Comparison of Caputo and Atangana–Baleanu fractional derivatives for the pseudohyperbolic telegraph differential equations

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Comparison of Caputo and Atangana–Baleanu fractional derivatives for the pseudohyperbolic telegraph differential equations

MAHMUT MODANLI

Departmet of Mathematics, University of Harran, ¸Sanliurfa 63300, Turkey E-mail: mmodanli@harran.edu.tr

MS received 30 May 2021; revised 29 July 2021; accepted 16 August 2021

Abstract. In this paper, numerical solution of partial differential equations of the so-called hyperbolic telegraph, which has different applications in many fields such as engineering and physics, is investigated. The numerical solutions of telegraph equation defined by Caputo fractional derivative and by Atangana–Baleanu fractional derivative are obtained by Dufort–Frankel difference scheme method. It is important to investigate the solution of this equation defined by these two fractional derivatives and to compare these solutions. Studying this problem for the derivatives of different fractional order makes this problem different from previous studies. The originality of this problem is illustrated by initially considering two types of problems with both the Caputo and Atangana–

Baleanu fractional derivatives. In addition, the approximate solution of these two problems with the Dufort–Frankel difference scheme method and their comparison indicate the originality of this study. Difference schemes are constructed for this equation defined by Caputo and Atangana–Baleanu fractional derivatives. Stability estimates are given for this difference scheme method. The error analysis is calculated by comparing the exact solutions of these two problems, which are defined by both Caputo and Atangana–Baleanu fractional derivatives. Present results show that this method is effective and suitable for these equations defined by Caputo and Atangana–Baleanu fractional derivative. From the simulations obtained using the Matlab program, it can be seen that the Dufort–Frankel difference scheme method is suitable for both types of problems and has approximate solutions close to the exact solution.

Keywords. Pseudohyperbolic telegraph differential equation; Caputo derivative; Atangana–Baleanu fractional derivative; Dufort–Frankel difference scheme method; numerical solution.

PACS Nos 02.30.Jr; 02.60.-x; 02.60.Cb; 02.70.Bf

1. Introduction

Fractional partial differential equations are applied to many phenomena in various fields of engineering and scientific disciplines such as economics, control theory, physics, chemistry, biology, mechanics and electro- magnetics. In theoretical physics, the explicit solutions of linear and nonlinear partial differential equations (PDEs) are usually very important to seek and construct. Therefore, the solution of this equation helps the researchers to understand the physical phenomena. Pseudohyperbolic equations are of a higher order class and mixed partial hyperbolic differential equations have derivatives according to time and place. In [1], Guo and Rui applied two least-squares Galerkin finite- element schemes for pseudohyperbolic equations. By

these two methods, they achieved approximate solutions with first-order and second-order accuracy in time incre- ment. Many researchers have used mixed finite-element methods for elliptic, parabolic and hyperbolic partial differential equations [2–7].

There are many methods for approximate solution of fractional partial differential equations. Some of these methods are difference scheme method, Galerkin method, homotopy perturbation method and modified double Laplace method. Aminikhah [8] studied Laplace transform and homotopy perturbation method for approximate solutions of homogeneous and inhomogeneous coupled Berger’s equation. In [9], modified double Laplace decomposition method was applied for coupled pseudoparabolic equations. Akgül and Modanli [10] studied Crank–Nicholson difference 0123456789().: V,-vol

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method and reproduced kernel function for third-order fractional differential equations in the sense of Atangana–

Baleanu Caputo derivative. Modanli studied implicit difference method and Dufort–Frankel method in [11].

The meshless method based on the radial basis func- tions (RBFs) and collocation approach was applied for the treatment of the time-fractional telegraph equation [12]. Residual power series method (RPSM) as a new approximation was applied for pseudohyperbolic partial differential equation [13]. Liuet al [14] studied pseudohyperbolic partial differential equation using a new splitting H1-Galerkin mixed method. Finally, two splitting definite mixed finite-element schemes were studied for the pseudohyperbolic equation and semidiscrete and fully discrete error estimates were given in [15,16].

