An analytical approach for Shehu transform on fractional coupled 1D, 2D and 3D Burgers’ equations
-
Mamta Kapoor
and
Varun Joshi
Abstract
Obtaining the numerical approximation of fractional partial differential equations (PDEs) is a cumbersome task. Therefore, more researchers regarding approximated-analytical solutions of such complex-natured fractional PDEs (FPDEs) are required. In this article, analytical-approximated solutions of the fractional-order coupled Burgers’ equation are provided in one-, two-, and three-dimensions. The proposed technique is named as Iterative Shehu Transform Method (ISTM). The simplicity and accurateness of the method are affirmed through five examples. Graphical representation and tabular discussion are provided to compare the exact and approximated results. The robustness of the proposed regime is also validated by error analysis. In the present work, approximated and exact solutions are compared to verify the validity of the proposed scheme. Error analysis is also provided through which the efficiency of the proposed scheme can be assured. Obtained errors are lesser than the compared results.
Graphical abstract
1 Introduction
Numerical regimes are cumbersome to implement, as it demands a lot of calculation, difficult programming, and discretization, especially, for the fractional order partial differential equations (FPDEs). Due to discretization in the numerical regimes, there is always a scope of error. Thus, in this research, a gist is provided on how some complex natured PDEs can be solved analytically. This study adopts Shehu transform because of its efficiency and easiness compared to other methods in the existing literature.
1.1 Fractional calculus
The notion of fractional calculus is a well-known concept such as fractional derivative and fractional integral. A letter was written by L’ Hospital to Leibnitz in 1695 regarding “How do we calculate
In recent years, fractional PDEs have emerged as the most important topic from the perspective of scientists and researchers due to their applicability in numerous fields of engineering and science. The degree of flexibility is very high for the fractional derivative in the associated models and which produces an excellent tool for describing the variable history and the hereditary characteristics of the various prototypes. Major scale research is done to develop the analytical and numerical results of linear and non-linear FPDEs. Burgers’ equation is considered one of the primary non-linear PDEs consisting of the diffusion properties and non-linear propagation. Burgers’ equation was established to tackle the model of the turbulent motion of the fluid. Dealing with the fluid movement of turbulent nature is a cumbersome task.
1.1.1 1D fractional coupled Burgers’ equation
1.1.2 2D fractional coupled Burgers’ equation
1.1.3 3D fractional coupled Burgers’ equation
Regarding the higher-order derivative, small coefficients, and non-linear terms, Burgers’ equation and Navier-Stokes equation are the same. The process of the unidirectional propagation of the weakly acoustic waves can be dealt with using the fractional Burgers’ equation (FBE). The contribution of Caputo is noticeable in the field of fractional calculus [30,31,32,33,34]. The major advantage of the Caputo fractional derivative is its capability to allow the traditional I.C. and B.C. in the formula for the given problem.
Different definitions of the fractional derivatives are provided in the literature, such as Riemann-Liouville, Caputo and Grunwald-Letnikov, Atangana-Baleanu [35], Caputo-Fabrizio [36], Liouville-Caputo [37], and conformable derivative [38].
A variety of analytical and numerical results are discussed by different researchers regarding FBEs. Akram et al. [39] provided the numerical technique for the solution of the time-fractional Burgers’ equation. Kurt et al. [40] provided the solution of Burgers’ equation with a new fractional derivative. Esen and Tasbozan [41] provided the numerical solution of the time-fractional Burgers’ equation using the quadratic Galerkin approach. Sulaiman et al. [42] gave the solution regarding the time-fractional Burgers’ equation using Laplace HPM. Onal and Esen [43] gave the Crank-Nicolson scheme for the approximation of the FBEs. Hassani and Naraghirad [44] provided the computational approach to the optimization regime for solving the time-fractional Burgers’ equation of the variable order. Esen and Tasbozan [45] implemented the notion of cubic B-spline finite element approach for the numerical solution of time-fractional Burgers’ equation. Kurt et al. [46] provided two reliable approaches for the solution of fractional coupled Burgers’ equation. Johnson et al. [47] gave the Laplace HPM for the Burgers’ equation of space and time-fractional orders. Prakash et al. [22] provided the numerical regime regarding the fractional coupled Burgers’ equation using fractional VIM. Eltayeb and Bachar [48] gave the notion regarding 2D fractional coupled Burgers’ equation and triple Laplace ADM. Singh et al. [49] provided the solution of the time and space fractional coupled Burgers’ equation using the Homotopy algorithm. Albuohimad and Adibi [50] provided the solution of the time-fractional coupled Burgers’ equation using the Chebyshev collocation regime. Ahmed et al. [51] gave the analytical solution of the space and time-fractional coupled Burgers’ equation. Chen and An [52] gave the numerical approximation of the coupled Burgers’ equation with time and space fractional derivative.
