data.

BFM1 BFM2

x σ = 0% σ = 1% σ = 3% σ = 0% σ = 1% σ = 3%

0.3 5.4× 10−4 2.8× 10−3 6.6× 10−3 4.8× 10−3 6.2× 10−3 1.4× 10−2

0.4 7.5× 10−4 2.6× 10−3 6.5× 10−3 1.7× 10−3 5.0× 10−3 9.8× 10−3

0.5 9.6× 10−4 2.3× 10−3 6.0× 10−3 1.7× 10−3 3.1× 10−3 3.9× 10−3

0.6 1.1× 10−3 1.9× 10−3 5.3× 10−3 4.8× 10−3 1.0× 10−3 1.4× 10−3

0.7 1.1× 10−3 1.5× 10−3 4.6× 10−3 6.5× 10−3 3.7× 10−4 4.9× 10−3

0.8 1.0× 10−3 1.2× 10−3 3.8× 10−3 6.3× 10−3 8.0× 10−4 5.3× 10−3

0.9 8.1× 10−4 1.0× 10−3 3.1× 10−3 3.9× 10−3 1.4× 10−5 2.2× 10−3

1.0 5.1× 10−4 9.7× 10−4 2.6× 10−3 1.3× 10−5 1.7× 10−3 3.7× 10−3



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 76

Table 7: The values of RMS(g) and RMS( f ) for various values of T with noisy data (σ = 1%)

and n= m= l = 12.

BFM1 BFM2

T RMS(g) RMS( f ) RMS(g) RMS( f )

1.01 0.0014006 0.0020455 0.0075399 0.0095838

3 0.0087923 0.0187542 0.0059202 0.0087923

5 0.0526107 0.1159522 0.0111424 0.0096519

7 0.0526152 0.1159473 0.0108588 0.0122928

10 0.0526188 0.1159588 0.0103002 0.0129093

20 0.0526188 0.1159509 0.0082595 0.0217635

Table 8: The values of RMS(g) and RMS( f ) for various values of m, n, l with noisy data

(σ = 1%) and T = 1.01.

BF M1 BF M2

n m l RMS(g) RMS( f ) RMS(g) RMS( f )

2 2 2 0.0161029 0.1020894 0.3490224 0.0347809

5 5 5 0.0014011 0.0036349 0.0080622 0.0813068

8 8 8 0.0014007 0.0023907 0.0068267 0.0221749

12 12 12 0.0014006 0.0020455 0.0073905 0.0095838

15 15 15 0.0014004 0.0019496 0.0082203 0.0091647

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.2

0.4

0.6

0.8

1

1.2

1.4

t

g(
t)

 

 

Exact
σ=0%

σ=1%

σ=3%

σ=5%

(a) BFM1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

0.2

0.4

0.6

0.8

1

1.2

1.4

t

g(
t)

 

 

Exact
σ=0%

σ=1%

σ=3%

σ=5%

(b) BFM2

Figure 4: The exact and approximate solution of g(t) when n = m = l = 6, T = 5 with

noiseless and noisy data.



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 77

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x

f(
x)

 

 

Exact
σ=0%

σ=1%

σ=3%

σ=5%

(a) BFM1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

x

f(
x)

 

 

Exact
σ=0%

σ=1%

σ=3%

σ=5%

(b) BFM2

Figure 5: The exact and approximate solution of f (x) when n = m = l = 5, T = 5 with

noiseless and noisy data.

Example 3. Consider the following problem [29]:

Ut =Ux x ; 0< x < 1, 0< t < 1, (54)

U(x , 0) =

¨
x2 + sin x; 0≤ x < 1

2 ,

f (x); 1
2 ≤ x ≤ 1,

(55)

U(1, t) =1+ 2t + exp(−t) sin 1; 0≤ t ≤ 1, (56)

U(
1

2
, t) =

1

4
+ 2t + exp(−t) sin(

1

2
); 0≤ t ≤ 1, (57)

Ux(
1

2
, t) =1+ exp(−t) cos(

1

2
); 0≤ t ≤ 1. (58)

The exact solution of this problem is

U(x , t) =x2 + 2t + exp(−t) sin x ,

g(t) =2t,

f (x) =x2 + sin x .

Tables 9 and 10 show the absolute values of error for noiseless and noisy (σ = 1%) data using

the methods proposed in Section 4 and compare the results with the result obtained using the

scheme (MFS) introduced in [29]. The values of RMS(g) and RMS( f ) with various noise level

are presented in Figures 6 and 7, respectively, which indicate that our proposed methods are

effective. From These tables and these figures we conclude that our numerical methods are more

accurate than the proposed method in [29].



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 78

Table 9: The absolute error of g(t) when n = m = l = 5, T = 1.01 with noiseless and noisy

data.

