Simulations of fractional time-derivative against proportional time-delay for solving and investigating the generalized perturbed-KdV equation

Marwan Alquran; Mohammed Ali; Kamel Al-Khaled; George Grossman

doi:10.1515/nleng-2022-0282

Article Open Access

Simulations of fractional time-derivative against proportional time-delay for solving and investigating the generalized perturbed-KdV equation

Marwan Alquran , Mohammed Ali , Kamel Al-Khaled and George Grossman

Published/Copyright: May 9, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

In this study, the Caputo-type fractional time-derivative is simulated by inserting a proportional time-delay into the field function of the perturbed-KdV equation. Two effective methods have been adapted to obtain analytical solutions for this model. Then, independently, the effect of the fractional derivative and the proportional delay on the topological shape of the pKdV propagation was extrapolated. The significant conclusions of the current article reveal that the fractional derivative plays the same role as the presence of a proportional delay in the time coordinate if it is assigned as a substitute for it. With this, from a practical mathematical point of view, we have provided one of the geometric explanations of the fractional derivative. Finally, via the obtained approximate solution, we studied the impact of the perturbed coefficient on propagating the waves of the proposed KdV model.

Keywords: perturbed-KdV; Caputo derivative; fractional power series; homotopy perturbation method

MSC 2010: 26A33; 35C10; 74H10; 35Q53

1 Introduction

The perturbed-KdV (pKdV) model is a nonlinear evolution partial differential equation that governs the physical mechanism of propagating sound in fluids and arises in the applications of acoustics, aerodynamics, and medical engineering. The general form of pKdV is given as follows:

(1.1) ψ t + α ψ x + β ψ ψ x + γ ψ x x x = 0 ,

where ψ = ψ ( x , t ) ∈ C 1 [ ℜ × ℜ + ] is the surface field, α is the perturbation coefficient, and β and γ are the nonlinearity and dispersion parameters. Recently, explicit solutions of the types lumps-(soliton/periodic), breathers, single stripes, and two-wave solitons have been extracted to Eq. (1.1) by means of the Hirota bilinear method [1].

In this work, we aim to further explore the physical properties of pKdV from the perspective of analytical mathematics. In particular, we revisit Eq. (1.1), where the time-derivative is of Caputo-type and the time coordinate is restricted with proportional delay. Thus, the new form of the governing problem is

(1.2) D t λ ψ ( x , μ t ) + α ψ x ( x , t ) + β ψ ( x , t ) ψ x ( x , t ) + γ ψ x x x ( x , t ) = 0 ,

where 0 < λ ≤ 1 refers to the order of the Caputo-derivative, 0 < μ ≤ 1 is the proportional delay factor, and D t λ ψ ( x , t ) is defined as follows [2,3]:

(1.3) D t λ ψ ( x , t ) = 1 Γ ( 1 − λ ) ∫ 0 t ∂ ψ ( x , τ ) ∂ τ ( t − τ ) λ d τ .

Since there are no mathematical approaches to find explicit solutions to nonlinear equations with a fractional derivative, we will resort to analytical methods to obtain analytic or numerical solutions to such fractional problems. To authors’ knowledge, the revised model is to be investigated for the first time in this work.

This article seeks to achieve three goals. First, we find closed-form solutions to pKdV by adapting the fractional power series (FPS) and the homotopy perturbation techniques. Second, we compare the effect of the fractional derivative against the time delay on the shape of the resulting waveform motion. Finally, a graphical analysis is performed to reveal the impact of the perturbed coefficient on the dynamics of the proposed KdV.

For the last three decades, there has been great effort in developing mathematical methods to accommodate the presence of fractional derivatives. There have been numerical and analytical methods for obtaining approximate solutions to fractional equations. Examples of such methods are, operational matrix method [4–6], collocation methods [7,8], finite-difference methods [9,10], and reproducing kernel approaches [11,12], different forms of FPS [13–20], the homotopy perturbation technique and its updates [21–25], combined Laplace transform and FPS [26–28], and many others [29,30]. With regard to the methods used in solving problems involving the time delay, we advise readers to view [31–37] and the references therein.