1.1.4 Literature survey
| References | Fractional coupled 1D Burgers’ equation | Fractional coupled 2D Burgers’ equation | Fractional coupled 3D Burgers’ equation | Error analysis |
|---|---|---|---|---|
| Prakash et al. [53] | YES | YES | NO | NO |
| Singh et al. [54] | NO | YES | YES | YES |
| Singh et al. [15] | YES | NO | NO | NO |
| Prakash et al. [22] | YES | NO | NO | YES |
| Sulaiman et al. [42] | YES | NO | NO | NO |
| Kurt et al. [46] | NO | YES | NO | YES |
| Eltayeb and Bachar [48] | NO | YES | YES | NO |
| Ahmed et al. [51] | NO | YES | NO | YES |
| Chen and An [52] | YES | NO | NO | NO |
| Veeresha and Prakasha [55] | NO | YES | NO | YES |
| Edeki et al. [56] | YES | NO | NO | NO |
| Prakasha et al. [57] | NO | YES | NO | YES |
1.2 Research gap
There are several schemes observed from the literature that deal with fractional coupled Burgers’ equation in one, two, or three dimensions, but no method is provided that tackles fractional coupled Burgers’ equation in all one, two, and three dimensions. Therefore, the authors have focused on developing a technique that proves the validity of the approximated-analytical solution of the mentioned equations in one, two, and three dimensions. Better results are obtained in Tables 1 and 2 which are compared with refs. [53,58] and refs. [55,59], respectively. Through Tables 3–5, the convergence of the proposed regime is affirmed.
Comparison of absolute errors regarding Example 1
| η 1 | t | Abs. err. ϕ | Abs. err. ψ | Abs. err. ϕ | Abs. err. ψ | Abs. err. ϕ | Abs. err. ψ |
|---|---|---|---|---|---|---|---|
| Present | Present | [53] | [53] | ADM [58] | ADM [58] | ||
| −10 | 0.002 | 1.11 × 10−16 | 1.11 × 10−16 | 7.07 × 10−10 | 7.07 × 10−10 | 1.99 × 10−3 | 1.82 × 10−6 |
| −10 | 0.003 | 1.11 × 10−16 | 1.11 × 10−16 | 2.44 × 10−9 | 2.44 × 10−9 | 2.98 × 10−3 | 4.11 × 10−6 |
| −10 | 0.004 | 1.11 × 10−16 | 1.11 × 10−16 | 5.82 × 10−9 | 5.82 × 10−9 | 3.97 × 10−3 | 7.30 × 10−6 |
| 10 | 0.002 | 1.11 × 10−16 | 1.11 × 10−16 | 7.07 × 10−10 | 7.07 × 10−10 | 1.99 × 10−3 | 1.82 × 10−6 |
| 10 | 0.003 | 1.11 × 10−16 | 1.11 × 10−16 | 2.44 × 10−9 | 2.44 × 10−9 | 2.99 × 10−3 | 4.11 × 10−6 |
| 10 | 0.004 | 1.11 × 10−16 | 1.11 × 10−16 | 5.82 × 10−9 | 5.82 × 10−9 | 3.99 × 10−3 | 7.30 × 10−6 |
Comparison of absolute errors regarding Example 1
| η 1 | t | Abs. err. ϕ | Abs. err. ψ | Abs. err. ϕ | Abs. err. ψ | Abs. err. ϕ | Abs. err. ψ |
|---|---|---|---|---|---|---|---|
| Present | Present | GDTM [59] | GDTM [59] | HPM [55] | HPM [55] | ||
| −10 | 0.002 | 1.11 × 10−16 | 1.11 × 10−16 | 1.08 × 10−6 | 1.08 × 10−6 | 1.99 × 10−3 | 1.82 × 10−6 |
| −10 | 0.003 | 1.11 × 10−16 | 1.11 × 10−16 | 2.44 × 10−6 | 2.44 × 10−6 | 2.98 × 10−3 | 4.11 × 10−6 |
| −10 | 0.004 | 1.11 × 10−16 | 1.11 × 10−16 | 4.34 × 10−6 | 4.34 × 10−6 | 3.97 × 10−3 | 7.30 × 10−6 |
| 10 | 0.002 | 1.11 × 10−16 | 1.11 × 10−16 | 1.08 × 10−6 | 1.08 × 10−6 | 1.99 × 10−3 | 1.82 × 10−6 |
| 10 | 0.003 | 1.11 × 10−16 | 1.11 × 10−16 | 2.44 × 10−6 | 2.44 × 10−6 | 2.99 × 10−3 | 4.11 × 10−6 |
