New wavelet based full-approximation scheme for the numerical solution of nonlinear elliptic partial differential equations

Alexandria Engineering Journal (2016) 55, 2797–2804

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Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

New wavelet based full-approximation scheme for the numerical solution of nonlinear elliptic partial differential equations S.C. Shiralashetti *, M.H. Kantli, A.B. Deshi Department of Mathematics, Karnatak University, Dharwad 580003, India Received 19 December 2015; accepted 19 July 2016 Available online 9 August 2016

KEYWORDS Daubechies wavelet filter coefficients; Full-approximation scheme; Elliptic partial differential equation; Elastohydrodynamic lubrication problems

Abstract Recently, wavelet analysis application has dragged the attention of researchers in a wide variety of practical problems, particularly for the numerical solution of nonlinear partial differential equations. Based on Daubechies filter coefficients, a modified method using wavelet intergrid operators known as new wavelet based full-approximation scheme (NWFAS) similar to multigrid fullapproximation scheme (FAS) is developed for the numerical solution of nonlinear elliptic partial differential equations. The present method gives higher accuracy in terms of better convergence with low CPU time. The results of tested examples of proposed method show better performance which is demonstrated through the illustrative examples. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Nonlinearity is essential in the majority of physical phenomena and engineering processes, resulting in nonlinear differential equations. In fact, these nonlinear equations are usually difficult to solve, since no general technique works globally. Hence each individual equation has to be studied as a separate problem. In the particular interest, the nonlinear elliptic type equations are encountered in transport problems, notably fluid flow problems. Solutions to this type of problems are usually required more CPU time with slow convergence. The nonlinear character of the partial differential equations that govern these problems reduces the analytical solution is difficult. In * Corresponding author. Fax: +91 836 347884. E-mail address: [email protected] (S.C. Shiralashetti). Peer review under responsibility of Faculty of Engineering, Alexandria University.

fact, the classical methods (for example finite difference method) are used to solve the problems with low accuracy in more CPU time. System of nonlinear equations is difficult to solve in general, usually, solving these equations using iterative methods such as Newton’s method, Jacobi iterative method, and Gauss-Seidel method. The FAS is largely applicable in increasing the efficiency of the iterative methods used to solve nonlinear system of algebraic equations. FAS is a well-founded numerical method for solving nonlinear system of equations for a approximating given differential equation. In the historical three decades the development of effective iterative solvers for nonlinear systems of algebraic equations has been a significant research topic in numerical analysis, computational science and engineering. Nowadays it is recognized that FAS iterative solvers are highly efficient for nonlinear differential equations introduced by Brandt [1]. A detailed treatment of FAS is given in Briggs et al. [2]. An introduction of FAS is

http://dx.doi.org/10.1016/j.aej.2016.07.019 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

2798 found in Hackbusch and Trottenberg [3], Wesseling [4] and Trottenberg et al. [5]. Many authors namely, Brandt [1] and Briggs et al. [2], applied the FAS to some classes of differential equations. Lubrecht [6], Venner and Lubrecht [7], Zargari et al. [8] and others have significant contributions in elastohydrodynamic lubrication (EHL) problems. Wavelets have several applications in approximation theory and have been widely used in the numerical approximation since from more than two decades. Wavelet based numerical methods are used for solving the system of equations with faster convergence and low computational cost. Some of the earlier works can be found in Dahmen et al. [9]. The many numerical methods employ various types of wavelets for the numerical solutions of different classes of differential equations; some of them are Haar wavelets by Bujurke et al. [10– 12] and Islam et al. [13], Ebolian and Fattahzdeh [14] used Chebyshev wavelets, B-Spline by Dehghan and Lakestani [15]. Due to desirable properties, researchers are now paying attention to Daubechies wavelet. Recently, Daubechies compactly supported orthogonal wavelet is widely used. Diaz et al. [16], Avudainayagam and Vani [17] and Bujurke et al. [18–20] and many researchers applied Daubechies wavelet method to some classes of differential equations. The main aim of the present paper was to introduce the new wavelet based FAS for the solution of nonlinear elliptic partial differential equations arising in science and engineering. The organization of this paper is as follows: Daubechies wavelet filter coefficients are given in Section 2. Section 3, presents the method of solution and intergrid operators. Numerical findings of the test problems are presented in Section 4. Finally, conclusions of the proposed work are discussed in Section 5. 2. Daubechies wavelet filter coefficients The class of compactly supported wavelet bases was introduced by Daubechies [21]. They are an orthonormal bases for functions in L2 ðRÞ. A family of orthogonal Daubechies wavelets with compact support has been constructed by [22]. Due to excellent properties of orthogonality and minimum compact support, Daubechies wavelets can be useful and convenient, providing guaranty of convergence and accuracy of the approximation in a wide variety of situations. In this paper, we use Daubechies filter coefficients for N ¼ 4 which are pffiffi pffiffi pffiffi pffiffi3, h1 ¼ 3þpffiffi3, h2 ¼ 3pffiffi3, Low pass filter coefficients: h0 ¼ 1þ 4 2 4 2 4 2 pffiffi pffiffi3 and h3 ¼ 1 4 2 pffiffi pffiffi pffiffi3, pffiffi3, High pass filter coefficients: g0 ¼ 1 g1 ¼  3 4 2 4 2 pffiffi pffiffi pffiffi3, g ¼  1þpffiffi3 . g2 ¼ 3þ 3 4 2 4 2

