Volume 2022, Issue 1 3666348

Research Article

Open Access

Numerical and Analytical Simulations of Nonlinear Time Fractional Advection and Burger’s Equations

Hassan Khan,

Corresponding Author

Hassan Khan

[email protected]

orcid.org/0000-0001-6417-1181

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

Department of Mathematics, Near East University TRNC, Mersin 10, Turkey neu.edu.tr

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Qasim Khan,

Qasim Khan

orcid.org/0000-0002-7572-1393

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong ied.edu.hk

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Fairouz Tchier,

Fairouz Tchier

orcid.org/0000-0001-7855-508X

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia ksu.edu.sa

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Ibrarullah,

Ibrarullah

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

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Evren Hincal,

Evren Hincal

orcid.org/0000-0001-6175-1455

Department of Mathematics, Near East University TRNC, Mersin 10, Turkey neu.edu.tr

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Gurpreet Singh,

Gurpreet Singh

orcid.org/0000-0003-2961-9379

School of Mathematical Sciences, Dublin City University, Ireland dcu.ie

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F. M. O. Tawfiq,

F. M. O. Tawfiq

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia ksu.edu.sa

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Shahbaz Khan,

Shahbaz Khan

orcid.org/0000-0003-3661-3303

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

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Hassan Khan,

Corresponding Author

Hassan Khan

[email protected]

orcid.org/0000-0001-6417-1181

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

Department of Mathematics, Near East University TRNC, Mersin 10, Turkey neu.edu.tr

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Qasim Khan,

Qasim Khan

orcid.org/0000-0002-7572-1393

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong ied.edu.hk

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Fairouz Tchier,

Fairouz Tchier

orcid.org/0000-0001-7855-508X

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia ksu.edu.sa

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Ibrarullah,

Ibrarullah

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

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Evren Hincal,

Evren Hincal

orcid.org/0000-0001-6175-1455

Department of Mathematics, Near East University TRNC, Mersin 10, Turkey neu.edu.tr

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Gurpreet Singh,

Gurpreet Singh

orcid.org/0000-0003-2961-9379

School of Mathematical Sciences, Dublin City University, Ireland dcu.ie

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F. M. O. Tawfiq,

F. M. O. Tawfiq

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia ksu.edu.sa

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Shahbaz Khan,

Shahbaz Khan

orcid.org/0000-0003-3661-3303

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan awkum.edu.pk

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First published: 06 October 2022

https://doi.org/10.1155/2022/3666348

Citations: 1

Academic Editor: Yusuf Gurefe

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Abstract

In this paper, the higher nonlinear problems of fractional advection-diffusion equations and systems of nonlinear fractional Burger’s equations are solved by using two sophisticated procedures, namely, the q-homotopy analysis transform method and the residual power series method. The proposed methods are implemented with the Caputo operator. The present techniques are utilised in a very comprehensive and effective manner to obtain the solutions to the suggested fractional-order problems. The nonlinearity of the problem was controlled tactfully. The numerical results of a few examples are calculated and analyzed. The tables and graphs are constructed to understand the higher accuracy and applicability of the current method. The obtained results that are in good contact with the actual dynamics of the given problem, which is verified by the graphs and tables. The present techniques require fewer calculations and are associated with a higher degree of accuracy, and therefore can be extended to solve other high nonlinear fractional problems.

1. Introduction

The most powerful tool for researchers to simulate various physical phenomena in applied sciences and nature is known as fractional partial differential equations (FPDEs). The following physical phenomena have been accurately modelled by FPDEs: optics [1], economics [2], fluid traffic [3], electrodynamics [4], hepatitis B virus [5], tuberculosis [6], air foil [7], modelling of earth quack nonlinear oscillation [8], propagation of spherical waves [9], Chaos theory [10], fractional COVID-19 model [11], finance [12], pine wilt disease [13], Zener [14], cancer chemotherapy [15], traffic flow model [16], Poisson-Nernst-Planck diffusion [17], diabetes [18], biomedical and biological [19], and many other numerous applications in various branches of applied mathematics (see [20–22]). Due to these numerous applications, FPDEs and factional calculus have gained more attention from researchers as compared to ordinary calculus.