The organization of this work is as follows. In Section 2, we find a closed-form solution to the revisited pKdV (1.2) via using the FPS method and then investigate the influence of the parameter β . In Section 3, the alternative homotopy perturbation approach will be used to solve Eq. (1.2) and study the influence of the parameter α . Furthermore, we compare the findings obtained by the proposed two approaches to validate their implementations and accuracy. Finally, some recommendations will be given in Section 4.

2 Approach I: FPS

In this section, we recall some preliminaries related to the topic of FPS, which will be used in this article.

Definition 1

The FPS in ( x , t ) -plane is given as follows:

(2.1) ∑ n = 0 ∞ A n ( x ) t n λ Γ ( n λ + 1 ) = A 0 ( x ) + A 1 ( x ) t λ Γ ( λ + 1 ) + A 2 ( x ) t 2 λ Γ ( 2 λ + 1 ) + …

Given that A n ( x ) is continuous on ( a , b ) and 0 ≤ t < R , where R represents the radius of convergence.

Theorem 1

Assume ψ ( x , t ) has an FPS representation

(2.2) ψ ( x , t ) = ∑ n = 0 ∞ A n ( x ) t n λ Γ ( n λ + 1 ) , a < x < b , 0 ≤ t < R .

Then, A n ( x ) = D t n λ ψ ( x , 0 ) .

Proof

Since t ∈ [ 0 , R ) , then ∑ n = 0 ∞ A n ( x ) t n λ Γ ( n λ + 1 ) converges absolutely. Therefore, it converges uniformly. Hence, we have the following:

(2.3) D t λ ψ ( x , t ) = D t λ ∑ n = 0 ∞ A n ( x ) t n λ Γ ( n λ + 1 ) = D t λ A 0 ( x ) + A 1 ( x ) t λ Γ ( λ + 1 ) + A 2 ( x ) t 2 λ Γ ( 2 λ + 1 ) + … = A 1 ( x ) + A 2 ( x ) t λ Γ ( λ + 1 ) + A 3 ( x ) t 2 λ Γ ( 2 λ + 1 ) + … D t 2 λ ψ ( x , t ) = D t λ ( D t λ ψ ( x , t ) ) = A 2 ( x ) + A 3 ( x ) t λ Γ ( λ + 1 ) + A 4 ( x ) t 2 λ Γ ( 2 λ + 1 ) + … . . D t n λ ψ ( x , t ) = A n ( x ) + A n + 1 ( x ) t λ Γ ( λ + 1 ) + A n + 2 ( x ) t 2 λ Γ ( 2 λ + 1 ) + …

Accordingly, D t n λ ψ ( x , 0 ) = A n ( x ) . We should point here to the fact that D t λ t μ = Γ ( μ + 1 ) Γ ( μ − λ + 1 ) t μ − λ : μ ≥ λ .

Theorem 2

Assume ψ ( x , t ) has an FPS representation, then

(2.4) D t λ ψ ( x , t ) = ∑ n = 0 ∞ A n + 1 ( x ) t n λ Γ ( n λ + 1 ) ,

(2.5) D t λ ψ ( x , μ t ) = ∑ n = 0 ∞ A n + 1 ( x ) μ ( n + 1 ) λ t n λ Γ ( n λ + 1 ) ,

(2.6) ψ x ( x , t ) = ∑ n = 0 ∞ A n ′ ( x ) t n λ Γ ( n λ + 1 ) ,

(2.7) ψ x x x ( x , t ) = ∑ n = 0 ∞ A n ‴ ( x ) t n λ Γ ( n λ + 1 ) .

Now, we proceed by assuming that the solution of Eq. (1.2) has an FPS form, i.e.,

(2.8) ψ ( x , t ) = ∑ n = 0 ∞ A n ( x ) t n λ Γ ( n λ + 1 ) .

Then, we plug Eqs (2.4)–(2.8) in Eq. (1.2) to obtain

(2.9) ∑ n = 0 ∞ A n + 1 ( x ) μ ( n + 1 ) λ t n λ Γ ( n λ + 1 ) + ∑ n = 0 ∞ α A n ′ ( x ) t n λ Γ ( n λ + 1 ) + ∑ n = 0 ∞ γ A n ‴ ( x ) t n λ Γ ( n λ + 1 ) + β ∑ n = 0 ∞ A n ( x ) t n λ Γ ( n λ + 1 ) ∑ n = 0 ∞ A n ′ ( x ) t n λ Γ ( n λ + 1 ) = 0 .

