A multiple-scale Pascal polynomial triangle solving elliptic equations and inverse Cauchy problems

Engineering Analysis with Boundary Elements 62 (2016) 35–43

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

A multiple-scale Pascal polynomial triangle solving elliptic equations and inverse Cauchy problems Chein-Shan Liu, Chung-Lun Kuo n Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan

art ic l e i nf o

a b s t r a c t

Article history: Received 23 April 2015 Received in revised form 31 July 2015 Accepted 11 September 2015

The polynomial expansion method is a useful tool to solve partial differential equations (PDEs). However, the researchers seldom use it as a major medium to solve PDEs due to its highly ill-conditioned behavior. We propose a single-scale and a multiple-scale Pascal triangle formulations to solve the linear elliptic PDEs in a simply connected domain equipped with complex boundary shape. For the former method a constant parameter R0 is required, while in the latter one all introduced scales are automatically determined by the collocation points. Then we use the multiple-scale method to solve the inverse Cauchy problems, which is very accurate and very stable against large noise to 20%. Numerical results confirm the validity of the present multiple-scale Pascal polynomial expansion method. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Multi-scale polynomial expansion Pascal triangle Elliptic PDEs Inverse Cauchy problems

1. Introduction There are many papers which are concerned with the numerical solutions of elliptic type boundary value problems (BVPs) for the computational applications in several areas, such as, Liu [1–6], and Pradhan et al. [7]. Nowadays the meshless and mesh reduction methods are the main stream of numerical computation methods, to name a few, Zhu et al. [8,9], Atluri and Zhu [10,11], Atluri et al. [12], Atluri and Shen [13], Cho et al. [14], Jin [15], and Li et al. [16]. These methods always lead to nonlinear algebraic equations (NAEs), when one applied them to solve nonlinear partial differential equations (PDEs). Many collocation techniques which are special cases of the meshless local Petrov–Galerkin (MLPG) method [10,11], which together with the expansions by different basis-functions were employed to solve the elliptic type BVPs; see, for example, Cheng et al. [17], Hu et al. [18], Algahtani [19], Tian et al. [20], Hu and Chen [21], and Libre et al. [22]. Li et al. [23] gave a very detailed description of the collocation Trefftz method. Basically, the above bases expansion methods are effective for linear problems. The purpose of this paper is to develop a quite powerful polynomial expansion algorithm, having the advantage of easy numerical implementation, and having a great flexibility applied to most linear elliptic type BVPs defined in arbitrary plane domain. n

Corresponding author. E-mail address: [email protected] (C.-L. Kuo).

http://dx.doi.org/10.1016/j.enganabound.2015.09.003 0955-7997/& 2015 Elsevier Ltd. All rights reserved.

We begin with the following linear elliptic equation:

Δuðx; yÞ ¼ Fðx; y; u; ux ; uy Þ; uðx; yÞ ¼ Hðx; yÞ;

ðx; yÞ A Γ ;

ðx; yÞ A Ω;

ð1Þ ð2Þ

where Δ is the Laplacian operator, Γ is the boundary of the problem domain Ω, and F and H are given functions with F being linear on ðu; ux ; uy Þ. The polynomial interpolation is an ill-posed problem and it makes the interpolation by higher-order polynomials not being easy to be numerically implemented. In order to overcome these difficulties, Liu and Atluri [24] have introduced a characteristic length into the high-order polynomials expansion, which improved the numerical accuracy for the applications to solve some ill-posed linear problems. At the same time, Liu et al. [25] have developed a multi-scale Trefftz-collocation Laplacian conditioner to deal with the ill-conditioned linear systems. This concept of multi-scale Trefftz-collocation method has been later employed by Chen et al. [26] to solve the sloshing wave problem. Liu [27] has proposed a multi-scale half-order polynomial interpolation method, and Kuo et al. [28] have used the modified two characteristic lengths Pascal triangle method to solve inverse heat source problem. In this paper we extend the work by Kuo et al. [28], and propose a new multiple-scale expansion technique by high-order polynomials, which can overcome the abovementioned ill-conditioned behavior. This paper is arranged as follows. In Section 2 we introduce a simple modification of the Pascal triangle expansion method by considering a characteristic length. Then according to the concept

36

C.-S. Liu, C.-L. Kuo / Engineering Analysis with Boundary Elements 62 (2016) 35–43

of equilibrated matrix we introduce a multiple-scale into the Pascal triangle expansion method, of which the introduced scales are fully determined by the collocation points. The numerical examples for the direct problems are solved in Section 4. The inverse Cauchy problems are described in Section 5, while the numerical examples are given in Section 6. Finally, we draw some conclusions in Section 7.

