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Original Research ARTICLE

Front. Phys., 31 January 2020 | https://doi.org/10.3389/fphy.2019.00241

Shape-Preservation of the Four-Point Ternary Interpolating Non-stationary Subdivision Scheme

Pakeeza Ashraf1, Mehak Sabir1, Abdul Ghaffar2, Kottakkaran Sooppy Nisar3 and Ilyas Khan4*
  • 1Department of Mathematics, Government Sadiq College Women University, Bahawalpur, Pakistan
  • 2Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
  • 3Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser, Saudi Arabia
  • 4Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

In this paper, we present the shape-preserving properties of the four-point ternary non-stationary interpolating subdivision scheme (the four-point scheme). This scheme involves a tension parameter. We derive the conditions on the tension parameter and initial control polygon that permit the creation of positivity- and monotonicity-preserving curves after a finite number of subdivision steps. In addition, the outcomes are generalized to determine conditions for positivity- and monotonicity-preservation of the limit curves. Convexity-preservation of the limit curve of the four-point scheme is also analyzed. The shape-preserving behavior of the four-point scheme is also shown through several numerical examples.

1. Introduction

Subdivision Schemes (SS) are iterative algorithms for constructing smooth curves/surfaces from a given control polygon/mesh. The advantages of such schemes are that they are easy to use, simple to investigate, and highly flexible. The popularity of SS is increasing in various applications such as in computer-aided geometric design, computer graphics, computer animation, signal processing, and commercial industry due to their attractive properties. Shape-preservation of the subdivision curve has significant importance in geometric shape design. Shape-preserving SS are extensively used in the design of curves to manage and predict their shape according to the shape of initial control points. Differential equations are used for mathematical modeling of many phenomena. Different techniques are being used to solve boundary value problems [1] and non-linear problems [2]. In the same way, SS can also be used to solve fractional differential equations such as [37].

Rham [8] was the first to present an SS with C0 continuity to attain a smooth curve. Afterward, Chaikin [9] introduced a corner-cutting approximating scheme with C1 continuity. Dyn et al. [10] developed a four-point binary interpolating scheme that is capable of generating a C1-continuous limit curve. Dyn et al. [11] formulated the convexity-preserving property of the famous four-point interpolatory scheme [10] by taking into account that the initial control points are convex. Kuijt and Damme [12] presented a series of local non-linear interpolating schemes that preserve monotonicity. With time, the research community started taking an interest in ternary SS because, by increasing arity from binary to ternary, one can improve the order of continuity of the limit curve without significantly increasing support width [see Beccari et al. [13]]. Hassan et al. [14] constructed a four-point ternary interpolatory scheme with a tension parameter. Cai [15] derived conditions on this parameter to ensure convexity preservation of the limit curve. Pitolli [16] examined the shape-preserving properties of a ternary scheme with bell-shaped masks.

Most of the SS offered in literature are stationary, but this limits the application of the schemes. To reproduce conics, spirals, and polynomial curves, one has to opt for non-stationary schemes. Beccari et al. [17] presented a C1 four-point binary non-stationary interpolating scheme. Akram et al. [18] analyzed the shape-preserving properties of this scheme [17]. Beccari et al. [19] also offered a four-point ternary non-stationary interpolatory scheme with a tension parameter. They showed that the proposed scheme can generate a variety of curves within the C2-continuous range of its tension parameter. Ghaffar et al. [20, 21] introduced odd and even point non-stationary binary SS with a shape parameter for curve design. Ghaffar et al. [22] also presented a new class of 2m−point non-stationary SS with some attractive properties such as torsion, continuity, monotonicity, curvature, and convexity preservation.

This research aims to completely explore the shape-preserving properties of the four-point ternary non-stationary interpolatory scheme [19] (the four-point scheme). We formulate the necessary conditions on the tension parameter of the scheme and initial control points that permit the creation of positivity- and monotonicity-preserving curves after finite iteration levels. Beccari et al. [19] visually demonstrated that, for an initial convex control polygon, the four-point scheme did not generate convex curves. In this regard, we establish the conditions on the tension parameter that prove that the four-point scheme does not generate convexity-preserving limit curves.