By using the fact ∑ n = 0 ∞ B n ∑ n = 0 ∞ C n = ∑ n = 0 ∞ ∑ m = 0 n B m C n − m , Eq. (2.9) is reduced to

(2.10) ∑ n = 0 ∞ A n + 1 ( x ) μ n λ t n λ Γ ( n λ + 1 ) + ∑ n = 0 ∞ α A n ′ ( x ) t n λ Γ ( n λ + 1 ) + ∑ n = 0 ∞ γ A n ‴ ( x ) t n λ Γ ( n λ + 1 ) + ∑ n = 0 ∞ ∑ m = 0 n β A m ′ ( x ) A n − m ( x ) Γ ( n λ + 1 ) Γ ( m λ + 1 ) Γ ( ( n − m ) λ + 1 ) t n λ Γ ( n λ + 1 ) = 0 .

We can now add the above four series to obtain

(2.11) ∑ n = 0 ∞ ( μ ( n + 1 ) λ A n + 1 ( x ) + α A n ′ ( x ) + γ A n ‴ ( x ) + ∑ m = 0 n β A m ′ ( x ) A n − m ( x ) Γ ( n λ + 1 ) Γ ( m λ + 1 ) Γ ( ( n − m ) λ + 1 ) t n λ Γ ( n λ + 1 ) = 0 .

By the fact that for a power series to vanish identically over any interval, each coefficient in the series must be zero. Thus, for Eq. (2.11) to be valid over its given domain, we deduce the following recurrence relation:

(2.12) A n + 1 ( x ) = − μ − ( n + 1 ) λ α A n ′ ( x ) + γ A n ‴ ( x ) + ∑ m = 0 n β A m ′ ( x ) A n − m ( x ) Γ ( n λ + 1 ) Γ ( m λ + 1 ) Γ ( ( n − m ) λ + 1 ) , n = 0 , 1 , 2 , …

Equation (2.12) can be utilized to determine A n ( x ) for n ≥ 1 by referencing a specified arbitrary A 0 ( x ) , which can be obtained by solving Equation (1.2) with a general initial condition of the form ψ ( x , 0 ) = f ( x ) . By taking into account equations (2.1) and (2.2), it can be established that A 0 ( x ) is equal to f ( x ) . Next, we introduce the first numerical example to validate our approach and explore the effect of the fractional order λ and the proportional delay μ acting independently on the propagation of pKdV.

2.1 Example 1

Consider the following initial value problem:

(2.13) D t λ ψ ( x , μ t ) + α ψ x ( x , t ) + β ψ ( x , t ) ψ x ( x , t ) + γ ψ x x x ( x , t ) = 0 , ψ ( x , 0 ) = − 12 γ β ( α + x ) 2 .

Let ϕ n ( x , t ) represents the first nth partial sum of the series solution of ψ ( x , t ) . By using the relation (2.12), the first few terms of { A n ( x ) } are provided in Table 1. To study the impact of λ and μ , we consider ϕ 4 ( x , t ) as the supportive approximate solution of Eq. (2.13). Figure 1(a) shows the propagation of pKdV for different values of the fractional order λ with no effect of the proportional delay, i.e., μ = 1 , whereas Figure 1(b) shows the propagation of pKdV for different values of the proportional delay μ vs λ = 1 , integer-time derivative. We note from this figure that one of the geometric explanations for the role of the fractional derivative is delaying the value of the rate of change of the function (whether it is increasing or decreasing), just like the effect of the proportional delay in the time coordinate. On the other side, regarding the pKdV’s coefficients acting on its propagation form, we study in particular the impact of the nonlinear parameter β . Figures 1 and 2 show that upon the change of β ’s sign, pKdV reverses the monotonicity of propagating its wave-solutions.