2.1. Discrete wavelet transform (DWT) matrix The matrix formulation of the discrete wavelet transforms (DWT) plays an important role in the wavelet method for the numerical computations. As we already know about the DWT matrix and its applications in the wavelet method and is given in [22] as,

S.C. Shiralashetti et al.

Using this matrix authors used restriction and prolongation operators W and WT respectively given in Section 3.2. 2.2. New discrete wavelet transform (NDWT) matrix Here, we developed NDWT matrix similar to DWT matrix in which by adding rows and columns consecutively with diagonal element as 1, which is built as,

Using W2 matrix, we introduced restriction and prolongation operators M and MT respectively such as wavelet multigrid operators given in Section 3.3. 3. Method of solution Consider the nonlinear partial differential equation. After discretizing the partial differential equation through the FDM, we get the system of nonlinear equations of the form, Fðui j Þ ¼ bi j ;

ð1Þ

where i; j ¼ 1; 2; . . . ; N, which have N  N equations with N  N unknowns. Solve Eq. (1) through iterative method Gauss Seidel (GS), we get approximate solution v. Approximate solution contains some errors, and therefore required solution equals to sum of approximate solution and error. There are many methods to minimize such error to get the accurate solution. Some of them are FAS, WFAS, NWFAS, etc. Now we are discussing the method of solution of the above mentioned methods as below. 3.1. Full-Approximation Scheme (FAS) The algorithm of FAS given by Briggs et al. [2] is as follows, Step – 1: From the system (1), we get the approximate solution v for u. Now we find the residual as rNN ¼ bNN  AðvÞNN :

ð2Þ

New wavelet based full-approximation scheme

2799

We reduce the matrices in the finer level to coarsest level and then construct the matrices back to finer level from the coarsest level using Restriction as well as, Prolongation operator, i.e.,

e44 ¼ P42 e22 R24 and so on we have, eNN ¼ PNN2 eN2 N2 RN2 N : Step – 6: Correct the solution with error. uNN ¼ vNN þ eNN . This is the required solution of the given partial differential equation.

and

3.2. Wavelet Full-Approximation Scheme (WFAS) The same procedure is applied as explained in the FAS. Instead of using R and P matrices, we use wavelet intergrid operators as,

Step – 2: rN2 N2 ¼ RN2 N rNN PNN2 :

and WT respectively.

Similarly,

3.3. New Wavelet Full-Approximation Scheme (NWFAS)

vN2 N2 ¼ RN2 N vNN PNN2 and     A vN2 N2 þ eN2 N2 þ A vN2 N2 ¼ rN2 N2 :

ð3Þ

Here also the same procedure is applied as explained in the above methods. Instead of using R and P matrices, we use modified wavelet intergrid operators as,

ð4Þ

and MT respectively.

Solve Eq. (3) with initial guess ‘0’, we get eN2 N2 . Step – 3: rN4 N4 ¼ RN4 N2 rN2 N2 PN2 N4 : Similarly, vN4 N4 ¼ RN4 N2 vN2 N2 PN2 N4 and     A vN4 N4 þ eN4 N4 þ A vN4 N4 ¼ rN4 N4 :

Solve Eq. (4) with initial guess ‘0’, we get eN4 N4 . Step – 4: The procedure continues up to the coarsest level, we have, r11 ¼ R12 r22 P21 : Similarly, v11 ¼ R12 v22 P21 and Aðv11 þ e11 Þ þ Aðv11 Þ ¼ r11 : Solve Eq. (5) we get, e11 . Step – 5: Interpolate error up to the finer level, i.e. e22 ¼ P21 e11 R12 ;

ð5Þ

4. Numerical experiment Here, we present some of the test examples, which show the efficiency of NWFAS. The error is defined as L1 ¼ max jue  ua j, where ue and ua are exact and approximate solution respectively. Rate of convergence Rc ðNÞ The rate of convergence is defined as Rc ðNÞ ¼

logðmaxðEðN=2ÞÞ= maxðEðNÞÞÞ : log 2

ð6Þ

Example I. First, we consider a nonlinear two dimensional elliptic partial differential equation,

2800

S.C. Shiralashetti et al.