BFM1 BFM2 MFS [29]

t σ = 0% σ = 1% σ = 0% σ = 1% σ = 0% σ = 1%

0.0 2.7× 10−13 8.0× 10−4 2.3× 10−5 1.0× 10−3 3.1× 10−3 2.0× 10−3

0.1 2.4× 10−13 7.4× 10−4 1.2× 10−4 6.4× 10−4 1.6× 10−2 1.6× 10−2

0.2 2.3× 10−13 6.8× 10−4 8.4× 10−5 4.9× 10−4 4.1× 10−5 4.2× 10−4

0.3 2.3× 10−13 6.4× 10−4 2.4× 10−5 4.8× 10−4 1.1× 10−3 1.6× 10−3

0.4 2.3× 10−13 5.9× 10−4 1.0× 10−4 5.2× 10−4 1.5× 10−3 1.0× 10−3

0.5 2.4× 10−13 5.6× 10−4 1.1× 10−4 5.5× 10−4 2.5× 10−3 2.0× 10−3

0.6 2.7× 10−13 5.3× 10−4 4.3× 10−5 5.6× 10−4 1.2× 10−3 6.7× 10−4

0.7 3.0× 10−13 5.0× 10−4 7.2× 10−5 5.2× 10−4 9.9× 10−4 1.5× 10−3

0.8 3.4× 10−13 4.9× 10−4 1.5× 10−4 4.8× 10−4 1.9× 10−3 2.4× 10−3

0.9 3.8× 10−13 4.7× 10−4 7.7× 10−5 4.8× 10−4 3.7× 10−4 1.4× 10−4

1.0 4.3× 10−13 4.6× 10−4 3.1× 10−4 6.0× 10−4 7.6× 10−3 7.1× 10−3

Table 10: The absolute error of f (x) when n = m = l = 5, T = 3 with noiseless and noisy

data.

BFM1 BFM2 MFS [29]

x σ = 0% σ = 1% σ = 0% σ = 1% σ = 0% σ = 1%

0.50 5.0× 10−11 1.0× 10−4 1.3× 10−4 1.0× 10−3 3.9× 10−3 6.7× 10−3

0.55 1.1× 10−10 1.1× 10−4 5.9× 10−5 9.5× 10−4 4.0× 10−3 6.7× 10−3

0.60 3.4× 10−10 1.1× 10−4 2.1× 10−5 9.1× 10−4 3.8× 10−3 6.6× 10−3

0.65 6.5× 10−10 1.1× 10−4 1.0× 10−4 8.8× 10−4 3.6× 10−3 6.2× 10−3

0.70 1.0× 10−9 1.2× 10−4 1.7× 10−4 8.6× 10−4 3.2× 10−3 5.8× 10−3

0.75 1.3× 10−9 1.2× 10−4 2.5× 10−4 8.5× 10−4 2.5× 10−3 5.1× 10−3

0.80 1.7× 10−9 1.3× 10−4 3.1× 10−4 8.5× 10−4 2.0× 10−3 4.3× 10−3

0.85 2.1× 10−9 1.3× 10−4 3.7× 10−4 8.7× 10−4 1.3× 10−3 3.4× 10−3

0.90 2.4× 10−9 1.3× 10−4 4.2× 10−4 9.0× 10−4 5.2× 10−4 2.8× 10−3

0.95 2.8× 10−9 1.4× 10−4 4.5× 10−4 9.4× 10−4 3.2× 10−4 1.4× 10−3

1.00 3.1× 10−9 1.4× 10−4 4.8× 10−4 1.0× 10−3 1.1× 10−3 3.4× 10−4



M. Rostamian, A. Shahrezaee / Eur. J. Pure Appl. Math, 9 (2016), 64-83 79

0 1 2 3 4 5 6 7
0

1

2

3

4

5

6
x 10

−3

σ

R
M

S
(g

)

 

 

BFM1
BFM2
MFS [17]

Figure 6: The values of RMS(g) with various noise level and n= m= l = 5, T = 1.01.

0 1 2 3 4 5 6 7
0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

σ

R
M

S
(f

)

 

 

BFM1
BFM2
MFS [17]

Figure 7: The values of RMS( f ) with various noise level and n= m= l = 5, T = 3.

6. Conclusion

In this paper, we have used two new numerical methods of using basis functions with

Tikhonov regularization scheme to solve a backward IHCP. Two unknown functions in this IHCP

are estimated. The methods provide space-time approximations for the temperature derived



REFERENCES 80

by expanding the required approximate solutions using collocation based on basis function

interpolation methods. Three numerical examples in the presence of various noise levels added

in the input boundary data, have been solved. On the other hand, for all given examples the

issue of numerical stability is discussed. It is shown that applying Tikhonov regularization

method with GCV criterion for solving the final systems of equation (36), resulted in quite

satisfactory findings that compare well with exact solutions.

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