Table 1

The first four coefficients of the FPS solution for Example 1

	α = β = γ = 1	α = 1 , β = − 1 , γ = 1
A 0 ( x )	− 12 ( x + 1 ) 2	12 ( x + 1 ) 2
A 1 ( x )	− 24 μ − λ ( x + 1 ) 3	24 μ − λ ( x + 1 ) 3
A 2 ( x )	− 72 μ − 3 λ ( x + 1 ) 4	72 μ − 3 λ ( x + 1 ) 4
A 3 ( x )	− 288 μ − 6 λ ( ( x 2 + 2 x + 13 ) Γ ( λ + 1 ) 2 − 6 μ λ Γ ( 2 λ + 1 ) ) ( x + 1 ) 7 Γ ( λ + 1 ) 2	288 μ − 6 λ ( ( x 2 + 2 x + 13 ) Γ ( λ + 1 ) 2 − 6 μ λ Γ ( 2 λ + 1 ) ) ( x + 1 ) 7 Γ ( λ + 1 ) 2

$Figure 1 Profile solutions of ϕ 4 ( x , t ) {\phi }_{4}\left(x,t) . (a) The effect of the fractional order λ \lambda when μ = 1 \mu =1 . (b) The effect of the proportional delay μ \mu when λ = 1 \lambda =1 . Where α = β = γ = 1 \alpha =\beta =\gamma =1 .$

Figure 1

Profile solutions of ϕ 4 ( x , t ) . (a) The effect of the fractional order λ when μ = 1 . (b) The effect of the proportional delay μ when λ = 1 . Where α = β = γ = 1 .

$Figure 2 Profile solutions of ϕ 4 ( x , t ) {\phi }_{4}\left(x,t) . (a) The effect of the fractional order λ \lambda when μ = 1 \mu =1 . (b) The effect of the proportional delay μ \mu when λ = 1 \lambda =1 , where α = 1 , β = − 1 \alpha =1,\beta =-1 , and γ = 1 \gamma =1 .$

Figure 2

Profile solutions of ϕ 4 ( x , t ) . (a) The effect of the fractional order λ when μ = 1 . (b) The effect of the proportional delay μ when λ = 1 , where α = 1 , β = − 1 , and γ = 1 .

3 Approach II: Homotopy perturbation

The advantage of this method is that a general homotopic form can be defined for fractional nonlinear problems, where it does not deal mainly with the involved fractional derivative but via considering its anti-derivative. Then, we write the solution as a power series in terms of an auxiliary parameter called the perturbation. Based on the defined homotopy form, an iterative relationship can be drawn to identify the terms of the series solution.

Now, we define the following homotopy form regarding the pKdV given in Eq. (1.2):

(3.1) D t λ ψ ( x , μ t ) = − p ( α ψ x ( x , t ) + β ψ ( x , t ) ψ x ( x , t ) + γ ψ x x x ( x , t ) ) ,

where 0 ≤ p ≤ 1 , is the perturbation parameter. Then, we decompose ψ ( x , t ) as a power series in p , i.e.,

(3.2) ψ ( x , t ) = ∑ i = 0 ∞ ψ i ( x , t ) p i = ψ 0 ( x , t ) + ψ 1 ( x , t ) p + ψ 2 ( x , t ) p 2 + …

Substitution of Eq. (3.2) in Eq. (3.1) provides

(3.3) ∑ i = 0 ∞ D t λ ψ i ( x , μ t ) p i = − α ∑ i = 0 ∞ ψ x , i ( x , t ) p i + 1 − β p ∑ i = 0 ∞ ψ x , i ( x , t ) p i ∑ i = 0 ∞ ψ i ( x , t ) p i − γ ∑ i = 0 ∞ ψ x x x , i ( x , t ) p i + 1 ,

where ψ x , i ( x , t ) = ∂ ψ i ( x , t ) ∂ x and ψ x x x , i ( x , t ) = ∂ 3 ψ i ( x , t ) ∂ x 3 . Applying the product of two infinite series, we write Eq. (3.3) as follows:

(3.4) ∑ i = 0 ∞ D t λ ψ i ( x , μ t ) p i = − α ∑ i = 0 ∞ ψ x , i ( x , t ) p i + 1 − β ∑ i = 0 ∞ ∑ n = 0 i ψ i − n ( x , t ) ψ x , n ( x , t ) p i + 1 − γ ∑ i = 0 ∞ ψ x x x , i ( x , t ) p i + 1 .