To obtain the approximate solutions to the above models, researchers use and develop a variety of analytical approaches. The frequently used methods are optimal homotopy asymptotic method (OHAM) [23], Iterative Laplace transform method [24], extended direct algebraic method (EDAM) [25], Adomian decomposition method (ADM) [26], the Finite difference method (FDM) [27], the homotopy perturbation transform technique along with transformation (HPTM) [28], the (G/G^′)-expansion method [29], the Haar wavelet method (HWM) [30], standard reductive perturbation method [31], the variational iteration procedure with transformation (VITM) [32], and the differential transform method (DTM) [33]. In this context, Hassan et al. have presented the solutions of some nonlinear FPDEs which can be seen in [34–36]. Some useful methods can be seen in [37–39].

Obtaining the analytical solutions of FPDEs and their systems has been a difficult task for researchers in recent years. In this circumstance, Prakash and Kaur use q-HATM [40] to solve the time-fractional Navier Stokes equation. Similarly, q-HATM was implemented by Kumar et al. and used to obtain the solutions of the fractional long-wave equations in the regularised form [41]. The Schrodinger generalized form of fractional order was solved by Veeresha et al. using q-HATM [42]. The q-HATM convergence was done by El-Tawil and Huseen [43]. The RPSM technique was used by Alquran to solve the drainage equation in [44] and the fractional-order Phi-4 equation in [45]. The Whitham-Broer-Kaup equation of fractional order was analysed by Wang and Chen by using RPSM [46]. RPSM has been used in [47] to find the solution of the fractional Biswas-Milovic equation of multidimensions. Komashynska et al. used the RPSM [48] to solve a system of multipantograph delay differential equations. In [49], RPSM was used to find an approximation solution for the fractional-order Sharma-Tasso-Olever equation. The solutions to the fractional order Schrodinger equations were determined using RPSM in [50]. The RPSM has been used to solve a variety of problems, including the gas dynamic equation [51], the Emden-Fowler equation, Berger-Fisher equation, and the Benney-Lin equation in its fractional format, solved by RPSM in [52]. Similarly, RPSM was used by Al-Smadi [53] to overcome the solutions to initial value problems.

In this paper, the solutions of systems of fractional Berger’s equations and nonlinear advection equations were examined by combining two analytical techniques, q-HAM and RPSM. The results are compared to each other as well as the exact solution to the given problems. The current methods are used to compare the solutions methodologies for the analysis of the higher nonlinear problems of fractional advection-diffusion equations and systems of nonlinear fractional Burger’s equations in the Caputo sense. The obtained approximate series solutions are fast converging towards the exact solutions of the targeted problems, according to quantitative analysis. It is found that the techniques under discussion are simple and effective for the solutions of FPDE systems. The graphs and tables for the solutions to the targeted problems via RPSM and q-HATM. It is confirmed that the obtained solutions are in good contact with the exact solution to the problems. The fractional order solutions are convergent towards the integer order solutions, which shows the reliability of the fractional solutions. The presented techniques have a wide range of applications and could be used to examine approximate analytical solutions of other nonlinear FPDEs and multidimensional systems of FPDEs in the future.

This paper will be formatted as follows: in the second section, some basic definitions are discussed. In Section 3, the methodology is described, and in Section 4, two separate techniques are used to compare certain numerical results. The conclusion and references are found in the fifth and final section of the paper.

2. Basic Definitions

In this section we will discuss some important definitions

2.1. Definition

The Caputo operator in [54, 55] is given as.

(1)

2.2. Definition

An expansion of power series (PS) at point t = t₀ is known as fractional PS and is given by [56].

(2)

note FPS can be expanded at point t₀ is

(3)

which is the Taylor’s series expansion form.

2.3. Laplace Transform (LT)

The LT for continuous function

is defined as [57]

(4)

where G(s) is the LT for the function

2.4. Definition

The LT

of Caputo fractional derivative is given by [57]

(5)

2.5. Definition

The LT of two function

is defined by [57]

(6)

, represent the product between k and

(7)

2.6. Definition

The LT of fractional derivative is defined as [57]

(8)

where the LT of

is denoted by G(s).