These equations describe heat, reaction–diffusion, mass transfer and nerve conduction, and other physical phenomena [17–20]. Recently, there has been a lot of work on real-life problems of fractional calculus [21,22].

Singh et al[23] studied the fractional model of guava for biological pest control with memory effect. How- ever, in the literature review, it was seen that there was no study on fractional-order telegraph-type pseudohyperbolic equations. Therefore, the approximate solution of this equation, which is defined by both Caputo and Atangana–Baleanu Caputo fractional derivatives, is solved using the Dufort–Frankel difference scheme method. In particular, it is aimed to examine these equations, which are defined separately for Caputo and Atangana–Baleanu Caputo fractional derivatives, using this method. In this work, we shall present Dufort–

Frankel finite-difference method for fractional-order pseudohyperbolic telegraph partial differential equation such as Caputo and Atangana–Baleanu derivatives.

For this, the following fractional-order initial-boundary value problem defined by the Atangana–Baleanu derivative

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u_tt(t,x)+₀^{A BC}D_t^αu(t,x)+u(t,x)=βu_{t x x}(t,x) +ux x(t,x)+ f(t,x),

0<x < L, 0<t<T,

u(0,x)=h(x),u_t(0,x)=g(x),0≤x ≤L, u(t,0)=u(t,L)=0, 0≤t≤ T,

0< α≤1, β >0,

(1)

and defined by the Caputo derivative

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

utt(t,x)+^C₀D_t^αu(t,x)+u(t,x)=βut x x(t,x) +ux x(t,x)+ f(t,x),

0<x <L, 0<t <T,

u(0,x)=h(x),u_t(0,x)=g(x),0≤ x ≤ L, u(t,0)=u(t,L)=0, 0≤t ≤T,

0< α≤1, β >0,

(2)

are studied. Here,h(x),g(x), f(t,x)are known func- tions,u(t,x)is an unknown function andβ >0. ₀^{A BC}D_t^α is the Atangana–Baleanu Caputo (ABC) derivative that is defined by the following definition:

DEFINITION 1

Suppose that f ∈ H¹(a,b),b > a, σ ∈ [0,1]. Then A BCfractional derivative is

A BC

0 D_t^σ(f(t))

= B(σ) 1−σ

_t

a

f(x)E_σ

−σ(t−x)^σ 1−σ

dx. (3)

Here,

E_σ

−σ(t−x)^σ 1−σ

=

∞ k=0

−σ⁽^t₁⁻_−σ^x⁾^σk

(σk+1) (4)

is the Mittag–Leffler function and B(σ)=1−σ + σ

(σ).

DEFINITION 2

Caputo fractional derivative is defined as

0CD^α_t f(t)= 1 (n−α)

_t

0

fⁿ(s)

(t−s)^α−ⁿ⁺¹ds, (5) wheren =1,2, . . .∈ N andn−1< α≤n.

Using Laplace transform method for eq. (2), it can be written as follows:

£ A BC

0 D_t^σ(f(t))

= B(σ) 1−σ

s^σu(s,x)−u(0,x) s^σ +₁_−σ^σ . (6) The Dufort–Frankel difference scheme method is suitable for these two problems under the conditions of stability estimates. However, the stability conditions of the difference schemes constructed for each equation defined for the fractional derivatives of Caputo and Atangana–Baleanu Caputo are different. Therefore, it is difficult to predict which of the equations defined with these two fractional derivative operators is better than the other in general terms. But for any value, it is clearly compared and the results are given as in the tables. Figures are given for exact solution and approximate solution of eqs (1) and (2). The figures obtained using the Dufort–Frankel finite-difference scheme for eqs (1) and (2) are compared with the exact solution for differentαandβ values.

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In the next section, we shall construct difference scheme for eqs (1) and (2). Then we shall prove stability estimates for these difference schemes.

2. Constructed difference scheme and its stability Now, let us introduce grids with uniform steps that are given as

W^h = {xn :xn =nh,n =0,1, . . . ,M},h= X M, W^τ = {tk:tk=kτ,k =0,1, . . . ,N}, τ = T

N.