2. A modified polynomial expansion method The use of polynomial expansion as a trial solution of PDE is simple and is straightforward to derive the required algebraic equations after a suitable collocation in the problem domain. However, it is seldom used as a major numerical tool to solve linear PDEs. The main reason is that the resultant linear algebraic equations (LAEs) are often highly ill-conditioned. How to reduce the condition number of the linear system becomes an important issue in the application of polynomials expansion to PDEs. The elements in following polynomial matrix: 2 3 1 y y2 ⋯ ym  1 ym 6 x xy xy2 ⋯ xym  1 xym 7 6 7 6 2 7 2 2 2 2 m1 2 m 7 6x x y x y ⋯ x y x y ð3Þ 6 7 6 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 7 4 5 xm xm y xm y2 ⋯ xm ym  1 xm ym are often used to expand the solution of uðx; yÞ. If the elements are restricted in the left-upper triangle then such an expansion is known as the Pascal triangle expansion:

uy ðx; yÞ ¼

2

Δuðx; yÞ ¼

x x y x y x y



i  j 

y R0

x R0

;

ð8Þ

i  j  2 

j  3 # :

y R0

j  1

ð9Þ

In order to obtain an accurate solution of the linear elliptic PDE and inverse Cauchy problem by using the modified Pascal triangle polynomial expansion method, we have to develop more effective and accurate solution method to solve these LAEs by reducing the condition numbers. Instead of the single-scale expansion we consider a novel multiple-scale Pascal triangle expansion of uðx; yÞ by m X i X

uðx; yÞ ¼

cij sij xi  j yj  1 ;

ð10Þ

i¼1j¼1

where the scales sij will be determined below. From Eq. (5) it is straightforward to write

ð4Þ 4

uy ðx; yÞ ¼ 5

m X i X

cij ði  jÞxi  j  1 yj  1 ;

ð11Þ

m X i X

cij ðj  1Þxi  j yj  2 ;

ð12Þ

i¼1j¼1

xy y

Δuðx; yÞ ¼

Therefore, the solution uðx; yÞ is expanded by uðx; yÞ ¼

 ði jÞði j  1Þ

j  2

3. A multiple-scale Pascal triangle

… … … … … … … …

m X i X

y R0

i¼1j¼1

x x3 y x2 y2 xy3 y4 2 3

i  j 

Inserting these equations into Eqs. (1) and (2), and selecting n1 and n2 collocation points on the boundary and in the domain, to satisfy the boundary condition and the field equation, respectively, we can obtain a system of LAEs to solve the n coefficients cij. The above method is a single-scale Pascal triangle expansion method.

2

x3 x2 y xy2 y3 3 2

x R0

" cij R20

x þ ðj  1Þðj 2Þ R0

ux ðx; yÞ ¼

4

4

m X i X i¼1j¼1

x xy y

5

 cij ðj  1ÞR0

i¼1j¼1

1 x y

m X i X

cij xi  j yj  1 ;

m X i X

h i cij ði jÞði  j 1Þxi  j  2 yj  1 þ ðj  1Þðj  2Þxi  j yj  3 :

i¼1j¼1

ð13Þ

ð5Þ

i¼1j¼1

where the coefficients cij are to be determined, whose number of all elements is n ¼ mðm þ 1Þ=2. The highest order of the above polynomial is m  1. Because x and y in the problem domain Ω may be an arbitrarily large quantity, the above expansion would lead to a divergence of the powers xm and ym. Thus according to the suggestion by Liu and Atluri [24] we can employ the following modified polynomial expansion method, involving a normalized length scale R0, to express the solution:  i  j  j  1 m X i X x y uðx; yÞ ¼ cij ; ð6Þ R R 0 0 i¼1j¼1 where the coefficients cij are to be determined, whose number of all elements is n ¼ mðm þ 1Þ=2. The highest order of the above polynomial is m  1. Here we use a modified Pascal triangle to expand the solution, where R0 4 0 is the characteristic length of the plane domain we consider. Basically we need Ω A ½  R0 ; R0  ½  R0 ; R0 . From Eq. (6) it is straightforward to write  i  j  1  j  1 m X i X x y ux ðx; yÞ ¼ cij ði  jÞR0 ; ð7Þ R0 R0 i¼1j¼1