The rest of the paper is designed as follows. In section 2, we present the four-point scheme and recall some of its important results. The positivity-preserving and monotonicity-preserving properties of the four-point scheme are proved in sections 3 and 4, respectively. In section 5, the convexity-preserving property of the four-point scheme is discussed. Some numerical examples are given in section 6 to analyze and demonstrate the shape-preserving properties of the four-point scheme. Conclusions are drawn in the last section.

2. The Four-Point Scheme

Beccari et al. [19] presented a four-point scheme involving a tension parameter. For given initial control polygon {(xi0,pi0)}i and for the set of control points at the jth refinement level {(xij,pij)}i, j ∈ ℕ0: = ℕ ∪ {0}, the control points at the (j + 1)th refinement level can be obtained by the rules:

{p3ij+1=pij,p3i+1j+1=160((90γij+11)pi1j+(90γij+1+43)pij                +(90γij+1+17)pi+1j+(90γij+1+1)pi+2j),p3i+2j+1=160((90γij+1+1)pi1j+(90γij+1+17)pij               +(90γij+1+43)pi+1j+(90γij+11)pi+2j),    (1)

where,

γij+1=-13(1-(βj+1)2)(1+βj+1),    (2)

and,

βj+1=2+βj,βj2 (βj1)j0.    (3)

The four-point scheme (1) generates C2-continuous limit curves for any choice of the initial tension parameter β0 in the interval [−2, +∞[\{−1}. For the initial parameter β0 ∈ [−2, +∞[\{−1}, the recurrence relation in (3) satisfies the property:

limj+βj=2.    (4)

Proposition 1.

Given the initial parameter β0 ∈ [−2, +∞[\{−1}, the parameter γij+1 given in (2) satisfies the property:

limj+γij+1=127.    (5)

3. Positivity Preservation

In this section, we discuss the positivity-preserving property of the four-point scheme (1), which can be obtained by taking fij=pi+1jpij and Fj=maxi{fij,1fij}, j ∈ ℕ0.

Lemma 2.

Let the initial control points {(xi0,pi0):i} be positive, i.e., pi0>0, i ∈ ℤ, for any j ∈ ℕ0, such that:

F0<1γij+1=αj    (6)

then pij>0, Fj < αj, j ∈ ℕ0, i ∈ ℤ, i.e., the control points generated by the four-point scheme (1) at the jth refinement level are also positive.

Proof.

As γij+1(1,)j0, we have:

αj=1γij+1>0.

The proof of Lemma 2 is obtained by induction on j.

• By hypothesis, the holds for j = 0, i.e., pi0>0,F0<αj,i.

• Suppose, by induction hypothesis pij>0 and Fj < αj, i ∈ ℤ and for some j ∈ ℕ. Now, we prove that pij+1>0 and Fj+1 < αj.

Obviously, 1αj<fij<αj and 1αj<1fij<αj.

By the definition of the four-point scheme (1), we have:

p3ij+1>0.    (7)

Consider

p3i+1j+1=160((90γij+11)pi1j+(90γij+1+43)pij                   +(90γij+1+17)pi+1j+(90γij+1+1)pi+2j)              =pij60((90γij+1+1)1fi1j+90γij+1+43                  +(90γij+190γij+1fi+1j)fij+(17+fi+1j)fij)              >pij60((90γij+1+1)αj+90γij+1+43+(90γij+1                  90γij+1αj)1αj+(17+1αj)1αj)              =pij60(αj)2(90γij+1(αj)3(αj)3+43(αj)2                  +90γij+1αj+17αj+1)               =pij60(γij+1)2(90γij+1(γij+1)31(γij+1)3+43(γij+1)2                 +90γij+1γij+1+17γij+1+1)               =pij60γij+1(91(γij+1)3+17(γij+1)247γij+11),

As we know that pij>0, it is also clear that 160γij+1[91(γij+1)3+17(γij+1)2-47γij+1-1]>0, for γij+1>0. This implies that:

p3i+1j+1>0.    (8)

In the same way, we can get p3i+2j+1>0, so we have pij+1>0.

