Numerical Solution of Abel′s Integral Equations using Hermite Wavelet

Open access

Abstract

A numerical method is developed for solving the Abel′s integral equations is presented. The method is based upon Hermite wavelet approximations. Hermite wavelet method is then utilized to reduce the Abel′s integral equations into the solution of algebraic equations. Illustrative examples are included to demonstrate the validity, efficiency and applicability of the proposed technique. Algorithm provides high accuracy and compared with other existing methods.

1 Introduction

Wavelets theory is a new emerging tool in applied mathematical research area. It is applicable in various fields, such as, signal analysis for waveform representation and segmentations, time-frequency analysis and Harmonic analysis. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms [1, 2]. Since 1991 the various types of wavelet methods have been applied for the numerical solution of different kinds of integral equations [3]. Namely, the Haar wavelets method [3], Legendre wavelets method [4], Rationalized haar wavelet [5], Hermite cubic splines [6], Coifman wavelet scaling functions [7], CAS wavelets [8], Bernoulli wavelets [9], wavelet preconditioned techniques [25, 26, 27, 28]. Some of the papers are found for solving Abel′s integral equations using the wavelet based methods, such as Legendre wavelets [10] and Chebyshev wavelets [11].

Abel′s integral equations have applications in various fields of science and engineering. Such as microscopy,

seismology, semiconductors, scattering theory, heat conduction, metallurgy, fluid flow, chemical reactions, plasma diagnostics, X-ray radiography, physical electronics, nuclear physics [12, 13, 14].

In 1823, Abel, when generalizing the tautochrone problem derived the equation:

0xy(t)(xt)dt=f(x),

where f (x) is a known function and y(x) is an unknown function to be determined. This equation is a particular case of a linear Volterra integral equation of the first kind. For solving eq. (1) different numerical based methods have been developed over the past few years, such as product integration methods [15], collocation method [16, 17], homotopy analysis transform method [18]. The generalized Abel′s integral equations on a finite segment appeared for the first time in the paper of Zeilon [19]. There are several numerical methods for approximating the solution of singular integral equations is known. Baker [20] studied the numerical treatment of integral equations. A numerical solution of weakly singular volterra integral equations was introduced in [21]. Babolian and Salimi [22] discussed an operational matrix method based on block-pulse functions for singular integral equations. In this paper, we introduced the Hermite wavelets based numerical method for solving Abel′s integral equations.

The article is organized as follows: In Section 2, formulation of Hermite wavelets and function approximation is presented. Section 3 is devoted the method of solution. In section 4, numerical results are demonstrated the accuracy of the proposed method by some of the illustrative examples. Lastly, the conclusion is given in section 5.

2 Properties of Hermite Wavelets

Wavelets constitute a family of functions constructed from dilation and translation of a single function called mother wavelet. When the dilation parameter a and translation parameter b varies continuously, we have the following family of continuous wavelets:

ψa,b(x)=|a|1/2ψ(xba),a,bεR,a0.

If we restrict the parameters a and b to discrete values as a=a0k,b=nb0a0k,a0>1,b0>0.We have the following family of discrete wavelets

ψk,n(x)=|a|1/2ψ(a0kxnb0),a,bεR,a0,

where ψk,n form a wavelet basis for L(R) . In particular, when a0 = 2 and b0 = 1 ,then ψk,n(x) forms an orthonormal basis. Hermite wavelets are defined as [24]

ψk,nx=2k+1/2πHm2kx2n+1,n12k1x<n2k10,otherwise

where m = 0,1, ...,M−1. Here is Hermite polynomials of degree m with respect to weight function W(x) = 1x2on the real line R and satisfies the following recurrence formula H0(x) = 1,H1(x) = 2x,

Hm+2(x)=2xHm+12(m+1)Hm(x)

wherem = 0,1,2, ..