Inserting these equations into Eqs. (1) and (2), and selecting n1 and n2 collocation points on the boundary and in the domain, to satisfy the boundary condition and the field equation, respectively, we can obtain a system of LAEs to solve the n coefficients cij. It is convenient to express the resulting LAEs in terms of a matrixvector product form: Ac ¼ b:

ð14Þ

Usually, Eq. (14) is an over-determined system for that we may collocate more points to generate more equations, which are used to find n coefficients in c with n⪡nc . First the coefficients cij used in the expansion (5) can be expressed as an n-dimensional vector c with components ck ; k ¼ 1; …; n. Then for a generic point ðx; yÞ A Ω the term uðx; yÞ can be expressed as an inner product of a vector a with c, i.e., 2 3 c1 6c 7 6 27  6 7 T 7 uðx; yÞ ¼ 1 x y x2 xy y2 x3 x2 y xy2 y3 … 6 ð15Þ 6 c3 7 ¼ a c: 6⋮7 4 5 cn Similarly, for a generic point ðx; yÞ A Ω the term Δuðx; yÞ can be expressed as an inner product of a vector d with c, where the

C.-S. Liu, C.-L. Kuo / Engineering Analysis with Boundary Elements 62 (2016) 35–43

components dk are in the form of dk ¼ ði jÞði j  1Þxi  j  2 yj  1 þ ðj 1Þðj 2Þxi  j yj  3 . Then, when we select n1 points ðxi ; yi Þ; i ¼ 1; …; n1 on the boundary Γ to satisfy the boundary condition, and n2 points ðxi ; yi Þ; i ¼ 1; …; n2 on the domain Ω to satisfy the field equation, for example, for the Laplace equation we have 2 3 2 3 aT1 Hðx1 ; y1 Þ 6 7 6 ⋮ 7 6 7 ⋮ 6 7 6 7 6 aT 7 6 7 6 n1 7 6 Hðxn1 ; yn1 Þ 7 6 7 6 7: A ¼ 6 T 7; b ¼ 6 ð16Þ 7 0 6 d1 7 6 7 6 7 6 7 6 ⋮ 7 4 5 ⋮ 4 T 5 0 dn2 Instead of Eq. (14), we can solve a normal linear system by the conjugate gradient method (CGM) Dc ¼ b1 ;

ð17Þ

where b1 ¼ AT b;

D ¼ AT A 4 0:

ð18Þ

Indeed the scaling of LAEs is an important topic that has a long history of development. A matrix is equilibrated if all its rows or columns have the same norm, and under this condition the matrix is better conditioned. The problem is the search of some suitable diagonal matrices Q and P, such that the condition number of QAP is reduced as much as possible. Liu [29] has proposed a simple procedure to find P and Q only through a few operations, which are derived explicitly. Previously, Liu [30] has used the concept of equilibrated matrix to choose the best source points for the method of fundamental solutions, and Liu and Atluri [31] have employed the concept of equilibrated matrix to find the best multiple-scale of the Trefftz method used in the solution of inverse Cauchy problems, whose resulting linear system is less illconditioned. Liu [32] has developed a general purpose optimally scaled vector regularization method to treat ill-conditioned linear problems, according to the idea of equilibrated matrix. If we demand the norm of each column of the coefficient matrix of A is equal, the multiple-scale sij is determined by k¼0 Do i ¼ 1; m Do j ¼ 1; i

ρðθÞ ¼ expð sin θÞ sin 2 ð2θÞ þ expð cos θÞ cos 2 ð2θÞ; where ρ is the radius function of closed-form solution is

Γ, and we suppose that the ð23Þ

First we note that when we use the original Pascal triangle expansion method with R0 ¼ 1 to solve this problem, the maximum error is large up to 10.925, which indicates that we cannot directly apply the Pascal triangle expansion method to solve linear elliptic problem, even when the polynomial exact solution is used. Under n1 ¼ 11, n2 ¼ 44 (nc ¼55), m ¼5, hence n ¼15, we employ R0 ¼ 4 and solve the resulting LAEs by using the CGM under the convergence criterion ε ¼ 10  8 . In Fig. 1 we plot the residual with respect to the number of iterations by solid line, which is convergence with 294 steps, and the error is plotted in Fig. 2(a), whose maximum error is 6:18  10  8 . On the other hand, the multiple-scale method is convergence with 35 steps as shown in Fig. 1(a), and it gives a highly accurate solution, whose error is shown in Fig. 2(b) with the maximum error being 7:95  10  13 . 4.2. Example 2 We consider the Laplace equation again; however, we suppose that the closed-form solution is more complex with uðx; yÞ ¼ ex cos y:

ð24Þ

Under n1 ¼ 50, n2 ¼ 200 (nc ¼250), and m ¼12, hence n¼ 55, we employ R0 ¼ 4 and solve the resulting LAEs by using the CGM under the convergence criterion ε ¼ 10  8 . However, the CGM for single-scale method does not converge within 20,000 steps. The error of u obtained by the single-scale method is plotted in Fig. 3 (a), whose maximum error is 1:48  10  3 . On the other hand, the multiple-scale method is convergence with 5764 steps and gives a more accurate solution, whose numerical error is shown in Fig. 3 (b) with the maximum error being 1:25  10  6 . The above two examples show that the multiple-scale method is much better than the single-scale method. Thus in the below we only use the multiple-scale method to solve the problems we consider. 1E+3

J c1 J sij ¼ J ck J

1E+2

ð19Þ

1E+1

where s11 ¼ 1 and ck denotes the kth column of A in Eq. (16). Such that in the new system

1E+0

Bc ¼ b;

1E-2

Multiple-scale method Single-scale method

1E-1

Residual

ð20Þ

the n column norms of the new coefficient matrix B are equal.

ð22Þ

u ¼ x2  y2 ¼ r 2 cos ð2θÞ:

k ¼ k þ1

Enddo;

37

1E-3 1E-4 1E-5

4. Numerical examples

1E-6

In this section we will apply the new methods to linear elliptic equations.

1E-7 1E-8

4.1. Example 1

1E-9

In this example we consider the Laplace equation in a complex amoeba-like irregular shape

Δuðx; yÞ ¼ 0;

ð21Þ

0

100

200

300

Number of iterations Fig. 1. For the solution of Laplace equation, comparing the residual curves obtained by the single-scale and multiple-scale methods.

C.-S. Liu, C.-L. Kuo / Engineering Analysis with Boundary Elements 62 (2016) 35–43

Absolute error

0

e-

4 0.0

00

02

02

0.0

10

00 0.

-0.5

2 0.000 4 0.0006

-1

-1 -1

-0.5

0

0.5

1

1.5

2

-1

2.5

-0.5

0

0.5

x

1.5

2

2.5

Absolute error 6 -0 -06 2e 4e

1e -13

1.5

13

1

y 0.5

1e

1e

-13

1e-

1e

-1

3

1.5

1

x

Absolute error

1

02 0.00004 0..00006 0

02

0.000

00

0

02

e1 4e 0 -10

-1

1. 2 5e e 5e1e-1-1010 -11 0 5e 1e-111 0

02

0

1.5

00. . 0000 0406

00

2e

06

0.

-10

-1

00

04 00 02 00 0.

5e

0.

2.

0.00

-0.5

0.

06

3.5

2

00

3e

y 0.5 0 -1100 4ee- -1 10 3.5 3e5e2.

0

0 .00 0. 0 00 04

0.

1e-10

y 0.5

4

1

00

10

004 0.0 02 0.00

004

0.

-11

1e-

0.0

0.000

1

5e

1.5

1.5

04 .00 02 00.00

2e-10 1.5e-10

-10 3e e-10 0 1 2.5 2e- -10 e 0 1.5 1e-1-11 5e

002

Absolute error

0.

38

3 -1 3e 3 -1 2e 3 -1

y 0.5

0 2e-06

3 -1

-13

3e

-0.5

2e

1e-1

3

0

3 - 1 13 13 3 4e 5e-6e-7e-1

-0.5 -1

-1 -1

-0.5

0

0.5

1

1.5

2

-1

2.5

-0.5

0

0.5

1

1.5

2

2.5

x

x Fig. 2. For the solution of Laplace equation, comparing the errors obtained by (a) the single-scale, and (b) the multiple-scale method.

Fig. 3. For the solution of example 2, comparing the errors obtained by (a) the single-scale method, and (b) the multiple-scale method.