In order to prove Fj+1 < αj, we show that fij+1<αj and 1fij+1<αj. For this, consider:

f3ij+1=p3i+1j+1p3ij+1            =160((90γij+1+1)1fi1j+90γij+1+43                +(90γij+190γij+1fi+1j)fij+(17+fi+1j)fij).

So, we have:

f3ij+1αj=160((90γij+1+1)1fi1j+90γij+1+43                          +(90γij+190γij+1fi+1j)fij+(17                          +fi+1j)fij60αj)                     <160((90γij+1+1)1αj+90γij+1+43                          +(90γij+190γij+1αj)αj+(17                          +αj)αj60αj)                     =160αj(90γij+11+90γij+1αj+43αj                          +90γij+1(αj)290γij+1αj+17(αj)2                          +(αj)360(αj)2)                     =160(γij+1)2(90(γij+1)4(γij+1)3                          +133(γij+1)243γij+1+1).

Since 160(γij+1)2>0, it is also clear that [-90(γij+1)4-(γij+1)3+133(γij+1)2-43γij+1+1]<0, for αj=1γij+1 and γij+1>0. This implies that f3ij+1-αj < 0. Thus, we have:

f3ij+1<αj.    (9)

Similarly, we can have f3i+1j+1<αj and f3i+2j+1<αj. Thus, it shows that fij+1<αj. In the same way, it can be shown that 1fij+1<αj when 1f3ij+1<αj,1f3i+1j+1<αj and 1f3i+2j+1<αj. Since, Fj+1=maxi{fij+1,1fij+1}, so Fj+1 < αj.

Lemma 2 examines the positivity-preservation of the four-point scheme (1) for the finite number of j subdivision steps. Henceforth, Theorem 3 is given to build up the positivity-preserving condition in the limiting case, as j → ∞. It can be observed that the parameter γij+1 given in (2) fulfills limjγij+1=127. Thus, limjαj=27 in Theorem 3, and the proof can be followed from Lemma 2 easily.

Theorem 3.

Suppose that the initial control points {(xi0,pi0):i} are positive, with the end goal that:

F0<27,

at that point, the limit curves generated by the four-point scheme (1) are positive.

4. Monotonicity Preservation

The monotonicity-preservation property of the four-point scheme (1) which can be obtained by defining the first-order divided difference by Dij=pi+1j-pij and taking qij=Di+1jDij,Qj=max{qij,1qij},j0,i examined in this section.

The next lemma is given to build the monotonicity-preserving condition for the finite number of j subdivision steps.

Lemma 4.

For j ∈ ℕ, suppose that the initial control points {(xi0,pi0):i} are strictly monotonically increasing, i.e., Di0>0,i, such that:

Q01γij+1=ηj.    (10)

Then Dij>0,Qjηj,i,j,i.e., the control points generated by the four-point scheme (1) at the jth subdivision step are still strictly monotonically increasing.

Proof.

First-order divided differences for the four-point scheme (1) can be obtained as:

 D3ij+1=(32γij+1+160)Dij+310Di+1j                  +(-32γij+1+160)Di+2j,D3i+1j+1=-130Dij+25Di+1j-130Di+2j,D3i+2j+1=(-32γij+1+160)Dij+310Di+1j                   +(32γij+1+160)Di+2j.

As γij+1(1,),j+, so it gives

ηj=1γij+1>0.

The proof of Lemma 4 proceeds by induction on j.

• By hypothesis, the assertion holds for j = 0, i.e., Di0>0,Q0ηj,i.

• Suppose by induction hypothesis Dij>0 and Qj ≤ ηj, i ∈ ℤ and for some j ∈ ℕ. Now we prove that Dij+1>0 and Qj+1 ≤ ηj.

To prove Dij+1>0, we show that:

D3ij+1>0, D3i+1j+1>0 and D3i+2j+1>0.

For this consider,

D3ij+1=(32γij+1+160)Dij+310Dijqij+(32γij+1                   +160)qijqi+1jDij           >Dij60(90γij+1+1+1ηj(1890γij+1ηj)                  +1(ηj)2)          =Dij60(ηj)2[(ηj)2+18ηj+1]          =Dij60(γij+1)2(1(γij+1)2+18γij+1+1)          =Dij60((γij+1)2+18γij+1+1).