| 10 | 0.004 | 1.11 × 10−16 | 1.11 × 10−16 | 4.34 × 10−6 | 4.34 × 10−6 | 3.99 × 10−3 | 7.30 × 10−6 |
Comparison of L ∞ errors for ϕ and ψ components at different grid points regarding Example 1
| t = 1 | t = 2 | |||
|---|---|---|---|---|
| N | L ∞ ϕ | L ∞ ψ | L ∞ ϕ | L ∞ ψ |
| 10 | 2.1244 × 10−7 | 2.1244 × 10−7 | 2.0051 × 10−4 | 2.0051 × 10−4 |
| 20 | 5.5511 × 10−17 | 5.5511 × 10−17 | 3.3097 × 10−13 | 3.3097 × 10−13 |
| 30 | 1.1102 × 10−16 | 1.1102 × 10−16 | 1.1102 × 10−16 | 1.1102 × 10−16 |
| 40 | 1.1102 × 10−16 | 1.1102 × 10−16 | 9.7145 × 10−17 | 9.7145 × 10−17 |
| 50 | 1.1102 × 10−16 | 1.1102 × 10−16 | 8.3267 × 10−17 | 8.3267 × 10−17 |
L ∞ errors of ϕ and ψ components at different grid points regarding Example 2
| t = 0.1 | t = 0.2 | |||
|---|---|---|---|---|
| N | L ∞ ϕ | L ∞ ψ | L ∞ ϕ | L ∞ ψ |
| 10 | 8.8818 × 10−16 | 1.7764 × 10−15 | 5.8666 × 10−11 | 8.7997 × 10−11 |
| 20 | 1.7764 × 10−15 | 1.7764 × 10−15 | 2.6645 × 10−15 | 2.6645 × 10−15 |
| 30 | 1.7764 × 10−15 | 1.7764 × 10−15 | 1.7764 × 10−15 | 2.6645 × 10−15 |
| 40 | 1.7764 × 10−15 | 1.7764 × 10−15 | 2.6645 × 10−15 | 2.6645 × 10−15 |
| 50 | 1.7764 × 10−15 | 1.7764 × 10−15 | 2.6645 × 10−15 | 2.6645 × 10−15 |
L ∞ errors of ϕ and ψ components at different grid points regarding Example 3
| t = 0.1 | t = 0.2 | |||
|---|---|---|---|---|
| N | L ∞ ϕ | L ∞ ψ | L ∞ ϕ | L ∞ ψ |
| 10 | 2.7311 × 10−14 | 2.7311 × 10−14 | 2.7456 × 10−11 | 2.7456 × 10−11 |
| 20 | 2.2204 × 10−16 | 2.2204 × 10−16 | 2.2204 × 10−16 | 2.2204 × 10−16 |
| 30 | 2.2204 × 10−16 | 2.2204 × 10−16 | 2.2204 × 10−16 | 2.2204 × 10−16 |
An iterative scheme is developed in the present research regarding the solution of fractional coupled Burgers’ equation in one, two, and three dimensions. The present scheme is easy to implement and needs no complex programming regarding numerical discretization. Developing the numerical programs for the fractional PDEs is not an easy task; therefore, developing such iterative schemes is the need of time to find the approximated-analytical solutions.
There are several transforms provided in the literature but from the calculation aspect, some transforms are easy to implement and some are not. Shehu transform is noticed as one of the easiest methods to implement integral transform among all existing integral transforms. From literature, it is observed that fractional Schrodinger equations have never been solved in one, two, and three dimensions with the aid of a single Integral transform. Therefore, due to importance of such equations, in this research, concentration is focused upon the solution for the same, which retains the novelty of the study. Moreover, error and convergence analyses are also incorporated in the article.
1.3 Shehu transform
Definition 1
Shehu transform is defined as follows [60]:
where S is considered a Shehu Transform operator.
Definition 2
where s and v are considered as Shehu Transform variables.
Lemma 1
Linearity property of Shehu transform [ 60 , 61 ]:
where α 1 and α 2 are the arbitrary constants.