From [24], we can write first-order difference scheme for eq. (1) as

A BC

0 D_t^σ(u(tk,xn))= 1 (σ)

k

j=0

u^k_n⁺¹−u^k_n

τ dj,k, (7) where

dj,k=(tj −tk+1)¹^−α−(tj −tk)¹^−α. Difference scheme for eq. (2) is given as

C

0 D_t^σ(u(tk,xn))= τ^−α (2−σ)

k

j=0

w^σ_j(u^k_n⁻^j⁺¹−u^k_n⁻^j), (8) where

w^σ_j =(j+1)¹^−σ −(j)¹^−σ.

Using Taylor expansion, the Dufort–Frankel difference formula is written as follows:

ux x(tk,xn)≈ u^k_n₊₁−(u^k_n⁻¹+u^k_n⁺¹)+u^k_n₋₁

h² . (9)

From [25],ut x x(tk,xk)can be obtained as ut x x(tk,xn)≈ 1

τ

u^k_n₊₁−2u^k_n⁺¹+u^k_n₋₁ h²

−u^k_n⁻₊¹₁−2u^k_n+u^k_n⁻₋¹₁ h²

, (10)

utt(tk,xn)≈

u^k_n⁺¹−2u^k_n+u^k_n⁻¹ τ²

. (11)

Using formulas (7), (9), (10) and (11), we obtain difference scheme method for eq. (1) as follows:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u^k_n⁺¹−2u^k_n+u^k_n⁻¹

τ² + 1

(σ) k j=0

u^k_n⁺¹−u^k_n

τ dj,k+u^k_n

= β τ

u^k_n₊₁−2u^k_n⁺¹+u^k_n₋₁ h²

−u^k_n+1⁻¹−2u^k_n+u^k_n−1⁻¹ h²

+u^k_n₊₁−(u^k_n⁻¹+u^k_n⁺¹)+u^k_n₋₁ h² + f_n^k, f_n^k = f(t_k,x_n),

1≤k≤ N, 1≤n≤ M−1 u⁰_n =h(xn),

u¹_n−u⁰_n

τ =g(x_n),0≤n≤ M,

u^k₀ =u^k_M =0, 0≤k ≤ N, 0< α≤1, β >0. (12)

Using formulas (8)–(11), we obtain difference scheme method for eq. (1) as follows:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u^k_n⁺¹−2u^k_n+u^k_n⁻¹

τ² + τ^−α

(2−σ) k j=0

w^σ_j(u^kn⁻^j⁺¹−u^kn⁻^j)

= β τ

u^k_n₊₁−2u^k_n⁺¹+u^k_n₋₁

h² −u^k_n⁻₊¹₁−2u^k_n+u^k_n⁻₋¹₁ h²

+u^k_n+1−(u^k_n⁻¹+u^k_n⁺¹)+u^k_n−1

h² +f_n^k, f_n^k=f(tk,xn), 1≤k≤N, 1≤n≤M−1

u⁰_n =h(x_n), u¹_n−u⁰_n

τ =g(xn),0≤n≤ M,

u^k₀=u^k_M =0, 0≤k ≤ N,0< α≤1, β >0. (13) Now, we shall prove the theorem of stability estimates for the difference scheme formula (12).

Theorem 1. Suppose thatσ →0, f₀¹→0and

−τ²h²+τ²+h²

2τ < β.

Then the stability estimates are satisfied for formula (12).

Proof. For the proof of the theorem, we shall use Von- Neumann analysis method. This method is known as (12)

u^k_n =r^ke^{i n}^φ. (14)

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Applying formula (14) to formula (12), and taking k =1,n=0,we obtain

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ 1

τ²+1 τ+2β

τh²+ 1 h²

r² +

1− 2

τ²−1

τ−2 cosφ β τh²+2β

τh²−2 cosφ 1 h²

r + 1

h² +2 cosφ β τh² + 1

τ² =0.