Unify the above four series in terms of its index-counter and the power of the parameter p , one can verify the following formula:

(3.5) 0 = D t λ ψ 0 ( x , μ t ) + ∑ i = 1 ∞ D t λ ψ i ( x , μ t ) + α ψ x , i − 1 ( x , t ) + β ∑ n = 0 i − 1 ψ i − 1 − n ( x , t ) ψ x , n ( x , t ) + γ ψ x x x , i − 1 ( x , t ) p i = 0 .

To determine the terms of Eq. (3.5) subject to ψ ( x , 0 ) = f ( x ) , we solve the following two problems:

(3.6) D t λ ψ 0 ( x , μ t ) = 0 , ψ 0 ( x , 0 ) = f ( x ) , D t λ ψ i ( x , μ t ) = − α ψ x , i − 1 ( x , t ) + β ∑ n = 0 i − 1 ψ i − 1 − n × ( x , t ) ψ x , n ( x , t ) + γ ψ x x x , i − 1 ( x , t ) ,

subject to the initial condition ψ i ( x , 0 ) = 0 , i ≥ 1 . From Eq. (3.6), one can verify the following outputs:

(3.7) ψ 0 ( x , t ) = f ( x ) , ψ 1 ( x , t ) = − μ − λ ( α f ′ ( x ) + β f ( x ) f ′ ( x ) + γ f ‴ ( x ) ) t λ Γ ( λ + 1 ) .

Now, we plug i = 2 in Eq. (3.6) to obtain

(3.8) ψ 2 ( x , μ t ) = − J t λ α ψ x , 1 ( x , t ) + β ∑ n = 0 1 ψ 1 − n ( x , t ) ψ x , n ( x , t ) + γ ψ x x x , 1 ( x , t ) = − α ψ x , 1 ( x , 1 ) + β ∑ n = 0 1 ψ 1 − n ( x , 1 ) ψ x , n ( x , 1 ) + γ ψ x x x , 1 ( x , 1 ) Γ ( λ + 1 ) t 2 λ Γ ( 2 λ + 1 ) ,

where J t λ is the anti-fractional derivative of the Caputo operator D t λ . We should point out here that D t r t s = Γ ( s + 1 ) Γ ( s − r + 1 ) t s − r and J t r t s = Γ ( s + 1 ) Γ ( s + r + 1 ) t s + r . By rescaling the time coordinate in Eq. (3.8), we obtain

(3.9) ψ 2 ( x , t ) = − α ψ x , 1 ( x , 1 ) + β ∑ n = 0 1 ψ 1 − n ( x , 1 ) ψ x , n ( x , 1 ) + γ ψ x x x , 1 ( x , 1 ) Γ ( λ + 1 ) t 2 λ μ − 2 λ Γ ( 2 λ + 1 ) .

Based on Eqs (3.6) and (3.9), the generalized ith-term of the homotopy series solution to pKdV takes the following form:

(3.10) ψ i ( x , t ) = − α ψ x , i − 1 ( x , 1 ) + β ∑ n = 0 i − 1 ψ i − 1 − n ( x , 1 ) ψ x , n ( x , 1 ) + γ ψ x x x , i − 1 ( x , 1 ) Γ ( ( i − 1 ) λ + 1 ) t i λ μ − i λ Γ ( i λ + 1 ) .

By Eq. (3.10), the closed form solution of pKdV is recognized.

3.1 Comparative analysis of FPS vs homotopy perturbation

Here, we validate the accuracy of the proposed methods in approximating the solution of pKdV. Let h k ( x , t ) represent the k t h -order of the homotopy solution against ϕ k ( x , t ) , the kth-order of the FPS. For the purpose of comparing our findings, we consider the following numerical test:

(3.11) D t λ ψ ( x , μ t ) + ψ x ( x , t ) + ψ ( x , t ) ψ x ( x , t ) + ψ x x x ( x , t ) = 0 , ψ ( x , 0 ) = − 12 ( x + 1 ) 2 .