4.3. Example 3

multiple-scale method by using the LU decomposition method, which is less robust than CGM. The comparison of the exact solution and the result obtained by the multiple-scale method is plotted in Fig. 5(a) and (b). Again we can see that the numerical result agrees very well with the exact solution. The error of u is plotted in Fig. 5(c), whose maximum error is 0.017. It is shown that the multiple-scale method reduces the ill-posedness effectively, so that we may solve the resultant LAEs by using any conventional linear solver.

We consider a more complex linear equation

Δuðx; yÞ ¼

1

σ ðx; yÞ

½f ðx; yÞ  σ x ðx; yÞux ðx; yÞ  σ y ðx; yÞuy ðx; yÞ;

ð25Þ

where σ ðx; yÞ ¼ 1 þ x2 þ y2 , and f ðx; yÞ ¼ 4ðx2  y2 Þ, such that Eq. (25) has a closed-form solution uðx; yÞ ¼ x2  y2 . Under n1 ¼ 50, n2 ¼ 200 (nc ¼250), and m ¼ 5, hence n ¼15, we solve the resulting LAEs from the multiple-scale method by using the CGM under the convergence criterion ε ¼ 10  12 . The CGM converges within 28 steps. The comparison of the exact solution and the result obtained by the multiple-scale method is plotted in Fig. 4(a). We can see that the numerical result coincides with the exact solution. The error of u is plotted in Fig. 4(b), whose maximum error is 3:19  10  14 . It is interesting that the multiple-scale method can provide very accurate solution. 4.4. Example 4 We consider a Helmholtz equation

Δuðx; yÞ þk2 uðx; yÞ ¼ 0; where uðx; yÞ ¼ sin

pkffiffiðx þyÞ 2

4.5. Example 5 We consider a nonhomogeneous modified Helmholtz equation

Δuðx; yÞ  uðx; yÞ ¼ f ðx; yÞ;

ð27Þ

defined in a unit disk, where uðx; yÞ ¼ tan ðxyÞ is a closed-form solution, and f ðx; yÞ can be obtained by inserting uðx; yÞ into Eq. (27). Under n1 ¼ 100, n2 ¼ 400, and m ¼20 we use the multiple-scale method to solve this problem. The error of u is plotted in Fig. 6, whose maximum error is 8:75  10  3 .

ð26Þ is a closed-form solution, where

ðx; yÞ A ½0; 12 . Under a quite large order of polynomial and wave number, that is, m ¼ 100, k ¼20, and equal spacing collocation points with Δx ¼ Δy ¼ 1=ð90  1Þ, we solve the resulting LAEs from the

5. Inverse Cauchy problems The detection of corrosion inside a pipe is a very important technique in engineering application. Here we consider a mathematical modelling of this problem and give an effective numerical

C.-S. Liu, C.-L. Kuo / Engineering Analysis with Boundary Elements 62 (2016) 35–43

39

Exact Multiple-scale

u(x,y) -2

-1

0

1.5

6

5

0

y 0.5

4

3

2

1

-1

1

7

0 0

-0.5 -1

-1

-1

-0.5

0

0.5

1

1.5

2

2.5

x Absolute error 1.5 14

-14

1e-14

2e

e-1 4 1e-14 5e-15

1.5

e-1

1

4

2.5e-

5e-15

1.5

y 0.5

1.5 e51ee--1144 15 5e-15 1e-14 1.5e-14 2e-14 4 e-1 2.5

5e-

15

0

5e-15

-0.5 -1 -1

-0.5

0

0.5

1

1.5

2

2.5

x Fig. 4. For the multiple-scale method of example 3, showing (a) the exact and the numerical solutions, and (b) numerical error.

algorithm for a method to detect the corrosion by an electrical field in the pipe. Given the Cauchy data uðx; yÞ and the Neumann data ∂u=∂nðx; yÞ at the point ðx; yÞ A R2 with an unit outward normal nðx; yÞ on the accessible part Γ 1 ≔fðr; θÞj r ¼ ρðθÞ; 0 r θ r βπ g of a noncircular contour, we consider an inverse Cauchy problem of the Laplace equation Δuðx; yÞ ¼ 0 in two dimensions to find the unknown function uðx; yÞ on an inaccessible part Γ 2 ≔f ðr; θÞj r ¼ ρðθÞ; βπ o θ o2π g where Γ 2 ¼ Γ =Γ 1 . Now we turn our attention to the inverse Cauchy problem of Eq. (1) and consider the following overspecified boundary conditions: uðρ; θÞ ¼ hðθÞ; un ðρ; θÞ ¼ gðθÞ;