(15) From formula (15), we can calculate

r1+r2= −1−_τ²2−¹_τ−2 cosφ_τ^β_h2+_τ²_h^β2−2 cosφ_h¹2

τ1²+¹_τ+_τ^2β_h2+_h¹2

= −1+_τ²2+¹_τ+2 cosφ_τ^β_h2−_τ²_h^β2+2 cosφ_h¹2

1

τ²+¹_τ+_τ²_h^β2+_h¹2

≤ −1+_τ²2 + ¹_τ +2_τ^β_h₂ −_τ²_h^β2 +2_h¹₂

τ1² +¹_τ + _τ²_h^β2 +_h¹2

≤1, (16) r1r2=

τ1² +_h¹2 +2 cosφ_τ^β_h2

1

τ² +¹_τ + _τ²_h^β2 +_h¹2

≤

1

τ² + _h¹2 +2_τ^β_h₂

τ1² +_τ¹ +_τ^2β_h2 +_h¹2

≤1. (17)

Using formulas (16) and (17), it appears to ber1 <1 andr2<1.Thus, we obtain|u^k_n| = |r^ke^{i n}^φ| ≤r^k<1. As a result, we have shown that formula (12) provides stability estimates.

Theorem 2. Suppose thatσ →0, f₀¹→0and 1

2(^h_τ² +τh²−3τ) < β.

Then the stability estimates are satisfied for formula (13).

Proof of this theorem can be given using the method in Theorem 1.

In the next section, we shall apply one test problem for the approximation solution of problem (1).

3. Numerical applications

In this section, we shall calculate one example for the pseudohyperbolic telegraph partial differential equations by Dufort–Frankel finite-difference method. We

apply the modified Gauss elimination method for difference eqs (12), (13). To obtain error analysis table, the following maximum norm is used for the numerical solution:

= max

n=0,1,...,M k=0,1,...,N

|u(t,x)−u^k_n|. (18) Here u^k_n = u(tk,xn) is the approximate solution, u(x,t)is the exact solution andis the maximum norm error.

Example 1. The fractional-order pseudohyperbolic telegraph partial differential equations defined by ABC derivative operator with initial boundary value condi- tion is as follows:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

utt(t,x)+₀^{A BC}D_t^αu(t,x)+u(t,x)=βut x x(t,x) +ux x(t,x)+ f(t,x), 0<x <L, 0<t <T,

f(t,x)=

t³+61−α

B(α)t²+ 12α

B(α)(α+3)t^α+² +1−α

B(α)t³+6t(1−α)

B(α) + 6α

B(α)(α+2)t^α+¹ + 6βα

B(α)(α+3)t^α+² + 3β(1−α)

B(α) t²)sinx, u(0,x)=0, 0≤ x ≤π,

u(t,0)=u(t, π)=0, 0≤t≤ T, 0< α≤1,0< β.

(19) Example 2. The fractional-order pseudohyperbolic telegraph partial differential equations defined by Caputo derivative operator with initial boundary value condi- tion is follows:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

u_tt(t,x)+^C₀D_t^αu(t,x)+u(t,x)=βu_{t x x}(t,x) +ux x(t,x)+ f(t,x),

0<x <L, 0<t <T, f(t,x)=

t³+61−α

B(α)t²+ 12α

B(α)(α+3)t^α+² +1−α

B(α)t³+ 6t(1−α)

B(α) + 6α

B(α)(α+2)t^α+¹ + 6βα

B(α)(α+3)t^α+2+3β(1−α) B(α) t²

sinx, u(0,x)=0, 0≤x ≤π,

u(t,0)=u(t, π)=0, 0≤t≤ T, 0< α≤1,0< β.

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Table 1. Error analysis table for the Atangana–Baleanu–Caputo derivative.