Table 1 Comparison of numerical solutions with exact solution for N ¼ 8 of Example I.

Table 2 Comparison of numerical solutions with exact solution for N ¼ 8 of Example II.

x

y

FDM

FAS

WFAS

NWFAS

Exact

x

y

FDM

FAS

WFAS

NWFAS

Exact

0.1111 0.2222 0.3333 0.4444 0.5556 0.6667 0.7778 0.8889 0.1111 0.2222 0.3333 0.4444 0.5556 0.6667 0.7778 0.8889

0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222

0.0063 0.0109 0.0140 0.0157 0.0160 0.0149 0.0121 0.0073 0.0109 0.0189 0.0242 0.0272 0.0278 0.0259 0.0210 0.0126

0.0066 0.0114 0.0149 0.0161 0.0164 0.0146 0.0121 0.0073 0.0114 0.0196 0.0257 0.0279 0.0285 0.0255 0.021 0.0126

0.0068 0.0119 0.0147 0.0166 0.0168 0.0149 0.0117 0.0073 0.0119 0.0213 0.0258 0.0291 0.0298 0.0259 0.0199 0.0126

0.007 0.0123 0.0156 0.0176 0.0178 0.0166 0.0133 0.008 0.0122 0.0215 0.0274 0.031 0.0313 0.0291 0.0234 0.0142

0.0098 0.0171 0.0219 0.0244 0.0244 0.0219 0.0171 0.0098 0.0171 0.0299 0.0384 0.0427 0.0427 0.0384 0.0299 0.0171

0.1111 0.2222 0.3333 0.4444 0.5556 0.6667 0.7778 0.8889 0.1111 0.2222 0.3333 0.4444 0.5556 0.6667 0.7778 0.8889

0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222

0.0014 0.0029 0.0045 0.0062 0.0082 0.0113 0.0174 0.0334 0.0056 0.0113 0.017 0.023 0.0292 0.0361 0.0439 0.0517

0.0014 0.0029 0.0044 0.0062 0.0082 0.0113 0.0174 0.0334 0.0056 0.0112 0.017 0.0229 0.0292 0.0361 0.0439 0.0517

0.0014 0.0029 0.0045 0.0062 0.0082 0.0113 0.0174 0.0334 0.0056 0.0113 0.017 0.0229 0.0292 0.0361 0.0439 0.0517

0.0014 0.0028 0.0043 0.0059 0.0078 0.0107 0.0163 0.032 0.0055 0.0111 0.0167 0.0225 0.0284 0.035 0.0418 0.0487

0.0014 0.0027 0.0041 0.0055 0.0069 0.0082 0.0096 0.011 0.0055 0.011 0.0165 0.022 0.0274 0.0329 0.0384 0.0439

 

 @2u @2u þ ueu ¼ 2½ðx  x2 Þ þ ðy  y2 Þ þ ðx  x2 Þ þ @x2 @y2

on unit square ð0; 1Þ  ð0; 1Þ with u ¼ 0 on the boundary. We obtained the results as the method explained in Section 3 and are presented in comparison with existing methods with exact solution uðx; yÞ ¼ ðx  x2 Þðy  y2 Þ in Table 1 and Fig. 1.

subjected to following Dirichlet boundary conditions uð0; yÞ ¼ 0; uð1; yÞ ¼ y þ a; uðx; 0Þ ¼ ax and uðx; 1Þ ¼ xðx þ aÞ. The analytical solution of (8) is uðx; yÞ ¼ xðxy þ aÞ. In the present computations, the constant a ¼ 0 simplicity, numerical solutions of Eq. (8) are presented in Table 2 and Fig. 2. Error analysis and the rate of convergence for different grids of first two examples are given in Table 3.