0 r θ r βπ ; 0 r θ r βπ ;

ð28Þ ð29Þ

where hðθÞ and gðθÞ are given functions and β r 1. The inverse Cauchy problem is specified as follows: to seek an unknown boundary function f ðθÞ on the part Γ2 of the boundary under Eqs. (1), (28) and (29) with the overspecified data on Γ1. This problem is to solve the Laplace equation under an overspecified Cauchy data on a partial noncircular boundary. Usually, it requires β Z 1, which is specified by Mera et al. [33] as a necessary condition for numerically solving the inverse Cauchy problem. However, in the present study we can let β o 1, even β ¼0.5, without losing the accuracy in the recovered boundary condition.

Fig. 5. For the multiple-scale method of example 4, showing (a) the exact solution, (b) the numerical solution, and (c) numerical error.

When the contour is a circle, Liu [34] has applied a modified Trefftz method to recover the unknown boundary data for the inverse Cauchy problem, but one needs to consider a regularization technique by truncating the higher-mode components of the given data. Then, Liu [35] employed the modified collocation Trefftz method with a single characteristic length to solve the inverse Cauchy problems in simply and doubly connected domains. Recently, Liu [36] has developed a very powerful optimally generalized regularization method to solve the Cauchy problem of Laplace equation by using the MFS, of which β ¼0.4 can be used.

40

C.-S. Liu, C.-L. Kuo / Engineering Analysis with Boundary Elements 62 (2016) 35–43

u(x,y)

1

Exact Multiple-scale

0

0.8

.5 -0 .4 -0

0.4

0

-0.2 0.3 0.4 0. 5

2 -0.

0

2 0.

-0.6

1 -0.

0.1

-0.4

3 -0. 4 -0. .5 -0

-0.8 -1 -1

-0.5

0

0.5

1

x Absolute error

8 7

0.

00 000 2.00

4 0.0

-0.8

8 0.006

0.

0.0

03

01

0.0

-0.5

0

0.008 06

0.0 0.001

Before embarking numerical tests of the present method to solve the inverse Cauchy problems, we are concerned with the stability of the multiple-scale Pascal triangle method, in the case when the boundary data are contaminated by random noise, which is investigated by adding a different level of random noise on the boundary data. We use the function RANDOM-NUMBER given in Fortran to generate the noisy data R(i), which are random numbers in ½  1; 1. Hence we use the simulated noisy data given by

0.

00

4

02

0.5

1

Fig. 6. For the multiple-scale method of example 5, showing (a) the exact and the numerical solutions, and (b) numerical error.

In the polar coordinates we can derive [35]   ∂uðρ; θÞ ρ0 ∂uðρ; θÞ un ðρ; θÞ ¼ ηðθÞ  2 ; ∂ρ ρ ∂θ

ρðθÞ ηðθÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: 2 ρ ðθÞ þ ½ρ0 ðθÞ2

^ θ Þ ¼ hðθ Þ þ sRðiÞ; hð i i

^ θi Þ ¼ gðθi Þ þsRðiÞ gð

ð35Þ

as the inputs, where s is the level of noise. 6.1. Example 6

ð30Þ

where ð31Þ

On the other hand, we can also express un in terms of ux and uy by     ρ0 ρ0 un ¼ ηðθÞ cos θ þ sin θ ux þ ηðθÞ sin θ  cos θ uy : ð32Þ

ρ

Inserting Eqs. (11) and (12) into the above equation we can obtain un. Now, besides the Cauchy type boundary data we need to consider the Neumann type boundary data on Γ1, which can be generated by the following way. For a generic point ðx; yÞ A Γ 1 the term un ðx; yÞ can be expressed as an inner product of a vector e with c, where the components ek are generated by k¼0 Do i ¼ 1; m Do j ¼ 1; i k ¼ kþ1

ð33Þ

6. Testing the inverse Cauchy problems

x

ρ



5

0.0 0 0.05 03

1 00 0.

0 0.

0. 0 00 .0 70

ρ cos θ ρ

Then we apply the same technique as that specified in Section 3 to generate a new coefficient matrix B with the multiple-scale given by Eq. (19).

02

05 0.004 0.0 .003 0 0.002 0.001 0.002 0.004 0.005 03

00

0.