τ = _N¹,h= _M^π α, β Exact solution Approximation =(Error)

α=0.01, β=0.5 1.009872192265541 1.038202290905170 0.0283 N=M =10 α=0.50, β=0.5 0.969083602352375 1.000777405430620 0.0316 α=0.99, β=0.5 0.262771714374492 0.273819564754884 0.0110 α=0.01, β=69.5 1.009872192265541 1.017289298494486 0.0074 N=M =100 α=0.50, β=67 0.969083602352375 0.966179818030101 0.0029 α=0.99, β=54.5 0.262771714374492 0.264291435205384 0.0015 α=0.01, β=238 1.009872192265541 1.009475816404156 3.9637×10⁻⁴ N=M =200 α=0.50, β=229.1 0.969083602352375 0.970027617538774 9.4402×10⁻⁴ α=0.99, β=187.2 0.262771714374492 0.262767903490127 3.8109×10⁻⁶ Calculated for eq. (19) by using Matlab program.

Table 2. Error analysis table for the Caputo derivative.

τ = _N¹,h= _M^π α, β Exact solution Approximation =(Error)

α=0.01, β=0.14 1.009872192265541 1.009273321940597 5.9887×10⁻⁴ N =M =10 α=0.50, β =0.14 0.969083602352375 0.974048863400875 0.0050

α=0.99, β=0.14 0.262771714374492 0.286736496451846 0.0240 α=0.01, β=0.01389314 1.009872192265541 1.009872150658027 4.1608×10⁻⁸ N =M =100 α=0.50, β =0.01389314 0.969083602352375 0.890017318142695 0.0791

α=0.99, β=0.01389314 0.262771714374492 0.211735555695084 0.0510 α=0.01, β=0.007 1.009872192265541 1.009772540139680 9.9652×10⁻⁵ N =M =200 α=0.50, β =0.00331 0.969083602352375 0.966908578426646 2.1819×10⁻⁶

α=0.99, β=0.0001 0.262771714374492 0.249901981974153 0.0129 Calculated for eq. (20) by using Matlab program.

Figure 1. Approximate solution of the Dufort–Frankel difference method forα=0.99,β=187.2 andN =M =200 when Atangana–Baleanu derivative is used.

Figure 2. Exact solution of the Dufort–Frankel difference method forα=0.99,β =187.2 andN = M =200 when Atangana–Baleanu derivative is used.

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Figure 3. Approximate solution of the Dufort–Frankel difference method forα=0.99,β=0.0001 andN =M =200 when Caputo derivative is used.

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

approximation sol exact solution

Figure 4. The comparison of exact and approximate solutions along the t-axis forα = 0.99,β = 187.2 and N = M =200 Atangana–Baleanu derivative.

Using formula (12) for eq. (19) and formula (13) for the eq. (20), we can obtain figures1–8.

These figures are very close to each other. Let us give error analysis tables (tables 1 and 2) for the detailed comparison and interpretation of these figures. In the tables, the approximate solution and exact solution of eqs (19) and (20) are given for different values of α. From formula (18), error analysis is calculated for eqs (19) and (20).

0 0.5 1 1.5 2 2.5 3 3.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure 5. The comparison of exact and approximate solutions along the x-axis for α = 0.99, β = 187.2 and N =M=200 Atangana–Baleanu derivative.

0 0.5 1 1.5 2 2.5 3 3.5

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×10^-3

Figure 6. The comparison of exact and approximate solutions along thex-axis forα =0.99,β =0.00001 andN = M =200 Caputo derivative.

4. Conclusion

In this paper, pseudohyperbolic telegraph partial differential equation with the sense of ABC derivative and Caputo derivative was studied. Dufort–Frankel difference scheme of these equations were separately constructed for two fractional derivatives. The stability estimates were proved for this first-order difference scheme method. Approximate solutions were found by the difference scheme method for the numerical exper-

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

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×10^-3

Figure 7. The comparison of exact and approximate solutions along the t-axis for α = 0.99, β = 0.00001 and N =M =200 Caputo derivative.