Function approximation: Here we approximating the solution y(x) of Abel’s integral equations using Hermite wavelet basis as follows:

y(x)=n=1m=0cn,mψn.m(x)

where ψn,m(x) is given in eq.(2). We approximate y(x) by truncating the series represented in eq.(4) as,

y(x)=n=12k1m=0M1cn,mψn,m(x)=CTΨ(x)

where C and Ψ(x) are 2k−1M×1 matrix,

CT=[c1,0,...c1,M1,c2,0,...c2M1,...c2k1,0,...c2k1,M1]
Ψx=ψ1,0,...ψ1,M1,ψ2,0,...ψ2,M1,...ψ2k1,0,...ψ2k1,M1

3 Convergence and Error Analysis

Theorem 3.1

The series solution of Hermite wavelet y(x)=n=1m=0cn,mψn,m(x)is converges to y(x).

Proof: Let L2(R) be the infinite dimensional Hilbert space and ψn,m is defined as eq.(2) forms an orthonormal basis.

Let y(x)=i=0M1cn,iψn,i(x)wherecn,i=y(x),ψn,i(x)for a fixed n

Let us denote the sequence of partial sums Sn of {cn,iψn,i(x)},

Let Sn and Sm be the partial sums with n ≥ m Now we have to prove Sn is a Cauchy sequence in Hilbert space L2(R).

Choose, Sn=i=0ncn,iψn,i(x),Nowy(x),Sn=y(x),i=0ncn,iψn,i(x)=i=m+1n|cn,i|2

We claim that ||SnSm||2=i=m+1n|cn,i|2,n>m

Now||i=m1+ncn,iψn,i(x)||2=i=m+1ncn,iψn,i(x),i=m+1ncn,iψn,i(x)=i=m+1n|cn,i|2,n>m

thus,i=m+1ncn,iψn,i(x)2=i=m+1n|cn,i|2,n>m

Since, Bessel’s inequality, we have i=m+1n|cn,i|2||y(x)||2is bounded and convergent.

Hence,||i=m+1ncn,iψn,i(x)||20asm,n.

This implies,||i=m+1ncn,iψn,i(x)||0.and

Therefore {Sp} is a Cauchy sequence and it converges to s (say).

We assert that y(x) = s

Now sy(x),ψn,i(x)=s,ψn,i(x)y(x),ψn,i(x)=s,ψn,i(x)limnSn,ψn,i(x)=0,

This implies,

Sy(x),ψn,i(x)=0

Hence y(x)=sandi=0ncn,iψn,i(x)converges to y(x) as n →∞ and proved.

Theorem 3.2

Suppose that y(x) ∈ Cm[0,1] and CTΨ(x) is the approximate solution using Hermite wavelet.

Then the error bound would be given by,

||E(x)||||2m!4m2m(k1)maxx[0,1]|ym(x)|||.

Proof: Applying the definition of norm in the inner product space, we have,

||E(x)||2=01[y(x)CTΨ(x)]2dx.

Divide interval [0, 1] into 2k−1 subintervals In=[n12k1,n2k1],n=1,2,3,...,2k1.

||E(x)||2=n=12k1n12k1n2k1[y(x)CTΨ(x)]2dx.

||E(x)||2n=12k1n12k1n2k1[y(x)Pm(x)]2dx.

where Pm(x) is the interpolating polynomial of degree m which approximates y (x) on In. By using the maximum error estimate for the polynomial on In, then

||E(x)||2n=12k1n12k1n2k1[2m!4m2m(k1)maxxIn|ym(x)|]2dx.

||E(x)||2n=12k1n12k1n2k1[2m!4m2m(k1)maxx[0,1]|ym(x)|]2dx.

||E(x)||2=01[2m!4m2m(k1)maxx[0,1]|ym(x)|]2dx

||E(x)||||2m!4m2m(k1)maxx[0,1]|ym(x)|||.

which, we have used the well-known maximum error bound for the interpolation.