As we know that Dij>0, and it is also clear that 160[(γij+1)2+18γij+1+1]>0, for ηj=1γij+1 and γij+1>0. This implies that,

D3ij+1>0.    (11)

In the same way, it can be proved that D3i+1j+1>0 and D3i+2j+1>0.

This implies that we have Dij+1>0. Moreover, to verify Qj+1 ≤ ηj, we show that qij+1ηj and 1qij+1ηj. For this, consider:

q3ij+1=D3i+1j+1D3ij+1            =-2+24qij-2qijqi+1j90γij+1+1+18qij-90γij+1qijqi+1j+qijqi+1j,

thus,

q3ij+1-ηj=-2+24qij-2qijqi+1j90γij+1+1+18qij-90γij+1qijqi+1j+qijqi+1j                     -ηj,q3ij+1-ηj=Nm1Dm1.    (12)

Using (11), as Dm1=90γij+1+1+18qij-90γij+1qijqi+1j+qi+1jqij>0. Further, Nm1 of (12) fulfills

Nm1=-2+24qij-2qijqi+1j-90γij+1ηj-ηj-18qijηj                 +90γij+1qijqi+1jηj-qijqi+1jηj           -2+ηj(24-2ηj)-90γij+1ηj-ηj-18ηjηj                 +ηj(90γij+1(ηj)2-ηjηj)         =90γij+1(ηj)3-90γij+1ηj+22ηj-22         =90γij+1(γij+1)3-90γij+1γij+1+22γij+1-22         =1(γij+1)2(-112(γij+1)2+22γij+1+90),

Since 1(γij+1)2>0, and it is clear that (-112(γij+1)2+22γij+1+90)<0, for ηj=1γij+1 and γij+1>0. Thus, from (12), we have q3ij+1-ηj0. This implies that:

q3ij+1ηj.    (13)

Similarly, it is easy to show that q3i+1j+1ηj and q3i+2j+1ηj, which leads to qij+1ηj.

In the same way, it can be proved that 1qij+1ηj by showing that 1q3ij+1ηj,1q3i+1j+1ηj and 1q3i+2j+1ηj. Since Qj+1=maxi{qij+1,1qij+1}, thus Qj+1 ≤ ηj. So, by induction Dij>0 and Qj ≤ ηj, i ∈ ℤ, for some j ∈ ℕ.

Lemma 4 examines the monotonicity preservation of the four-point scheme (1) for the finite number of j subdivision steps. Henceforth, Theorem 5 is given to build up the monotonicity-preserving condition in the limiting case, as j → ∞. It can be observed that the parameter γij+1 given in (2) fulfills limjγij+1=127. Thus, limjαj=27 in Theorem 5 and note that the proof can be followed from Lemma 4.

Theorem 5.

Assume that the initial control points {(xi0,pi0):i} are strictly monotonicallly increasing, with the end goal that

Q027,

at that point, the limit curves generated by the four-point scheme (1) are strictly monotonically increasing.

5. Convexity Preservation

In this section, we examine the convexity-preserving property of the four-point scheme (1). Basically, a subdivision scheme satisfies the convexity-preserving property if, for an initial convex control polygon, the limit curves generated by the scheme preserve the convexity of the initial data. For a subdivision scheme, the convexity-preserving property is attained if, at each refinement level, the second-order divided differences of the scheme are all positive. Specifically, for a given jth-level sequence of real values {pij,i} located at regularly spaced parameter values {xij=i3j,i}, the second-order divided difference of the scheme is defined by dij=32j2(pi-1j-2pij+pi+1j) and, for convexity preservation, {dij>0,i,j0} holds.

Beccari et al. [19] showed that, for an initial convex control polygon, the four-point scheme (1) fails to generate a convex limit curve when choosing different values of the initial tension parameter β0 in the interval [−2, +∞[\{−1}. In Figures 1A–D, dotted lines show the initial convex polygon and solid lines represent curves generated by the four-point scheme (1) after one iteration level. It is clear from the figure that the scheme does not preserve convexity.