0 0.2 0.4 0.6 0.8 1

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06

Figure 8. The enlarged exact and approximate solutions in the Matlab program and their comparison along the t-axis for α = 0.99,β = 187.2 and N = M = 200 Atangana–

Baleanu derivative.

iment. Matlab program was used for the numerical computations. Error analysis was made by comparing the exact solution and the approximate solution for these equations with two fractional-order derivatives. Sepa- rate simulations were given for exact and approximate solutions. Figures1and3of the approximate solution for problems (1) and (2) were compared with figure2 which is the exact solution of these equations. It was seen that these figures were quite close to each other for different values. The exact and approximate solutions were compared along thet- and x-axes forα = 0.99,

β = 187.2 and N = M = 200 of the Atangana–

Baleanu derivative in figures 4 and 5. The exact and approximate solutions were compared along thet- and x-axes forα =0.99,β =0.00001 andN = M =200 of the Caputo derivative in figures6 and7. The exact and approximate solutions were enlarged in the Mat- lab program and compared forα = 0.99,β = 187.2 and N = M = 200 along the t-axis of Atangana–

Baleanu derivative in figure 8. From tables 1 and 2, it is seen that the margins of errors for values of the Caputo and Atangana–Baleanu derivatives are different from each other. The reason for this is that the stability conditions of the difference schemes defined for both Caputo and Atangana–Baleanu are different from each other. The error values from the tables differ for different values. However, it is seen that the exact and approximate solutions in figures are close to each other.

Examining the solution of this differential equation, which is defined by both the Caputo derivative and the Atangana–Baleanu derivative, using the finite difference method makes this study different. In future studies, approximate solutions of these fractional-order problems can be done with the homotopy perturbation method or other similar iterative methods.

References

[1] W M Wan and Y C Liu, Acta Math. Appl. Sin.22(2), 311 (1999)

[2] J Douglas, R Ewing and M Wheeler, RAIRO Model.

Math. Anal. Numer.17, 249 (1983)

[3] C Johson and V Thomee, RAIRO Model. Math. Anal.

Numer.15, 41 (1981)

[4] Z W Jiang and H Z Chen,Northeast Math. J.17, 301 (2001)

[5] Z D Luo and R X Liu, SIAM J. Numer. Anal.36, 89 (1998)

[6] J S Zhang and D P Yang,Numer. Methods Partial Differ.

Equ.25, 622 (2009)

[7] L C Cowsar, T F Dupont and M F Wheeler, SIAM J.

Numer. Anal.33, 492 (1996)

[8] H Aminikhah,Appl. Math. Model.37, 5979 (2013) [9] H E Gadain,Acta Math. Sci. Ser. B38, 333 (2018) [10] A Akgül and M Modanli,Chaos Solitons Fractals127,

10 (2019)

[11] M Modanlı,Adv. Differ. Equ.2018, 333 (2018) [12] A Mohebbi, M Abbaszadeh and M Dehghan, Int.

J. Numer. Methods Heat Fluid Flow 24(8), 1636 (2014)

[13] M Modanli, S T Abdulazeez and A M Husien,Numer.

Meth. Partial Diff. Equ.37(3), 2235 (2021)

[14] Y Liu, J Wang, H Li, W Gao and S He,Int. J. Math.

Comput. Sci.5(3), 413 (2011)

(8)

[15] H Guo and H X Rui, Appl. Math. Comput.189, 425 (2007)

[16] Y Liu, H Li, J F Wang and S He,Numer. Meth. Partial Diff. Equ.28(2), 670 (2012)

[17] J Nagumo, S Arimoto and S Yoshizawa,Proc. IRE50, 91 (1962)

[18] C V Pao,J. Math. Anal. Appl.52, 105 (1975)

[19] R Arima and Y Hasegawa, Proc. Jpn. Acad. 39, 721 (1963)

[20] G Ponce,Nonlinear Anal.9, 399 (1985)

[21] E Bonyah, A K Sagoe, D Kumar and S Deniz,Ecol.

Complex.45, 100880 (2021)

[22] O Abu Arqub, J Singh, B Maayah and M Alhodaly, Math. Methods Appl. Sci.2021, 1 (2021)

[23] J Singh, B Ganbari, D Kumar and D Baleanu,J. Adv.

Res.32, 99 (2020)

[24] R T Alqahtani,J. Nonlinear Sci. Appl.9(6), 3647 (2016) [25] J Jachimaviˇcien˙e, M Sapagovas, A Štikonas and O Štikonien˙e,Nonlinear Anal.: Model. Control19(2), 225 (2014)