4 Method of Solution

Consider the Abel′s integral equation of the form,

λy(x)=f(x)+0xy(t)xtdt,0xt1

where λ = 0 or λ = 1. We first approximate y(x) as truncated series defined in eq.(4). That is,

y(x)n=12k1n=0M1cn,mψn,m(x)=CTΨ(x)

where C and Ψ(x) are defined in eq.(6) and eq.(9) in eq.(8), we get

λCTΨ(x)=f(x)+0xCTΨ(t)xtdt

Next, assume eq.(10) is precise at following collocation points xi=2i12kM,i=1,2,...,2k1M.Then we obtain

λCTΨ(xi)=f(xi)+0xiCTΨ(t)xitdt

Next, we obtain the system of algebraic equations with 2k−1M unknown coefficients. By solving this system of equations, we get Hermite wavelet coefficients and then substituting these coefficients in eq.(9), we get the approximate solution of eq.(8).

5 Numerical Examples

In this section, we present Hermite wavelets method for the numerical solution of Abel′s integral equation to demonstrate the capability of the present method.

Errorfunction=||ye(xi)ya(xi)||=i=1n(ye(xi)ya(xi))2

where ye and ya are the exact and approximate solution respectively.

Example 1. Consider the Abel′s integral equation of first kind [22],

2105x(10556x2+48x3)=0xy(t)xtdt,0x1.

We apply the present method to solve eq.(12) with k = 1 and M = 4. Then we get truncating approximate solution with unknowns as,

y(x)m=03c1,mψ1,m(x)=CTΨ(x)

Then applying the procedure discussed in the section 3. We get a system of four algebraic equations with four unknowns and solving this system, we obtain the Hermite wavelet coefficients as, c1,0=12571513,c1,1=695000,c1,2=27710000,c1,3=695000,and substituting in eq.(13), we obtain: y(x)=12571513ψ10(x)695000ψ11(x)+27710000ψ12(x)+695000ψ13(x)On simplifying, we get y(x) = x3 −x2 +1, which is exact solution of eq.(12). Numerical results with exact solutions are shown in table 1 and graphically shown in figure 1.

Fig. 1
Fig. 1

Comparison of numerical solutions with exact solutions of example 1.

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00017

Table 1

Numerical results of example 1.

xExact solutionPresent method (k = 1, M = 8)Abs. Error
0.10.99100.99101.61e-12
0.20.96800.96804.71e-13
0.30.93700.93702.27e-12
0.40.90400.90403.04e-12
0.50.87500.87502.52e-12
0.60.85600.85609.67e-13
0.70.85300.85309.64e-13
0.80.87200.87202.36e-12
0.90.91900.91902.36e-12

Example 2. Consider the Abel′s integral equation of the first kind [22, 23],

x=0xy(t)xtdt,0x1.

which has the exact solution y(x)=2πx.We solved the eq.(14) using the present method and obtained approximate solution is compared with exact and other existing methods which reflects in table 2 and figure 2. Error analysis is shown in table 3 and figure 3.

Fig. 2
Fig. 2

Comparison of numerical solutions, exact solutions and existing methods of example 2.

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00017

Fig. 3
Fig. 3

Comparison of error analysis of example 2.

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00017

Table 2

Numerical results of example 2.

xExact solutionPresent methodMethod [22]Method [23]
(k = 1, M = 10)(k = 1, M = 8)(m=16)
0.10.2013170.2008420.201280.200460
0.20.2847050.2846670.2860920.297987
0.30.3486910.3486280.3473940.337588
0.40.4026340.4026090.4041610.405769
0.50.4501580.4501290.4495680.464014
0.60.4931240.4931130.4927040.490550
0.70.5326340.5326070.5323150.539721
0.80.5694100.5694400.5691560.562698
0.90.6039510.6036900.6037420.606044
Table 3

Error analysis of example 2.

xPresent methodMethod [22]Method [23]
(k = 1, M = 10)(k = 1, M = 8)(m=16)
0.14.73e-041.18e-038.57e-04
0.23.77e-051.38e-031.32e-02
0.36.21e-051.29e-031.11e-02
0.42.37e-051.52e-033.13e-03
0.52.85e-055.90e-041.38e-02
0.69.87e-064.19e-042.57e-03
0.72.72e-053.19e-047.08e-03
0.83.05e-052.54e-046.71e-03
0.92.59e-042.08e-042.09e-03

Example 3. Consider the Abel′s integral equation of the second kind [22, 23],

4y(x)=4x+1arcsin(1x1+x)+π20xy(t)xtdt,0x1.

which has the exact solution y(x)=1x+1.Applying the Hermite wavelet method for solving eq.(15), then obtained approximate solution is compared with the exact solution and method[23] are shown in table 4 and figure 4. Error analysis is shown in table 5. Example 4. Consider the Abel′s integral equations of the second kind [22, 23],

Fig. 4
Fig. 4

Comparison of Numerical solutions of example 3.