-0.6

7 00 0. 5 00

-0.4

-1 -1

8

.005 0.0 00.0 03 04 0.002 0.001 0.002 0.00034 0.00.005

0.

0 -0.2

0.

00

y

00

30.0

sin θ 

1 00 0. 4 00 0.

0.2

0.

0.001

0.4

0.006

06 0.0.007 0

0.6

0.00

0.2003 0.00 4 1 0 00 .0 0. 0

5

0.8

0.00

1

ρ

0

Then when we select n1 =2 points ðxi ; yi Þ; i ¼ 1; …; n1 on the boundary Γ1 to satisfy the boundary conditions, and n2 points ðxi ; yi Þ; i ¼ 1; …; n2 on the domain Ω to satisfy the field equation, for example, for the Laplace equation we can derive 2 T 3 2 3 a1 hðθ1 Þ 6 7 6 ⋮ 7 6 7 ⋮ 6 6 7 7 6 aT 7 6 7 6 n1 =2 7 6 hðθn =2 Þ 7 1 6 6 7 7 T 6 e 7 6 7 6 1 7 6 gðθ1 Þ 7 6 6 7 7 ⋮ 6 6 7: 7 A¼6 ð34Þ ⋮ 7 7; b ¼ 6 6 eT 7 6 7 6 n1 =2 7 6 gðθn1 =2 Þ 7 6 6 7 7 6 dT 7 6 7 0 6 1 7 6 7 6 6 7 7 ⋮ 6 ⋮ 7 4 5 4 5 T 0 dn2

0.1

0 0



Enddo:

2

1 -0.

0.2

y

0.

.2 -0

þ ðj 1Þxi  j yj  2

0. 5 0. 0.4 3

-0 .3

0.6

   ρ0 ek ¼ ηðθÞ ði jÞxi  j  1 yj  1 cos θ þ sin θ

In this example we consider the inverse Cauchy problem of Laplace equation in a complex amoeba-like irregular shape as that given in Eqs. (21)–(23). We take β ¼ 1, n1 ¼ 50, n2 ¼ 200 (nc ¼250), and m ¼5, hence n¼ 15, and solve the resulting LAEs by using the CGM which converges with 26 steps under ε ¼ 10  12 . If s¼ 0, the maximum error in the whole domain and on the lower half boundary are both 1:497  10  12 . It can be seen that highly accurate results are obtained. Let s¼0.2. The CGM converges with 27 steps under ε ¼ 10  12 . We show the numerical error in Fig. 7(a) with the maximum error being 0.137. The recovered data on the lower half boundary is compared with the exact one in Fig. 7(b), which are almost coincident, with the maximum error being 0.139. To the best knowledge of authors, this accuracy is never seen before, when one knows that the imposed noise is large up to 20%. 6.2. Example 7 In this example we consider the inverse Cauchy problem of Laplace equation in a complex amoeba-like irregular shape as that

C.-S. Liu, C.-L. Kuo / Engineering Analysis with Boundary Elements 62 (2016) 35–43

41

2

Absolute error 1.5

1

1

2

0.06

0.04 0.06

2 0.0

0.02

0.04

0.0

0

f ( θ)

y 0.5 0

0.08

0.

08

-0.5 -1 -1

-0.5

1

0.

0

0.5

1

1.5

2

-1

2.5

x

Exact Numerical with s=0.002 and

8

=0.5

-2 Numerical with s=0.2

1.2

Exact

6

2.0

2.4

2.8

3.2

3.6

4.0

4.4

4.8

5.2

5.6

6.0

6.4

θ Fig. 9. Using the multiple-scale method to solve an inverse Cauchy problem of Laplace equation in example 7 with β¼ 0.5, comparing recovered data and exact solution.

f ( θ)

4

2

0

-2 2.8

3.2

3.6

4.0

4.4

4.8

5.2

5.6

6.0

6.4

θ Fig. 7. Using the multiple-scale method to solve an inverse Cauchy problem of Laplace equation in example 6, (a) showing the numerical error, and (b) comparing recovered data and exact solution.