2.7. Theorem

Assume that the series solution

is convergent to the solution u for a prescribed value of h. If the truncated series

(9)

is used as an approximation to the solution u(x, t) of problem, then an upper bound for the error, E_m is estimated as

(10)

Proof. See [43]

2.8. Theorem

suppose that u(t) ∈ C[t₀, t₀ + R] and fork = 0, 1, 2, ⋯, n + 1,

where 0 < α ≤ 1. Then u could be represented by:

(11)

Proof. See [58].

2.9. Theorem

on t₀ ≤ t ≤ d, where 0 < α ≤ 1, then the reminder R_n(t) of the generalized Taylor’s series will satisfies the inequality:

(12)

Proof. See [58].

2.10. Theorem

Suppose that u has a Fractional power series representation at t₀ of the form

(13)

where R is the radius of convergence. Then u is analytic in (t₀, t₀ + R).

Proof. See [58].

3. RPSM and Q-HATM Procedures

To understand main concept of RPSM and q-HATM [59] for system of FPDEs, we consider the following FPDEs system,

(14)

with initial condition,

(15)

where

the Caputo fractional derivative, R is linear and N is nonlinear operator, respectively, in Eq. (14).

3.1. RPSM Procedure for FPDEs System [60]

Eq. (14) can be simplified as

(16)

where υ(x, y, z, t), ϑ(x, y, z, t), and ϖ(x, y, z, t) is the k^th truncated series of form

(17)

In RPSM the zero^th approximate solution of υ(x, y, z, t), ϑ(x, y, z, t), and ϖ(x, y, z, t) is given by

(18)

Eq. (17), implies that,

(19)

for Eq. (14), the residual function is given by

(20)

so, the k^th residual function becomes

(21)

As in [61, 62], it is clear that Res(x, y, z, t) = 0 and . Therefore, , , and . In the Caputo definition, the constant has zero derivative therefore , k = 0, 1, ⋯, n which mean that of Res_υ(x, y, z, t), Res_ϑ(x, y, z, t), Res_ϖ(x, y, z, t), and Res_k(x, y, z, t) are t = 0 matching at n = 0, 1, ⋯, k; .

To determine f₁(x, y, z), f₂(x, y, z), f₃(x, y, z), ⋯, g₁(x, y, z), g₂(x, y, z), g₃(x, y, z), ⋯, and h₁(x, y, z), h₂(x, y, z), h₃(x, y, z), ⋯, we substitute k = 0, 1, ⋯, in Eq. (17), and then the obtained results are put in Eq. (19). In the final step, we apply

on both sides we obtained the following

(22)

3.2. Q-HATM Procedure for FPDEs System [63]

Using LT, Eq. (14) can be simplified as

(23)

so Eq. (23), implies that

(24)

We define the nonlinear operator is

(25)

where q ∈ [0, (1/n)], θ(x, y, z, t; q) is real function of x, y, z, t, and q.

Homotopy can be constructed as

(26)

In Eq. (26), the auxiliary parameter, nonzero auxiliary function and embedding parameter are h ≠ 0, H(x, y, z, t) and n ≥ 1, q ∈ [0, (1/n)], respectively.

denotes Laplacian operator and υ₀, ϑ₀, and ϖ₀ are the initial conditions. The following results are obtained at q = 0 and q = 1/n,

(27)

By using Taylor theorem θ(x, y, z, t; q) can be expressed as

(28)

where

(29)

After simplification, we have

(30)

Taking the m-times derivatives of Eq. (26), put q = 0, implies the zero^th order solution

(31)

where the vectors are defined as

(32)

The following recursive formula is obtained by taking inverse LT of Eq. (31),

(33)

Where

(34)

(35)

Eq. (33) and Eq. (35) are known q-HATM series solutions for the given system.