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00017

Table 4

Numerical results of example 3.

xExact solutionPresent methodMethod [23]
(k = 1, M = 10)(m=16)
0.10.9534625892455920.9534626044535200.95646081381695
0.20.9128709291752770.9128709284827000.90601007037324
0.30.8770580193070290.8770580211058820.88361513925322
0.40.8451542547285170.8451542553083540.84340093819493
0.50.8164965809277260.8164965819764080.80822420481499
0.60.7905694150420950.7905694154217440.79221049469412
0.70.7669649888473700.7669649899960250.76284677221990
0.80.7453559924999300.7453559915059690.74933888037055
0.90.725476250110120.7254762589759920.72434536240934
Table 5

Error analysis of example 3.

xPresent method (k = 1, M = 10)Method [23] (m=16)
0.11.52e-082.99e-03
0.26.92e-106.86e-03
0.31.79e-096.55e-03
0.45.79e-101.75e-03
0.51.04e-098.27e-03
0.63.79e-101.64e-03
0.71.14e-094.11e-03
0.89.93e-103.98e-03
0.98.86e-091.13e-03
y(x)=2x0xy(t)xtdt,0x1.

which has the exact solution y(x)=1exp(πt)erfc(πt).We solved the eq.(16) by the present method, we obtain the approximate solution and is compared with exact and other existing methods as shown in table 6 and figure 5. Error analysis is shown in table 7 and figure 6.

Fig. 5
Fig. 5

Comparison of Numerical solutions of example 4.

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00017

Fig. 6
Fig. 6

Comparison of Error analysis of example 4.

Citation: Applied Mathematics and Nonlinear Sciences 4, 1; 10.2478/AMNS.2019.1.00017

Table 6

Numerical results of example 4.

xExact solutionPresent methodMethod [22]Method [23]
(k = 1, M = 10)(k = 0, M = 16)(m=16)
0.10.4140590.4112290.4156890.402472
0.20.5083520.5075720.5055280.519751
0.30.5643090.5636850.5662050.554755
0.40.6033470.6029260.6019080.605031
0.50.6328680.6325210.6341880.640487
0.60.6563230.6560590.6551090.654785
0.70.6756010.6753580.6765880.678700
0.80.6918420.6916900.6915960.688860
0.90.7057870.7053980.7043770.706495
Table 7

Error analysis of example 4.

xPresent methodMethod [22]Method [23]
(k = 1, M = 10)(k = 0, M = 16)(m=16)
0.12.82e-031.62e-031.15e-02
0.27.79e-042.82e-031.13e-02
0.36.23e-041.89e-039.55e-03
0.44.21e-041.43e-031.68e-03
0.53.46e-041.32e-037.61e-03
0.62.63e-041.21e-031.53e-03
0.72.42e-049.86e-043.09e-03
0.81.50e-042.45e-042.98e-03
0.93.87e-041.40e-037.08e-04

6 Conclusion

The Hermite wavelet method is applied for the numerical solution of Abel′s integral equations. The present method reduces an integral equation into a set of algebraic equations. Obtained results are higher accuracy with exact ones and existing methods [22, 23], which can be observed in section 5. The numerical results shows that

the accuracy improves with increasing the values of M for better accuracy. Convergence theorem reveals that existence of solution.

Communicated by Juan Luis García Guirao

References

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    Ü. Lepik E. Tamme Application of the Haar wavelets for solution of linear integral Equations Ant. Turk–Dynam. Sys. Appl. Proce. (2005) 395–407.

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    K. Maleknejad M.T. Kajani Y. Mahmoudi Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelets J. Kybernet. 32 (2003) 1530-1539.

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    K. Maleknejad F. Mirzaee Using rationalized haar wavelet for solving linear integral equations App. Math. Comp. 160 (2005) 579–587.