2

1

f (θ)

1.6

given in Eqs. (21) and (22), and with the exact solution given by u ¼ cos x cosh y þ sin x sinh y. We take β ¼ 1, n1 ¼ 40, n2 ¼ 200 (nc ¼240), and m ¼7, hence n ¼28, and solve the resulting LAEs by using the CGM, which converges with 60 steps under ε ¼ 10  4 . Let s ¼0.1. The recovered data on the lower half boundary is compared with the exact one in Fig. 8, with the maximum error being 0.101. This accuracy is much better than that calculated by Liu [37], when one knows that the imposed noise is large up to 10% much larger than 1% used in the above paper. In Fig. 9, by comparing the numerical solution with exact solution for β ¼0.5 and s¼ 0.001, we can see that the present multiple-scale method with n1 ¼ 60, n2 ¼ 180, m ¼8, of which the CGM converges with 843 steps under ε ¼ 10  12 , is stable to recover the unknown boundary data. Amazingly, although the data are only over-specified on a 25% of the overall boundary with β ¼0.5 and only with data measured at 60 points, we can still recover other boundary data with an excellent accuracy with the maximum error being 0.0597. Previously, Liu [34] used the modified Trefftz method can treat Cauchy problem with β ¼0.5 and s¼0. To our best knowledge, in the open literature there exist no other numerical methods that can treat this type Cauchy problem with β ¼0.5 and under s¼0.002. This accuracy is much better than that calculated by Liu and Atluri [31].

0

6.3. Example 8

Exact Numerical with s=0.1

In this example we consider the inverse Cauchy problem with the following governing equation and closed-form solution [38]:

-1

uxx þ 2uyy þ uxy þ ux  uy ¼ 0;

ð36Þ

uðx; yÞ ¼ ex cos y;

ð37Þ 2

-2 2.8

3.2

3.6

4.0

4.4

4.8

5.2

5.6

6.0

6.4

θ Fig. 8. Using the multiple-scale method to solve an inverse Cauchy problem of Laplace equation in example 7, comparing recovered data and exact solution.

where ðx; yÞ A ½0; 1 . The Dirichlet boundary conditions are given on the right and left boundaries, while the overspecified boundary condition is imposed on the top boundary, and the bottom boundary condition is unknown. We take s ¼0.1, n1 ¼ 88, n2 ¼ 784 (nc ¼872), and m ¼6, hence n ¼21. Also in this example, we solve the resultant LAEs by

42

C.-S. Liu, C.-L. Kuo / Engineering Analysis with Boundary Elements 62 (2016) 35–43

1

In the inverse Cauchy problems, it was revealed that the unknown data can be recovered very accurately by utilizing the present method, although the overspecified data were provided on only a 25% (partial) of the boundary, and the imposed noise was large up to 20%. However, this paper does not consider the problems defined in the doubly and multiply connected domain. One may need to introduce the negative power polynomial expansion to deal with the singularity in the hole.

1.3

0.

9

2. 1

1.7

0

First, the authors appreciate the anonymous referee's comments, which improved the quality of this paper. The Project NSC102-2221-E-002-125-MY3 and the 2011 Outstanding Research Award from the Ministry of Science and Technology of Taiwan, and the 2014 ISI Highly Cited Researcher Award granted to the first author are highly appreciated.

1.7

Exact Multiple-scale

0

Acknowledgments

2.5

1.3

y 0.5

0.5

1

x 3

References

u(x,y=0)

2 Analytical solution s=0 s=1 s=2 s=3 s=5 s=10

1

0

0

1

2

x Fig. 10. Using the multiple-scale method to solve the inverse Cauchy problem in example 8, comparing (a) the exact and the numerical solutions and (b) recovered data and exact solution.

using the LU decomposition method. The comparison of the exact solution and the result obtained by the multiple-scale method is plotted in Fig. 10(a). We can see that the result is very good, of which the maximum error is 0.04. Also we compare the exact solution with the recovered unknown data under various levels of noise in Fig. 10(b). The multiple-scale method still can obtain very accurate result while the imposed noise is large up to 10%. The result is better than that obtained by Fan et al. [38].

7. Conclusions In this paper we have proposed two novel Pascal triangles endowing with a single scale and a multiple-scale to solve the direct and inverse Cauchy problems for the linear elliptic equations defined in arbitrary single domain with complex boundary shapes. In contrast to the previous multiple-scale methods, where one needs to judiciously to select some suitable values of R0 or Rk to obtain accurate solution, in the presently proposed multiple-scale Pascal triangle method, we could base the selection of the better values of sij on the concept of equilibrated matrix to derive the multiple-scale sij resorted in a closed-form, which are fully determined by the collocation points. Such that the new techniques in the solution of linear PDEs by using the high-order Pascal polynomials are very accurate and easy numerical implementation.

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