The exact solution of Eq. (3.11) is ψ ( x , t ) = − 12 ( − t + x + 1 ) 2 . Now, by using Approaches I and II, and for the case of λ = 3 4 and μ = 1 2 , we obtain

(3.12) λ = 3 4 , μ = 1 2 : h 3 ( 2 , t ) = − 3.18074 t 3 ∕ 2 − 1.62658 t 3 ∕ 4 − 17.9834 t 9 ∕ 4 − 4 3 = ϕ 3 ( 2 , t ) . ( λ = μ = 1 ) : h 3 ( 2 , t ) = − 16 t 3 81 − 4 t 2 9 − 8 t 9 − 4 3 = ϕ 3 ( 2 , t ) = ψ ( 2 , t ) − O ( t ) 4 .

Based on the last numerical example (3.11) and (3.12), we can say that both methods produce the same analytical solution, which indicates the correctness of their implementation. In comparison with the explicit solution of the pKdV in the absence of both the fractional derivative and the time delay, all three solutions are identical, which is a strong evidence of the effectiveness of these methods.

3.2 Influence of the perturbation’s coefficient

In this part, by considering the third-order homotopy solution h 3 ( x , t ) , we wish to study the effect of the perturbation coefficient α on the dynamics of pKdV with/without the presence of both fractional derivative and time delay. In order to achieve this goal, we draw profile solutions for different values of α within three stages; without the effect of both the fractional derivative and the time delay, under the effect of fractional derivative only, and under the effect of the time delay only, see Figure 3. We conclude from this graphical analysis that the perturbation coefficient affects the wave’s height of the pKdV.

$Figure 3 Profile solutions of h 3 ( x , t ) {h}_{3}\left(x,t) . (i) λ = 1 , μ = 1 \lambda =1,\mu =1 . (ii) λ = 0.75 , μ = 1 \lambda =0.75,\mu =1 . (iii) λ = 1 , μ = 0.75 \lambda =1,\mu =0.75 , where α = β = γ = 1 \alpha =\beta =\gamma =1 .$

Figure 3

Profile solutions of h 3 ( x , t ) . (i) λ = 1 , μ = 1 . (ii) λ = 0.75 , μ = 1 . (iii) λ = 1 , μ = 0.75 , where α = β = γ = 1 .

4 Conclusion

In conclusion, we have presented the perturbed-KdV under the influence of both time-fractional derivative and the proportional time-delay. The same closed-form solution has been obtained to the revisited pKdV via using two different approaches. Based on the supportive approximate power series solution, we studied the effect of the fractional derivative only ( μ = 1 ), then we have studied the effect of the time delay only ( λ = 1 ). With the aid of graphical analysis, we conclude that the fractional derivative can be approximated by exposing the time coordinate with a proportional delay, i.e., D t λ ψ ( x , t ) ≈ ψ ( x , λ t ) . This finding has been observed based on some numerical examples. Thus, an interesting topic for future research would be to provide theoretical justifications. Finally, the dynamics of pKdV has been visualized graphically by studying the impact of both perturbation and nonlinearity parameters.

Acknowledgments

We would like to express our sincere gratitude to the editor and the reviewers for their time and efforts in providing valuable feedback on our work. Their insightful comments and suggestions have significantly improved the quality of our manuscript, and we are extremely grateful for their expertise and dedication.

Funding information: No funding is received for this work.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: Not applicable.

References

[1] Alquran M, Alhami R. Analysis of lumps, single-stripe, breather-wave, and two-wave solutions to the generalized perturbed-KdV equation by means of Hirota’s bilinear method. Nonlinear Dyn. 2022;109:1985–92. 10.1007/s11071-022-07509-0Search in Google Scholar

[2] Caputo M. Linear models of dissipation whose Q is almost frequency independent. Ann Geophys. 1966;19(4):383–9. Search in Google Scholar

[3] Caputo M. Linear model of dissipation whose Q is almost frequency independent-II. Geophys J Int. 1967;13(5):529–39. 10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar

[4] Zamanpour I, Ezzati R. Operational matrix method for solving fractional weakly singular 2D partial Volterra integral equations. J Comput Appl Math. 2023;419:114704. 10.1016/j.cam.2022.114704Search in Google Scholar

[5] Syam M, Sharadga M, Hashim I. A numerical method for solving fractional delay differential equations based on the operational matrix method. Chaos Solitons Fractals. 2021;147:110977. 10.1016/j.chaos.2021.110977Search in Google Scholar