Example II. Next, we consider generalized nonlinear poisson equation is defined in a square domain of length L ¼ 1. The governing equation is [23],

Example III. Finally, we consider the dimensionless Reynolds equation of Elasto-Hydrodynamic lubrication point contact problem [24],

 2 @2u @2u @uðx; yÞ þ þ ¼ 2y  x4 @x2 @y2 @y

    @ @P @ @P @ e þ a2 e  ðqHÞ ¼ 0; @X @X @Y @Y dX

 ðy  y2 Þeðxx

2 Þðyy2 Þ

ð7Þ

ð8Þ

0.1

WFAS

FAS

0.1 0.05 0 20

y

0.05 0 20

20

10 0

0

20

10

10

y

x

10 0 0

x

0.1

Exact

0.1

MWFAS

ð9Þ

0.05 0 20

20

10 y

Figure 1

10 0

0

x

0.05 0 20

20

10

y

10 0 0

x

Comparison of numerical solutions with exact solution for N ¼ 16 of Example I.

New wavelet based full-approximation scheme

2801

1

WFAS

FAS

1 0.5 0 10

y

0 10

10

5 0

0

y

x

Exact

MWFAS

5 0

0

x

1

0.5 0 10

y

0.5 0 10

10

5

Table 3

10

5

5

1

Figure 2

0.5

0

0

10

5

5

y

x

5 0

0

x

Comparison of numerical solutions with exact solution for N ¼ 16 of Example II.

Error analysis and the rate of convergence for different N.

Test Problems

FDM

N

FAS

WFAS

NWFAS

Error(L1 )

RC

Error(L1 )

RC

Error(L1 )

RC

Error(L1 )

RC

I

8 16 32 64

1.4297E05 6.1376E06 3.8480E08 7.2618E09

– 1.22 7.32 2.41

1.3832E05 6.0736E06 3.8375E08 7.2568E09

– 1.19 7.31 2.40

1.3206E05 6.0327E06 3.8309E08 7.2536E09

– 1.13 7.30 2.40

1.0636E05 5.9394E06 3.8123E08 7.2441E09

– 8.41 7.28 2.40

II

8 16 32 64

2.2478E02 1.4678E02 8.3387E03 4.4395E03

– 0.61 0.82 0.91

2.2053E02 1.3091E02 8.0622E03 4.0162E03

– 0.75 0.70 1.01

2.1254E02 1.2782E02 7.4061E03 3.8194E03

– 0.73 0.79 0.96

2.1099E02 1.1211E02 7.1031E03 3.3103E03

– 0.91 0.66 1.10

where e ¼ qH , PðX; YÞ and HðX; YÞ are unknown pressure and kg film thickness, X is the dimensionless coordinate (X ¼ xa, a is the half length of the elliptic contact area in the x direction), Y is the dimensionless coordinate (Y ¼ yb, b is the half length of the elliptic contact area in the y direction), k a dimensionless 3

contact region. The Film thickness equation is given, in integral form as Z xb Z y b X2 þ Y2 2 HðX; YÞ ¼ H00 þ þ 2 p xa ya 2 PðS; TÞdSdT  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðX  SÞ2 þ ðY  TÞ2

2

0 URx , g is dimensionless lubricant visspeed parameter, k ¼ 12g a3 ph K2ex   g q cosity g ¼ g0 , q ¼ q0 is dimensionless lubricant density   q ¼ qq0 , Kex is the elliptic coefficient of the surface in the x   direction, a is a ratio parameter a ¼ ab , P is dimensionless   pressure P ¼ pph , ph is the maximum Hertzian contact stress   x . The and H is the dimensionless film thickness H ¼ hR a2 boundary conditions of (10) are, Inlet boundary condition PðXa ; YÞ ¼ 0, Outlet boundary conditions PðXb ; YÞ ¼ 0 and

@PðXb ; YÞ ¼ 0; Side boundary conditions PðX; Ya Þ @X ¼ PðX; Ya Þ ¼ 0

ð10Þ

where Xa and Xb are the dimensionless coordinates of the inlet and outlet, Xa is given but Xb should be determined by the outlet boundary conditions and Y ¼ 1 are the two sides of the

ð11Þ

where H00 is the central offset film thickness, the second term defines the undeformed contact shape and the integral term represents the elastic deformation of the contact. The dimensionless Force balance equation, given by Z yb 2p ð12Þ PðX; YÞ dX dY ¼ ; 3 xa ya

Z

xb

represents the balance between the applied load and the total internal pressure in the lubricant. The dimensionless form for viscosity g, which was established by Roelands [25], and density q, which was presented by Dowson and Higginson [26], are  z g ¼ exp ½lnðg0 Þ þ 9:67 1 þ ð1 þ 5:1  109 P  ph Þ ð13Þ and

2802

S.C. Shiralashetti et al. x 10 -4

x 10 -4 4

FAS

FDM

4 2

0 100

0 100

100

50

2

0

y

0

MWFAS

WFAS

2

2

0

Figure 3

0

100

50

50

y

50

y

x

0

0

x

Pressure numerical solutions comparison for N ¼ 64 of equation (9).