Lemma 2
Linearity property of inverse Shehu transform [ 60 , 61 ]:
If
Then,
Definition 3
Shehu transform of Caputo fractional derivative (C.F.D) [ 61 , 62 ]
Definition 4
Mittag-Leffler function considered for two parameters was given in ref. [63,64].
where E 1,1(n) = exp(n) and E 2,1(n 2) = cos(n)
Chart regarding Shehu transform [ 61 ]
| Q(t) | S[Q(t)] = J(s, v) | |
|---|---|---|
| 1. | 1 |
|
| 2. | t |
|
| 3. | t m , m ∈ N |
|
| 4. | t m , m > −1 |
|
| 5. | e at |
|
| 6. | sin(mt) |
|
| 7. | cos(mt) |
|
| 8. | sinh(mt) |
|
| 9. | cosh(mt) |
|
Chart regarding inverse Shehu transform [ 61 ]
| J(s, v) | Q(t) = S −1[J(s, v)] | |
|---|---|---|
| 1. |
|
1 |
| 2. |
|
t |
| 3. |
|
|
| 4. |
|
|
| 5. |
|
e at |
| 6. |
|
sin(mt) |
| 7. |
|
cos(mt) |
| 8. |
|
sinh(mt) |
| 9. |
|
cosh(mt) |
1.4 Outline of the article
For a better understanding of the proposed work of this research, the whole article is divided into different sections mentioned as follows:
In Section 2, the proposed Methodology is discussed for fractional coupled Burgers’ equation in one dimension.
In Section 3, applications for the proposed scheme are provided, as well as graphical representation and tabular form of the results are provided.
In Section 4, graphical discussion and error analysis are provided regarding the present work done.
In Section 5, concluding remarks are provided regarding this research.
The generalized formula regarding the solutions of fractional coupled Burgers’ equation in two and three dimensions are provided in Appendix A and Appendix B.
2 Methodology
2.1 General formula regarding solution of fractional coupled 1D Burgers’ equation
The form of the fractional coupled equation in one dimension could be considered as follows:
Applying Shehu transform in Eq. (10):
Applying Shehu transform in Eq. (11):
From Eq. (12):
From Eq. (13):
From Eq. (14):
From Eq. (15):
From Eq. (16):
From Eq. (17):
From Eq. (18):
From Eq. (19):
where
From Eq. (20):
Extracted formulae from Eq. (22):
and
for r = 1, 2, 3, …
Similarly,
Extracted formulae from Eq. (26):
for r = 1, 2, 3, …
3 Applications to fractional coupled Burgers’ equations in one, two, and three dimensions
In the present section, five numerical examples are solved regarding the implementation of the iterative Shehu transform method. The first example is related to the fractional 1D coupled Burgers’ equation. Second–fourth examples are related to the fractional 2D coupled Burgers’ equation. The fifth example is related to the fractional 3D coupled Burgers’ equation.
Example 1
Fractional coupled Burgers’ equation in one dimension is as follows [ 20 ]:
where
Considered θ = 1:
where
And
where
Similarly,
considering α = 1:
considering θ = 1:
where
and
where
Similarly,
considering
Example 2
Considered 2D coupled fractional Burgers’ equations are as follows [20]:
considering
where
where
considering
considering θ = 1:
where
and
where
and
considering
Example 3
Considered fractional coupled 2D Burgers’ equations as follows [21]:
where
considering θ = 1:
where
where
and
considering α = 1:
considering θ = 1:
where
where
and
considering α = 1:
Example 4
Considered fractional 2D coupled Burgers’ equation as follows [21]:
where
considering θ = 1:
where
and
where
where α = 1:
considering θ = 1:
where
and
where
where α = 1:
Example 5
Considered fractional 3D coupled Burgers’ equation as follows [54]:
where
where
Considering θ = 1:
where
Where,
considering θ = 1:
where
where
considering θ = 1:
where
where
4 Graphical discussion and error analysis
In Figures 1 and 2, a comparison of approximated and exact ϕ and ψ components are provided at t = 1 and t = 2 for N = 11 for Example 1. In Table 6, compatibility of approximated and exact solutions is shown at t = 1 for N = 11 regarding Example 1. In Table 3, a comparison of L ∞ errors for ϕ and ψ components is mentioned at different grid points for Example 1. In Figure 3, 3D plot for comparison of approximated and exact solutions of ϕ and ψ components is provided at t = 0.1 for N = 11 for Example 2. In Figure 4, the contour plot for compatibility of approximated and exact solutions of ϕ and ψ components is provided at t = 0.1 for N = 11 regarding Example 2. In Table 7, the approximated and exact results are compared at different mesh points regarding Example 2. In Table 4, L ∞ errors of ϕ and ψ components at different grid points are mentioned in case of Example 2. In Figures 5 and 6, the 3D and contour plots for comparison of approximated and exact results are provided at t = 0.1 with ρ = 1 and N = 11 for ϕ and ψ components regarding Example 3. In Table 8, approximated and exact results are compared at different mesh points for Example 3. In Table 5, L ∞ errors of ϕ and ψ components are mentioned at different grid points regarding Example 3. In Figures 7 and 8, 3D and contour plots for the comparison of approximated and exact solutions at t = 1 with ρ = 0.1 and N = 11 for ϕ and ψ components are provided regarding Example 4. In Table 9, the comparison of approximated and exact results at different mesh points is given for Example 4. In Table 10, L ∞ errors of ϕ and ψ components at different grid points are provided regarding Example 4.