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    K. Maleknejad M. Yousefi Numerical solution of the integral equation of the second kind by using wavelet bases of hermite cubic splines App. Math. Comp. 183 (2006) 134–141.

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    S. Sohrabi Comparison Chebyshev wavelets method with BPFs method for solving Abel’s integral equation Ain Shams Eng. J. 2 (2011) 249–254.

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    A. M. Wazwaz Linear and nonlinear integral equations: methods and applications Berlin: Higher Education Beijing Springer 2011.

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    A. M. Wazwaz A first course in integral equations Singapore: World Scientific Publishing 1997.

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    C.T.H. Baker The numerical treatment of integral equations Clarendon Press Oxford 1977.

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    S.C. Shiralashetti S. Kumbinarasaiah Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane–Emden type equations Appl. Math. Comput. 315 (2017) 591–602.

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    H. Brunner Collocation methods for Volterra integral and related functional differential equations Cambridge Monographs on Applied and Computational Mathematics Cambridge Univ. Press Cambridge 2004.

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    S. Noeiaghdam E. Zarei H. B. Kelishami Homotopy analysis transform method for solving Abel’s integral equations of the first kind Ain Shams Eng. J. 7 (2016) 483–495.

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    N. Zeilon Sur quelques points de la theorie de l’equation integrale d’Abel Arkiv Mat Astr Fysik 18 (1924) 1–19.

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    C.T.H. Baker The numerical treatment of integral equations Oxford: Clarendon Press 1977.

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    P. Baratella A. P. Orsi A new approach to the numerical solution of weakly singular Volterra integral equations J. Comput. Appl. Math. 163(2) (2004) 401–418.

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    E. Babolian A. Salimi Shamloo Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions J. Comp. Appl. Math. 214 (2008) 495–508.

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    A. Shahsavaran M. R. Moazami Goudarzi O. Moradtalab Solving Abel’s integral equation of the first kind using piecewise constant functions and Taylor expansion by collocation method In: 40th Annual Iranian mathematics conference.

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    A. Ali M.A. Iqbal S.T. Mohyud-Din Hermites wavelets method for Boundary Value problems Inter. J. Modern Appl. Phy. 3(1) (2013) 38–47.

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    M.H. Kantli S.C. Shiralashetti Finite difference Wavelet–Galerkin method for the numerical solution of elastohydrodynamic lubrication problems Journal of Analysis 26(2) (2018) 285–295.

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    N.M. Bujurke M. H. Kantli B.M. Shettar Wavelet preconditioned Newton-Krylov method for elastohydrodynamic lubrication of line contact problems Applied Mathematical Modelling 46 (2017) 285–298.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    C. K. Chui Wavelets: A Mathematical Tool for Signal Analysis SIAM Philadelphia PA. 1997.

  • [2]

    G. Beylkin R. Coifman V. Rokhlin Fast wavelet transforms and numerical algorithms I Commun. Pure Appl. Math. 44 (1991) 141–183.

    • Crossref
    • Export Citation
  • [3]

    Ü. Lepik E. Tamme Application of the Haar wavelets for solution of linear integral Equations Ant. Turk–Dynam. Sys. Appl. Proce. (2005) 395–407.

  • [4]

    K. Maleknejad M.T. Kajani Y. Mahmoudi Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelets J. Kybernet. 32 (2003) 1530-1539.

    • Crossref
    • Export Citation
  • [5]

    K. Maleknejad F. Mirzaee Using rationalized haar wavelet for solving linear integral equations App. Math. Comp. 160 (2005) 579–587.

    • Crossref
    • Export Citation
  • [6]

    K. Maleknejad M. Yousefi Numerical solution of the integral equation of the second kind by using wavelet bases of hermite cubic splines App. Math. Comp. 183 (2006) 134–141.

    • Crossref
    • Export Citation
  • [7]

    K. Maleknejad T. Lotfi Y. Rostami Numerical computational method in solving fredholm integral equations of the second kind by using coifman wavelet App. Math. Comp. 186 (2007) 212–218.