50

FAS

50

FDM

x

0 100

100

50

0 -50 100

y

0 -50 100

100

50 0

0

100

50

50

y

x

50 0

0

x

50

MWFAS

50

WFAS

0

4

0 100

0 -50 100

100

50

y Figure 4

 0:6P 1 þ 1:7P

0

x 10 -4

4

 1þ

50

y

x

x 10 -4

q¼

100

50

50

0 -50 100

0

0

100

50

50

y

x

50 0

0

x

Film thickness numerical solution comparison for N ¼ 64 of equation (11).

ð14Þ

where z ¼ 0:68 is the viscosity index, g0 ¼ 1:98E þ 08 is the ambient pressure and ph ¼ 1:84E þ 09 is the maximum Hertzian pressure, a is the ratio of a and b. Kex is the relative curvature in the x direction. For the equivalent curvature of the point contact, Kex ¼ 1. The following discussions are based on a ¼ 1 and Kex ¼ 1. The finite difference discretization of Eq. (9) can be written as

(e þe  i;j i1;j

) Piþ1;j  ð2ei;j þ ei1;j þ eiþ1;j ÞPi;j DX2 (e þe  ) e þe  i;j i;j1 Pi;j1 þ i;j 2i;jþ1 Pi;jþ1  ð2ei;j þ ei;j1 þ ei;jþ1 ÞPi;j 2 2 þa DY2

 qi;j Hi;j  qi1;j Hi1;j ¼0 ð15Þ  DX 2

Pi1;j þ

ei;j þeiþ1;j  2

New wavelet based full-approximation scheme

2803

Table 4 Comparison of residual with iterations of different schemes of the Example III. N 8 16 32 64 128

Residual FDM

FAS

WFAS

NWFAS

6.6418E04 1.8024E04 9.9443E05 9.7355E05 9.6619E05

4.1317E04 9.9986E05 5.3109E05 1.9871E05 9.0938E06

1.4521E04 5.4239E05 9.9091E06 6.6498E06 1.0317E06

9.1840E05 3.9621E05 7.8916E06 3.9818E06 9.2189E07

Table 5 The comparison of CPU time (in seconds) for the different methods. Examples ðN ¼ 64Þ

Method

Setup time

Running time

Total time

I

FDM FAS WFAS NWFAS

18.40 10.80 8.11 7.97

1.13 0.98 0.59 0.36

19.53 11.78 8.70 8.33

II

FDM FAS WFAS NWFAS

18.56 10.89 8.97 7.72

1.02 0.99 0.63 0.70

19.58 11.88 9.60 8.42

III

FDM FAS WFAS NWFAS

20.13 20.09 18.09 18.09

8.10 5.24 2.02 1.06

28.23 25.33 20.11 19.15

The film thickness equation becomes Hi j ¼ H00 þ

n X n X2i þ Y2j 2X Dkl Pkl þ 2 2 p k¼1 l¼1 ij

ð16Þ

where Dkl ij is the stiffness coefficient of the elastic deformation       h h log Xi  Xj þ   1 Dij ¼ Xi  Xj þ 2 2       h h log Xi  Xj    1  Xi  Xj  ð17Þ 2 2 for i ¼ 0; 1; 2; . . . ; n and j ¼ 0; 1; 2; . . . ; n. The force balance equation which as DXDY

n X n X 2p Pij ¼ 3 i¼1 j¼1

ð18Þ

three non-dimensional physical parameters that characterize the point contact problem are velocity (U), load force (W) and elasticity (G). As in the previous examples, we obtained the results are presented pressure solution in Fig. 3 and film thickness solution in Fig. 4 and residual with iterations for different N is presented in Table 4. The comparison of CPU time for different schemes is given in Table 5. 5. Conclusions In this paper, we introduced a new wavelet based fullapproximation scheme for the numerical solution of some classes of elliptic type nonlinear partial differential equations,