Comparison of approximated and exact ϕ(η 1, t) at t = 1 and t = 2 for N = 11 regarding Example 1.
Comparison of approximated and exact ψ(η 1, t) at t = 1 and t = 2 for N = 11 regarding Example 1.
Comparison of approximated and exact solutions at t = 1 for N = 11 regarding Example 1
| η 1 | Approximated | Exact | Approximated | Exact |
|---|---|---|---|---|
| ϕ | ϕ | ψ | ψ | |
| 0.1 | 0.036726664 | 0.036726662 | 0.036726664 | 0.036726662 |
| 0.2 | 0.073086367 | 0.073086362 | 0.073086367 | 0.073086362 |
| 0.3 | 0.108715815 | 0.108715808 | 0.108715815 | 0.108715808 |
| 0.4 | 0.143259011 | 0.143259002 | 0.143259011 | 0.143259002 |
| 0.5 | 0.17637081 | 0.176370799 | 0.17637081 | 0.176370799 |
3D plot for comparison of approximated and exact solutions of ϕ and ψ components at t = 0.1 for N = 11 regarding Example 2.
Contour plot for comparison of approximated and exact solutions of ϕ and ψ components at t = 0.1 for N = 11 regarding Example 2.
3D plot for comparison of approximated and exact results at t = 0.1 with ρ = 1 and N = 11 for ϕ and ψ components regarding Example 3.
Comparison of approximate and exact results at different mesh points regarding Example 2
| (η 1, η 2) | Approx. ϕ | Exact ϕ | Approx. ψ | Exact ψ |
|---|---|---|---|---|
| (1.26, 1.25) | 2.308109 | 2.308109 | −0.25646 | −0.25646 |
| (1.89, 1.88) | 3.462163 | 3.462163 | −0.38468 | −0.38468 |
| (2.51, 2.51) | 4.616218 | 4.616218 | −0.51291 | −0.51291 |
Contour plot for comparison of approximated and exact results at t = 0.1 with ρ = 1 and N = 11 for ϕ and ψ components regarding Example 3.
Comparison of approximate and exact results at different mesh points regarding Example 3
| (η1, η2) | Approx. ϕ | Exact ϕ | Approx. ψ | Exact ψ |
|---|---|---|---|---|
| (1.25, 1.25) | −0.48124 | −0.48124 | 0.481238 | 0.481238 |
| (1.88, 1.88) | 0.481238 | 0.481238 | −0.48124 | −0.48124 |
| (2.51, 2.51) | 0.778659 | 0.778659 | −0.77866 | −0.77866 |
3D plot for comparison of approximated and exact solutions at t = 1 with ρ = 0.1 and N = 11 for ϕ and ψ components regarding Example 4.
Contour plot for comparison of approximated and exact solutions at t = 1 with ρ = 0.1 and N = 11 for ϕ and ψ components regarding Example 4.