    • Crossref
    • Export Citation
  • [8]

    S. Yousefi A. Banifatemi Numerical solution of Fredholm integral equations by using CAS wavelets App. Math. Comp. 183 (2006) 458–463.

    • Crossref
    • Export Citation
  • [9]

    S. C. Shiralashetti R. A. Mundewadi Bernoulli Wavelet Based Numerical Method for Solving Fredholm Integral Equations of the Second Kind J. Inform. Comp. Sci. 11(2) (2016) 111–119.

  • [10]

    S. A. Yousefi Numerical solution of Abel’s integral equation by using Legendre wavelets Appl. Math. Comp. 175 (2006) 574–80.

    • Crossref
    • Export Citation
  • [11]

    S. Sohrabi Comparison Chebyshev wavelets method with BPFs method for solving Abel’s integral equation Ain Shams Eng. J. 2 (2011) 249–254.

    • Crossref
    • Export Citation
  • [12]

    A. M. Wazwaz Linear and nonlinear integral equations: methods and applications Berlin: Higher Education Beijing Springer 2011.

  • [13]

    A. M. Wazwaz A first course in integral equations Singapore: World Scientific Publishing 1997.

  • [14]

    R. Gorenflo S. Vessella Abel integral equations analysis and applications In: Lecture notes in mathematics Heidelberg: Springer 1991.

  • [15]

    C.T.H. Baker The numerical treatment of integral equations Clarendon Press Oxford 1977.

  • [16]

    S.C. Shiralashetti S. Kumbinarasaiah Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane–Emden type equations Appl. Math. Comput. 315 (2017) 591–602.

  • [17]

    H. Brunner Collocation methods for Volterra integral and related functional differential equations Cambridge Monographs on Applied and Computational Mathematics Cambridge Univ. Press Cambridge 2004.

  • [18]

    S. Noeiaghdam E. Zarei H. B. Kelishami Homotopy analysis transform method for solving Abel’s integral equations of the first kind Ain Shams Eng. J. 7 (2016) 483–495.

    • Crossref
    • Export Citation
  • [19]

    N. Zeilon Sur quelques points de la theorie de l’equation integrale d’Abel Arkiv Mat Astr Fysik 18 (1924) 1–19.

  • [20]

    C.T.H. Baker The numerical treatment of integral equations Oxford: Clarendon Press 1977.

  • [21]

    P. Baratella A. P. Orsi A new approach to the numerical solution of weakly singular Volterra integral equations J. Comput. Appl. Math. 163(2) (2004) 401–418.

    • Crossref
    • Export Citation
  • [22]

    E. Babolian A. Salimi Shamloo Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions J. Comp. Appl. Math. 214 (2008) 495–508.

    • Crossref
    • Export Citation
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    A. Shahsavaran M. R. Moazami Goudarzi O. Moradtalab Solving Abel’s integral equation of the first kind using piecewise constant functions and Taylor expansion by collocation method In: 40th Annual Iranian mathematics conference.

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    A. Ali M.A. Iqbal S.T. Mohyud-Din Hermites wavelets method for Boundary Value problems Inter. J. Modern Appl. Phy. 3(1) (2013) 38–47.

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    M.H. Kantli S.C. Shiralashetti Finite difference Wavelet–Galerkin method for the numerical solution of elastohydrodynamic lubrication problems Journal of Analysis 26(2) (2018) 285–295.

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    N.M. Bujurke M.H. Kantli B.M. Shettar Jacobian free Newton-GMRES method for the solution of elastohydrodynamic grease lubrication in line contact using wavelet based pre-conditioners Proceedings of the National Academy of Sciences India Section A: Physical Sciences 88(2) (2018) 247–265.

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    N.M. Bujurke M. H. Kantli B.M. Shettar Wavelet preconditioned Newton-Krylov method for elastohydrodynamic lubrication of line contact problems Applied Mathematical Modelling 46 (2017) 285–298.

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    S.C. Shiralashetti M.H. Kantli A.B. Deshi A new wavelet multigrid method for the numerical solution of elliptic type differential equations Alexandria Engineering Journal 57 (2018) 203–209.

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