FIGURE 1
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Figure 1. The convexity-preserving limit curves generated by the proposed scheme with the control polygon.

Now, we check whether the condition {dij>0,i,j0} is satisfied by the four-point scheme (1) or not. By taking yij=di+1jdij,Yj=max{yij,1yij},j0,i, we establish the following result.

Proposition 6.

For j ∈ ℕ, suppose that the initial control points {(xi0,pi0):i} are strictly convex, i.e., di0>0,i, such that

Y01γij+1=δj    (14)

then dij0, i.e., the points generated by the four-point scheme (1) at the jth subdivision step are not strictly convex.

Proof.

The second-order divided difference of the four-point scheme (1) can be obtained as:

{d3ij+1=(32γij+1+120)dij+(32γij+1-120)di+1j,                        d3i+1j+1=(32γij+1-120)dij+(32γij+1+120)di+1j,                         d3i+2j+1=(-32γij+1+160)dij+(-3γij+1+310)di+1j                 +(-32γij+1+160)di+2j.                            

As γij+1(1,),j, so it gives δj=1γij+1>0. The proof of Proposition 6 proceeds by induction on j.

• By hypothesis, the assertion holds for j = 0, i.e., di0>0,Y0δj,i.

• Suppose by induction hypothesis dij>0 and Yj ≤ δj, i ∈ ℤ and for some j ∈ ℕ. Now we show that dij+1>0. Also, simply, we have 1δjyijδj and 1δj1yijδj.

To prove dij+1>0, it is sufficient to show that:

d3ij+1>0, d3i+1j+1>0 and d3i+2j+1>0.

From (15), we have:

d3ij+1=32γij+1dij+120dij+32γij+1yijdij-120yijdij            >dij60[90γij+1+3+90γij+11δj-3δj]            =dij60γij+1[90(γij+1)3+90(γij+1)2+3(γij+1)-3].

As we know that dij>0, and it is also clear that 160γij+1[90(γij+1)3+90(γij+1)2+3(γij+1)-3]>0, for δj=1γij+1 and γij+1>0. So, we have:

d3ij+1>0.    (15)

Now consider from (15)

d3i+1j+1=(32γij+1-120)dij+(32γij+1+120)yijdij               >dij60(90γij+1-3+90γij+11δj+31δj)               =dijγij+160(90-3γij+1+90γij+1+3)               =dij60(90(γij+1)2+93γij+1-3).

As we know that dij>0, and it is clear that 160[90(γij+1)2+93γij+1-3]>0, for δj=1γij+1 and γij+1>0. This implies that:

d3i+1j+1>0.    (16)

Now consider,

d3i+2j+1=(32γij+1+160)dij+(3γij+1+310)dijyij                 +(32γij+1+160)dijyijyi+1j              >dij60(90γij+1+1+(180γij+1+1δj)1δj                +(1890γij+1δj)1δj)             =dij60(179(γij+1)2162γij+1+1),

As we know that dij>0, and it is also clear that dij60[-179(γij+1)2-162γij+1+1]<0, for δj=1γij+1 and γij+1>0. This implies that:

d3i+2j+1<0.    (17)

By combining (15), (16), and (17), we have dij+10, which shows that the four-point scheme (1) does not preserve strict convexity. Some numerical examples are presented to verify and examine the conditions of shape preserving for the 4-point ternary scheme (1). In Examples 1 − 4, the initial set of values is displayed by dotted line segments while the limit curves are marked by solid lines, such that the limit curves generated by the four-point scheme (1) satisfy the shape-preserving condition.

Example 1.

There are several important meteorological data parameters that scientists use for dealing with different climate challenges. Wind velocity data (WVD) is one of them. These data always have a positive value, and the minimum value is ~0. In this example, we choose WVD from Wu et al. [23], as given in Table 1. We use these WVD to demonstrate the positivity-preserving property of the four-point scheme (1). In Figure 2A, the dotted line represents WVD (which is positive) and the solid curve is generated by the four-point scheme (1), which is also positive.

TABLE 1
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Table 1. Wind data (positive data) [23].