[6] Usman M, Hamid M, Zubair T, Haq RU, Wang W, Liu MB. Novel operational matrices-based method for solving fractional order delay differential equations via shifted Gegenbauer polynomials. Appl Math Comput. 2020;372:124985. 10.1016/j.amc.2019.124985Search in Google Scholar

[7] Rawashdeh EA. Numerical solution of fractional integro-differential equations by collocation method. Appl Math Comput. 2006;176(1):1–6. 10.1016/j.amc.2005.09.059Search in Google Scholar

[8] Bhrawy AH, Alzaidy JF, Abdelkawy MA, Biswas A. Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrodinger equations. Nonlinear Dyn. 2016;84(3):1553–67. 10.1007/s11071-015-2588-xSearch in Google Scholar

[9] Dwivedi KD, Singh J. Numerical solution of two-dimensional fractional order reaction advection sub-diffusion equation with finite-difference Fibonacci collocation method. Math Comput Simulat. 2021;181:38–50. 10.1016/j.matcom.2020.09.008Search in Google Scholar

[10] AbuArqub O, Edwan R, Al-Smadi M, Momani S. Solving space-fractional Cauchy problem by modified finite-difference discretization scheme. Alexandr Eng J. 2020;59(4):2409–17. 10.1016/j.aej.2020.03.003Search in Google Scholar

[11] AbuArqub O, Al-Smadi M, AbuGdairi R, Alhodaly M, Hayat T. Implementation of reproducing kernel Hilbert algorithm for pointwise numerical solvability of fractional Burgers’ model in time-dependent variable domain regarding constraint boundary condition of Robin. Results Phys. 2021;24:104210. 10.1016/j.rinp.2021.104210Search in Google Scholar

[12] AbuArqub O, Osman MS, Park C, Lee JR, Alsulami H, Alhodaly M. Development of the reproducing kernel Hilbert space algorithm for numerical pointwise solution of the time-fractional nonlocal reaction-diffusion equation. Alexandr Eng J. 2022;61(12):10539–50. 10.1016/j.aej.2022.04.008Search in Google Scholar

[13] Ali M, Alquran M, Jaradat I. Explicit and approximate solutions for the Conformable-Caputo time-fractional diffusive predator-prey model. Int J Appl Comput Math. 2021;7:90. 10.1007/s40819-021-01032-3Search in Google Scholar

[14] Ali M, Jaradat I, Alquran M. New computational method for solving fractional Riccati equation. J Math Comput Sci. 2017;17(1):106–14. 10.22436/jmcs.017.01.10Search in Google Scholar

[15] Jaradat A, Noorani MSM, Alquran M, Jaradat HM. A novel method for solving Caputo-time-fractional dispersive long wave Wu-Zhang system. Nonlinear Dyn Syst Theory. 2018;18(2):182–90. Search in Google Scholar

[16] Abu Arqub O. Application of residual power series method for the solution of time-fractional Schrödinger equations in one-dimensional space. Fundam Inform. 2019;166(2):87–110. 10.3233/FI-2019-1795Search in Google Scholar

[17] Ali M, Alquran M, Jaradat I, AbuAfouna N, Baleanu D. Dynamics of integer-fractional time-derivative for the new two-mode Kuramoto-Sivashinsky model. Rom Rep Phys. 2020;72:103. Search in Google Scholar

[18] Alquran M, Jaradat I, Momani S, Baleanu D. Chaotic and solitonic solutions for a new time-fractional two-mode Korteweg-de Vries equation. Rom Rep Phys. 2020;72:117. Search in Google Scholar

[19] Makhadmih M, Jaradat I, Alquran M, Baleanu D. A new analytical method to simulate the mutual impact of space-time memory indices embedded in (1+2)-physical models. Nonlinear Eng. 2022;11(1):522–38. 10.1515/nleng-2022-0244Search in Google Scholar

[20] Alquran M. The amazing fractional Maclaurin series for solving different types of fractional mathematical problems that arise in physics and engineering. Partial Differ Equ Appl Math. 2023;7:100506. 10.1016/j.padiff.2023.100506Search in Google Scholar