particularly application for elasto-hydrodynamic lubrication with point contact problem. The results obtained by the proposed technique show the better performance in terms of better convergence with low computational cost than the existing methods. The efficiency of the proposed method was demonstrated through the illustrative examples. References [1] A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comput. 31 (1977) 333–390. [2] W.L. Briggs, V.E. Henson, S.F. McCormick, A Multigrid Tutorial, second ed., SIAM, Philadelphia, 2000. [3] W. Hackbusch, U. Trottenberg, Multigrid Methods, SpringerVerlag, Berlin, 1982. [4] P. Wesseling, An Introduction to Multigrid Methods, John Wiley, Chichester, 1992. [5] U. Trottenberg, C. Oosterlee, A. Schuller, Multigrid, Academic Press, London, San Diego, 2001. [6] A.A. Lubrecht, Numerical solution of the EHL line and point contact problem using multigrid techniques (Ph.D. thesis), University of Twente, Enschede, The Netherlands, 1987. [7] C.H. Venner, A.A. Lubrecht, Multilevel Methods in Lubrication, Elsevier, 2000. [8] E.A. Zargari, P.K. Jimack, M.A. Walkley, An investigation of the film thickness calculation for elastohydrodynamic lubrication problems, Int. J. Numer. Meth. Fluids (2007) 1–6. [9] W. Dahmen, A. Kurdila, P. Oswald, Multiscale Wavelet Methods for Partial Differential Equations, Academic Press, 1997. [10] N.M. Bujurke, S.C. Shiralashetti, C.S. Salimath, Numerical solution of stiff systems from non-linear dynamics using single term haar wavelet series, Nonlinear Dyn. 51 (2008) 595–605. [11] N.M. Bujurke, S.C. Shiralashetti, C.S. Salimath, Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets, J. Comput. Appl. Math. 219 (2008) 90–101. [12] N.M. Bujurke, S.C. Shiralashetti, C.S. Salimath, An application of single term haar wavelet series in the solution of non-linear oscillator equations, J. Comput. Appl. Math. 227 (2010) 234– 244. [13] S. Islam, I. Aziz, B. Sarler, The numerical solution of secondorder boundary-value problems by collocation method with the haar wavelets, Math. Comput. Model. 52 (2010) 1577–1590. [14] E. Babolian, F. Fattahzdeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput. 188 (2007) 417–426. [15] M. Dehghan, M. Lakestani, Numerical solution of nonlinear system of second-order boundary value problems using cubic Bspline scaling functions, Int. J. Comput. Math. 85 (2008) 1455– 1461. [16] L.A. Diaz, M.T. Martin, V. Vampa, Daubechies wavelet beam and plate finite elements, Finite Elem. Anal. Des. 45 (2009) 200– 209. [17] A. Avudainayagam, C. Vani, Wavelet based multigrid methods for linear and nonlinear elliptic partial differential equations, Appl. Math. Comput. 148 (2004) 307–320. [18] N.M. Bujurke, C.S. Salimath, R.B. Kudenatti, S.C. Shiralashetti, A fast wavelet-multigrid method to solve elliptic partial differential equations, Appl. Math. Comput. 185 (1) (2007) 667–680. [19] N.M. Bujurke, C.S. Salimath, R.B. Kudenatti, S.C. Shiralashetti, Wavelet-multigrid analysis of squeeze film characteristics of poroelastic bearings, J. Comput. Appl. Math. 203 (1) (2007) 237–248. [20] N.M. Bujurke, C.S. Salimath, R.B. Kudenatti, S.C. Shiralashetti, Analysis of modified Reynolds equation using

2804 the wavelet-multigrid scheme, Numer. Meth. PDEs 23 (3) (2007) 692–705. [21] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, MA, 1992. [22] I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math. 41 (1988) 909–996. [23] G.C. Bourantas, V.N. Burganos, An implicit meshless scheme for the solution of transient non-linear poisson-type equations, Eng. Anal. Bound. Elem. 37 (2013) 1117–1126.

S.C. Shiralashetti et al. [24] E. Feyzullahoglu, Isothermal elastohydrodynamic lubrication of elliptic contact, J. Balkan Tribol. Assoc. 15 (3) (2009) 438–446. [25] C.J.A. Roelands, Correlational aspects of the viscositytemperature-pressure relationship of lubricating oils (Ph.D. thesis), Technische Hogeschool Delft, V.R.B., Groningen, The Netherlands, 1996. [26] D. Dowson, G.R. Higginson, Elasto-hydrodynamic lubrication, The Fundamentals of Roller and Gear Lubrication, Pergaman Press, Oxford, Great Britain, 1966.