4. Numerical Results

4.1. Example

Consider nonlinear advection-diffusion equation of the form [64]

(36)

with ICs

(37)

and exact solution at δ = 1

(38)

4.1.1. RPSM-Solution

(1) 1st Iteration. Using RPSM, the k^th truncated series of Eq. (36), can obtain

(39)

the 1st approximate is given as

(40)

Eq. (39) should be written as

(41)

put k = 1 in Eq. (41), we get

(42)

where υ(x, 0) = f(x) = e^−x,

(43)

The residual function of Eq. (36), is given by

(44)

The k^th residual function Resυ(x, t) is given by,

(45)

put k = 1 in the Eq. (45), we obtain

(46)

we have

(47)

Eq. (46), will become

(48)

(2) 2nd Iteration. for k = 2 Eq. (41), can be written as

(49)

where f(x) = e^−x, and f₁(x) = e^−x,

(50)

put k = 2 in Eq. (45), we obtain

(51)

(52)

as we know that

(53)

for k = 2 Eq. (53), become as

(54)

using

on Eq. (52), we get

(55)

put

in Eq. (55), we obtain

(56)

(3) 3rd Iteration. Put k = 3 in Eq. (41), we obtain

(57)

where f(x) = e^−x, f₁(x) = e^−x, and f₂(x) = e^−x,

(58)

put k = 3 in Eq. (45) we obtain

(59)

(60)

put k = 3 in Eq. (53), we obtain

(61)

applying

on both sides of the Eq. (60), we get

(62)

put

in Eq. (62), we obtain

(63)

The RPSM solution of Eq. (36), is given as

(64)

4.1.2. Q-HATM Solution

(1) 1st Iteration. Taking LT of Eq. (36), and simplifying

(65)

The nonlinear term N is defined as

(66)

Using the procedure of q-HATM

(67)

put m = 1 in the Eq. (67), we obtain

(68)

(69)

put m = 1 in Eq. (69), we obtain

(70)

put in Eq. (68), we obtain

(71)

(2) 2nd Iteration. Put m = 2 in the Eq. (67), we obtain

(72)

put m = 2 in Eq. (69), we obtain

(73)

put in Eq. (72), we obtain

(74)

(3) 3rd Iteration. Put m = 3 in the Eq. (67), we obtain

(75)

put m = 3 in Eq. (69), we obtain

(76)

put in Eq. (75), we obtain

(77)

The q-HATM solution of Eq. (36), is given as

(78)

4.2. Example

The system of 3D Burgers’ equation are [65, 66]

(79)

the initial condition of Eq. (79), are

(80)

The exact solution of the Eq. (79) at δ = 1, are

(81)

4.2.1. RPSM-Solution

(1) 1st Iteration. The k^th truncated series of the solution of Eq. (79), using RPSM, we obtain

(82)

the zero^th RPSM approximate solution of Eq. (79), is given by

(83)

the Eq. (82), should be written as

(84)

put k = 1 in Eq. (84), we get

(85)

where

(86)

The residual function of Eq. (79), is given by

(87)

where the k^th residual function of Resυ(x, y, t), Resϑ(x, y, t), and Resϖ(x, y, t) is given by

(88)

put k = 1 in the Eq. (88), we obtain

(89)

(90)

we know that

(91)

put in Eq. (90), we obtain

(92)

(2) 2nd Iteration. Put k = 2 in Eq. (84), we obtain

(93)

where

(94)

put k = 2 in the Eq. (88), we obtain

(95)

(96)

we know that

(97)

put k = 2 in Eq. (97), we obtain

(98)

applying

, on both sides of the Eq. (96), we get

(99)

we know that

(100)

put in Eq. (99), we obtain

(101)

The solution of Eq. (79), in term of RPSM is given by

(102)

4.2.2. Q-HATM Solution

(1) 1st Iteration. Taking LT of Eq. (79), and simplifying we obtain

(103)

The nonlinear term N is defined as

(104)

Use the procedure of q-HATM

(105)

put m = 1 in the Eq. (105), we obtain

(106)

(107)

put m = 1 in Eq. (107), we get

(108)

Put in Eq. (106), we obtain

(109)

(2) 2nd Iteration. Put m = 2 in the Eq. (105), we obtain

(110)

put m = 2 in Eq. (107), we obtain

(111)

after simplification we obtain

(112)

Put Eq. (112) in Eq. (110), we obtain

(113)

The solution of Eq. (79), in term of q-HATM is given by

(114)