[21] He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput. 2003;135:73–9. 10.1016/S0096-3003(01)00312-5Search in Google Scholar

[22] Ali M, Alquran M, Mohammad M. Solitonic solutions for homogeneous KdV systems by homotopy analysis method. J Appl Math. 2012;2012:569098. 10.1155/2012/569098Search in Google Scholar

[23] Jaradat I, Alquran M, Momani S, Baleanu D. Numerical schemes for studying biomathematics model inherited with memory-time and delay-time. Alexandr Eng J. 2020;59(5):2969–74. 10.1016/j.aej.2020.03.038Search in Google Scholar

[24] Abu-Irwaq I, Alquran M, Jaradat I, Noorani MSM, Momani S, Baleanu D. Numerical investigations on the physical dynamics of the coupled fractional Boussinesq-Burgers system. Rom J Phys. 2020;65:111. Search in Google Scholar

[25] Sakar MG, Uludag F, Erdogan F. Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method. Appl Math Model. 2016;40:6639–49. 10.1016/j.apm.2016.02.005Search in Google Scholar

[26] Alquran M, Ali M, Alsukhour M, Jaradat I. Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics. Results Phys. 2020;19:103667. 10.1016/j.rinp.2020.103667Search in Google Scholar

[27] Eriqat T, El-Ajou A, Oqielat MN, Al-Zhour Z, Momani S. A new attractive analytic approach for solutions of linear and nonlinear Neutral fractional Pantograph equations. Chaos Solitons Fractals. 2020;138:109957. 10.1016/j.chaos.2020.109957Search in Google Scholar

[28] Alquran M, Alsukhour M, Ali M, Jaradat I. Combination of Laplace transform and residual power series techniques to solve autonomous n-dimensional fractional nonlinear systems. Nonlinear Eng. 2021;10(1):282–92. 10.1515/nleng-2021-0022Search in Google Scholar

[29] Subramanian M, Manigandan M, Tunç C, Gopal TN, Alzabut J. On system of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order. J Taibah Univ Sci. 2022;16(1):1–23. 10.1080/16583655.2021.2010984Search in Google Scholar

[30] Batool A, Talib I, Riaz MB, Tunç C. Extension of lower and upper solutions approach for generalized nonlinear fractional boundary value problems. Arab J Basic Appl Sci. 2022;29(1):249–57. 10.1080/25765299.2022.2112646Search in Google Scholar

[31] Alquran M, Jaradat I. Delay-asymptotic solutions for the time-fractional delay-type wave equation. Phys A Stat Mech Appl. 2019;527:121275. 10.1016/j.physa.2019.121275Search in Google Scholar

[32] Alquran M, Jaradat I, Baleanu D, Syam M. The Duffing model endowed with fractional time derivative and multiple pantograph time delays. Rom J Phys. 2019;64:107. Search in Google Scholar

[33] Yaghoobi S, Moghaddam BP, Ivaz K. An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dyn. 2016;87(2):815–26. 10.1007/s11071-016-3079-4Search in Google Scholar

[34] Shahmorada S, Ostadzada MH, Baleanu D. A Tau-like numerical method for solving fractional delay integro-differential equations. Appl Numer Math. 2020;151:322–36. 10.1016/j.apnum.2020.01.006Search in Google Scholar

[35] Alquran M. Investigating the revisited generalized stochastic potential-KdV equation: fractional time-derivative against proportional time-delay. Rom J Phys. 2023;68:106. Search in Google Scholar

[36] Bohner M, Tunç O, Tunç C. Qualitative analysis of Caputo fractional integro-differential equations with constant delays. Comp Appl Math. 2021;40:214. 10.1007/s40314-021-01595-3Search in Google Scholar

[37] Tunç O, Tunç C. Solution estimates to Caputo proportional fractional derivative delay integro-differential equations. Rev Real Acad Cienc Exactas Fis Nat Ser A-Mat. 2023;117:12. 10.1007/s13398-022-01345-ySearch in Google Scholar

Received: 2022-11-05

Revised: 2023-02-05

Accepted: 2023-02-09

Published Online: 2023-05-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0282

Keywords for this article

perturbed-KdV; Caputo derivative; fractional power series; homotopy perturbation method

Creative Commons

BY 4.0