FIGURE 2
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Figure 2. The positivity-preserving curves generated by the four-point scheme (1) for positive initial data.

Example 2.

In this example, we consider experimental data that are quoted from Sarfraz et al. [24]. The proposed data are positive and represent the volume of NaOH vs. HCl in a beaker, as stated in the experimental procedures. These experimental data are presented in Table 2. Figure 2B presents the positivity preservation of the curve generated by the four-point scheme (1). In this figure, the dotted line represents the positive data (which are given in Table 2) and the solid curve is generated by the four-point scheme (1). It is clear that the curve generated by the scheme is also positive.

TABLE 2
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Table 2. Positive data from Sarfraz et al. [24].

Example 3.

The data given in Table 3 are also experimental data. These data represent the oxygen level from an experiment conducted in the laboratory and are quoted from Butt and Brodlie [25]. We use the proposed data in Figure 3A. In this figure, we find that, by imposing the condition of positivity on the initial data, the four-point scheme (1) is capable of producing a positive curve.

TABLE 3
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Table 3. Positive data from Butt and Brodlie [25].

FIGURE 3
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Figure 3. The positivity-preserving curves generated by the four-point scheme (1) for positive initial data.

Example 4.

The data in Table 4 are obtained from Hussain and Ali [26]. These data represent the depreciation of the valuation of the market price of computers installed at City Computer Center. The x-coordinate corresponds to the time in years, and the y-coordinate corresponds to the computer price in Rs. 10,000. Figure 3B generated by the four-point scheme (1) indicates the positivity preservation of the curve generated by the scheme.

TABLE 4
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Table 4. Positive data from Hussain and Ali [26].

Example 5.

The data given in Table 5 represents monotonic data that are obtained from a monotonic function. From Figure 4A we find that by imposing the condition of monotonicity on the initial data, the four-point scheme (1) is capable of producing a monotonically increasing curve.

TABLE 5
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Table 5. Monotonic data.

FIGURE 4
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Figure 4. The monotonicity-preserving curves generated by the four-point scheme (1) with monotonic initial data.

Example 6.

In this example, we again consider monotonic data from a monotonic function. These data are presented in Table 6. Figure 4B displays the curve generated by the four-point scheme (1). It is clear from the figure that, for an initial monotonic dataset, the scheme produces a monotonic curve.

TABLE 6
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Table 6. Monotonic data.

6. Conclusion

In this paper, we have presented the shape-preserving properties of the four-point scheme (1). We have derived the necessary conditions on the initial control points and tension parameter of the scheme to show that the four-point scheme (1) generates positivity- and monotonicity-preserving curves after a finite number of subdivision steps. We have also shown that, for initial convex data, the proposed scheme does not generate a convex curve. Further, we have generalized these results for the positivity- and monotonicity-preservation of the limit curves. Finally, the discussion is followed by several numerical examples. By using this technique, one can analyze the shape-preserving properties of higher arity interpolation and also approximating schemes.

Data Availability Statement

All datasets generated for this study are included in the article/supplementary files.

Author Contributions

PA, AG, and KN: conceptualization. PA, MS, AG, and KN: writing the original manuscript: MS and IK: formal analysis: IK: methodology and supervision: KN and IK: writing review and editing: PA, MS, AG, KN, and IK: software.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: interpolating, non-stationary, shape-preservation, subdivision scheme, ternary

Citation: Ashraf P, Sabir M, Ghaffar A, Nisar KS and Khan I (2020) Shape-Preservation of the Four-Point Ternary Interpolating Non-stationary Subdivision Scheme. Front. Phys. 7:241. doi: 10.3389/fphy.2019.00241

Received: 16 October 2019; Accepted: 18 December 2019;
Published: 31 January 2020.

Edited by:

Mustafa Inc, Firat University, Turkey

Reviewed by:

Amin Jajarmi, University of Bojnord, Iran
Sunil Kumar, National Institute of Technology, Jamshedpur, India

Copyright © 2020 Ashraf, Sabir, Ghaffar, Nisar and Khan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Ilyas Khan, ilyaskhan@tdtu.edu.vn