5. Results and Discussion

Here, we will discuss the numerical solutions. In Figure 1, 3D plots of (a) RPSM, (b) Exact, and (c) q-HATM ψ−solutions at δ = 1 of Example 4.1 are presented. Which are in closed contact with the exact solution of Example 4.1. In Figure 2, 2D plots of (a) RPSM, (b) Exact, and (c) q-HATM ψ−solutions at δ = 1 of Example 4.1 are presented, from which the validity of the proposed methods are confirmed. While in Figure 3, 3D plots of Example 4.1 for fractional order δ = 1,0.9 are plotted. Numerical values of Example 4.1 are presented in Table 1. In Figure 4, 2D-solutions plots and Figure 5 of (a) RPSM, (b) Exact, and (c) q-HATM, at δ = 1 of Example 4.2 are plotted. The RPSM and q-HATM solutions are very closed to the exact solution. The RPSM and q-HATM solutions graphs at different fractional order are plotted in Figure 6.

Details are in the caption following the image — **Figure 1 (a)**
Open in figure viewer PowerPoint

3D plots of (a) RPSM, (b) Exact, and (c) q-HATM υ−solutions at δ = 1 of Example 4.1.

Table 1. Solution comparison of RPSM, q-HATM, and Exact with different time and spaces of Example 4.1.

x	t	RPSM	RPSM	q-HATM	q-HATM	Exact
x	t	δ = 0.9	δ = 1	δ = 0.9	δ = 1	δ = 1
0.25	0.001	0.7804183	0.7795799	0.7804183	0.7795799	0.7795799
0.50		0.6077903	0.6071374	0.6077903	0.6071374	0.6071374
0.75		0.4733476	0.4728391	0.4733476	0.4728391	0.4728391
1		0.3686435	0.3682475	0.3686435	0.3682475	0.3682475
0.25	0.005	0.7857118	0.7827045	0.7857118	0.7827045	0.7827045
0.50		0.6119130	0.6095709	0.6119130	0.6095709	0.6095709
0.75		0.4765583	0.4747342	0.4765583	0.4747342	0.4747342
1		0.3711440	0.3697234	0.3711440	0.3697234	0.3697234

6. Conclusion

In this paper, the solutions of nonlinear systems of fractional Burger’s equations and advection diffusion equation are calculated by using q-HATM and RPSM. The proposed methods provide the results with higher degree of accuracy and using very few terms of their series solutions. The solutions comparison of both techniques are compared with the actual solutions of each problem. The comparison has shown the best compromise between the solutions of the suggested techniques. The fractional-order solutions are calculated successfully and shown to be convergent towards the integer-order solutions. The novelty of this paper is given in the nonlinear fractional solutions which provide the more accurate results as compared to other studies in literature. In the future, the suggested techniques can be utilised easily for the solutions of higher dimensional and nonlinear FPDEs and their systems because of their simple and straightforward implementation.

Disclosure

This work is performed as part of the employment of the authors.

Conflicts of Interest

No competing interests are declared.

Authors’ Contributions

Hassan Khan was responsible for supervision, Qasim khan was responsible for methodology and draft writing, Fairouz Tchier was the project administrator, Ibrar Ullah was responsible for the methodology, Evren Hincal was responsible for funding and draft writing, Gurpreet Singh was responsible for draft writing, F. M. O. Tawfiq was responsible for the investigation, and Shahbaz Khan was responsible for the methodology and draft writing.

Acknowledgments

Researchers Supporting Project (no. RSP-2021/401), King Saud University, Riyadh, Saudi Arabia.

Open Research

Data Availability

No data were used to support this study.

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Numerical and Analytical Simulations of Nonlinear Time Fractional Advection and Burger’s Equations

Abstract

1. Introduction

2. Basic Definitions

2.1. Definition

2.2. Definition

2.3. Laplace Transform (LT)

2.4. Definition

2.5. Definition

2.6. Definition

2.7. Theorem

2.8. Theorem

2.9. Theorem

2.10. Theorem

3. RPSM and Q-HATM Procedures

3.1. RPSM Procedure for FPDEs System [60]

3.2. Q-HATM Procedure for FPDEs System [63]

4. Numerical Results

4.1. Example

4.1.1. RPSM-Solution

4.1.2. Q-HATM Solution

4.2. Example

4.2.1. RPSM-Solution

4.2.2. Q-HATM Solution

5. Results and Discussion

6. Conclusion

Disclosure

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

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