Comparison of approximate and exact results at different mesh points regarding Example 4
| (η 1, η 2) | Approx. ϕ | Exact ϕ | Approx. ψ | Exact ψ |
|---|---|---|---|---|
| (1.25, 1.25) | −15.0785639 | −15.0785639 | 15.0785639 | 15.0785639 |
| (1.88, 1.88) | −52.9798252 | −52.9798252 | 52.97982519 | 52.97982519 |
| (2.51, 2.51) | −186.14915 | −186.14915 | 186.149152 | 186.149152 |
L ∞ errors of ϕ and ψ components at different grid points regarding Example 4
| t = 0.1 | t = 0.2 | |||
|---|---|---|---|---|
| N | L ∞ ϕ | L ∞ ψ | L ∞ ϕ | L ∞ ψ |
| 10 | 2.2737 × 10−13 | 2.2737 × 10−13 | 1.1369 × 10−13 | 1.1369 × 10−13 |
| 20 | 2.2737 × 10−13 | 2.2737 × 10−13 | 1.1369 × 10−13 | 1.1369 × 10−13 |
| 30 | 2.2737 × 10−13 | 2.2737 × 10−13 | 1.1369 × 10−13 | 1.1369 × 10−13 |
In Table 3, obtained approximated results are matched in good compatibility. In Table 3, it is noticed that on increasing the number of grid points from N = 10 to N = 20, error is reduced by more than double order, which indicates that the produced results will lead to the convergence of the solutions. In Table 1, the absolute error of ϕ and ψ components are matched with [53] and [58], which shows improved results over [53] and [58]. In Table 2, absolute errors of ϕ and ψ components are compared with [59] and [55]. In terms of the absolute errors, present results are better than the obtained results. In Table 4, it is noticed that on changing N = 10 to N = 20, the error is not reduced at t = 0.1. At t = 0.2, on changing N = 10 to N = 20, error is reduced by a good amount. So, as a crux, it can be said that at t = 0.2 or at the very closed time level to t = 0.2, the scheme will converge rapidly. In Table 5, at t = 0.1, when N was changed from N = 10 to N = 20, scheme error was reduced by order 2. At t = 0.2, when N was changed from N = 10 to N = 20, error was reduced by order 4. It means that on increasing the time level, the proposed regime will converge rapidly.
5 Concluding remarks
A study of the iterative Shehu transform method is provided in the present article to solve fractional-order coupled Burgers’ equation in one, two, and three dimensions. ISTM is used to deal with an approximation of results. Accuracy of results is compared by the graphical comparison as well as with tabular comparison of approximated and exact results. The proposed method is considerably effective and easy to implement when it is compared with the numerical methods for the generation of approximated results of linear and non-linear fractional differential equations. In Table 3, an error is reduced by more than double order on changing the value of N, which indicates that the produced results will lead to the convergence of the solutions. In Table 1, the absolute error of ϕ and ψ components are better than that in ref. [53] and ref. [58]. In Table 2, absolute errors of ϕ and ψ components are better than that in ref. [59] and ref. [56]. In Table 5, at t = 0.1, when N was changed from N = 10 to N = 20, scheme error was reduced by order 2. At t = 0.2, when N was changed from N = 10 to N = 20, error was reduced by order 4. It means that on increasing the time level, the proposed regime will show rapid convergence. This study will be a good technique to get improved outcomes for the researchers regarding the analytical solutions of complex-natured fractional PDEs as well as in the field of partial integro-differential equations.
Acknowledgment
Authors acknowledge all reviewers whose valuable comments helped us to modify the manuscript significantly.
-
Funding information: The authors did not receive support from any organization for the submitted work.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Conflict of interest: The authors have no competing interests to declare that are relevant to the content of this article.
Appendix A
General formula regarding solution of fractional coupled 2D Burgers’ equation
Form of the fractional coupled equation in two dimensions could be considered as follows:
Applying Shehu transform in Eq. (A1):
Applying Shehu transform in Eq. (A2):
From Eq. (A3):
From Eq. (A4):
From Eq. (A5):
From Eq. (A6):
From Eq. (A7):
From Eq. (A8):
From Eq. (A9):
From Eq. (A10):
where
From Eq. (A11):
Extracted formulae from Eq. (A13):
and
for
Similarly,
Extracted formulae from Eq. (A17):
and
for
Appendix B
General formula regarding solution of fractional coupled 3D Burgers’ equation
Form of fractional coupled equation in three dimensions could be considered as follows:
Applying Shehu transform in Eq. (B1):
Applying Shehu transform in Eq. (B2):
From Eq. (B3):
From Eq. (B4):
From Eq. (B5):
From Eq. (B6):
From Eq. (B7):
From Eq. (B8):
From Eq. (B9):
From Eq. (B10):
where
From Eq. (B11):
Extracted formulae from Eq. (B13):
and
for
Similarly,
Extracted formulae from Eq. (B17):
and
for
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- Fractal approach to the fluidity of a cement mortar
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- The role of relaxation and retardation phenomenon of Oldroyd-B fluid flow through Stehfest’s and Tzou’s algorithms
- Damage identification of wind turbine blades based on dynamic characteristics
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