The present article correlates with a fuzzy hybrid technique combined with an iterative transformation technique identified as the fuzzy new iterative transform method. With the help of Atangana-Baleanu under generalized Hukuhara differentiability, we demonstrate the consistency of this method by achieving fuzzy fractional gas dynamics equations with fuzzy initial conditions. The achieved series solution was determined and contacted the estimated value of the suggested equation. To confirm our technique, three problems have been presented, and the results were estimated in fuzzy type. The lower and upper portions of the fuzzy solution in all three examples were simulated using two distinct fractional orders between 0 and 1. Because the exponential function is present, the fractional operator is nonsingular and global. It provides all forms of fuzzy solutions occurring between 0 and 1 at any fractional-order because it globalizes the dynamical behavior of the given equation. Because the fuzzy number provides the solution in fuzzy form, with upper and lower branches, fuzziness is also incorporated in the unknown quantity. It is essential to mention that the projected methodology to fuzziness is to confirm the superiority and efficiency of constructing numerical results to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

1. Introduction

Fuzzy set theory is also an effective methodology for analyzing unpredictable scenarios. These ambiguities can arise in each part of a fractional equation, such as initial and boundary conditions. As a solution, determining fractional models in real-world settings allows for implementing interval or fuzzy formulations. The fuzzy set theory has been widely applied in many domains, i.e., topology, fixed-point theory, integral inequalities, fractional calculus, bifurcation, image processing, consumer electronics, control theory, artificial intelligence, and operations research. The area of fractional calculus, which includes fractional-order integrals and derivatives, has gotten a lot of interest from scholars and scientists in the recent several decades. Fractional calculus has many applications in modern physical and biological processes, with precise and accurate conclusions. In fractional differential calculus, the integral (differential) operators have a larger level of freedom. As a result, scholars have taken a keen interest in this field. A plethora of research articles, monographs, and books have been produced in recent years, covering a wide range of topics such as existence theory and analytical results [1–6]. Modern calculus has been implemented to a wide range of topics in applied sciences where data is uncertain. Zadeh [7] proposed the fuzzy set to deal with similar issues. Fuzzy relations and fuzzy control were defined further by Chang and Zadeh [8]. Fixed point theory, topology, algebra, control systems, and fuzzy logic, among other fields, use fuzzy set theory. The investigators developed elementary fuzzy calculus, fuzzy set-based [9, 10]. Dobius and Prada established the fundamental idea of fuzzy integral equations [11]. Fractional integral and differential equations have gained appeal among scholars in recent years. As a result, fuzzy calculus was prolonged to fuzzy fractional integral equations and fuzzy fractional differential, with several biological and physical implications. It is best to use a fuzzy idea to specify the parameters rather than a crisp number when studying problems where pieces of information are uncertain. As a result, numerous scholars have focused on studying fuzzy fractional differential equations in diverse directions. In [12–15], several basic difficulties are investigated. The Atangana-Baleanu Caputo derivative [16], a nonsingular fractional derivative suggested by Attangana, Balaneau, and Caputo, has recently gained prominence among scholars [17–20]. Fuzzy derivatives can be used to calculate fractional differential equations under the ABC derivative, leading to a variety of fascinating conclusions and new directions for young researchers.

Fuzzy fractional calculus theory is a prominent topic of mathematics that spans a wide range of mathematical structures in both theoretical and practical applications. In this arena, traditional derivatives are highly reliant on Caputo-Liouville or Riemann-Liouville issues. The common fault of these two qualifiers is the nonlocality and singularity of the kernel function noticed in the integral operator’s side by side with the normalizing function occurring alongside the integral ticks. Indeed, the real-world core replicating dynamic fractional systems, which cannot be denied, must yield a more productive and straightforward definition. This orientation introduces a new fractional fuzzy derivative construct, Atangana-Baleanu Caputo, which is used to generate and communicate new concrete fuzzy mathematical ideas. Because the kernel is based on the nature of exponential decay, the new fuzzy fractional Atangana-Baleanu Caputo derivative appears to be releasing singularity with the local kernel function, making fuzzy fractional differential equations more genuine in the creation of diverse uncertain models [21–25].

In various scientific fields, such as control theory, plasma oscillation, fluid flow, quantum mechanics, electromagnetic, biomolecular dynamics, nonlinear optics, and population growth, fractional differential equations have recently gained much attention as appealing mathematical tools for modeling several complex nonlinear phenomena. Caputo-Katugampola, Grunwald, Coimbra, conformable, Caputo, Riesz, Riemann-Liouville, and Caputo Fabrizio’s fractional ideas are among the many local and nonlocal fractional notions in the literature. Nonlocal derivatives are more intriguing than local derivatives since most physical implementations rely on historical and nonlocal features. Some of these operations, such as Caputo and Riemann-Liouville, have been proposed based on singular kernels. Recently, the fractional derivatives based on nonsingular kernels given by Caputo-Fabrizio [26] and Atangana-Baleanu [27] have been presented to represent physical dissipative phenomena better and reduce numerical collision when in comparison to other traditional fractional derivatives.

Gas dynamics are mathematical formulations of conservation rules found in engineering, i.e., mass conservation, momentum conservation, and energy conservation. Jafari et al. [28], Jawad et al. [29], Elizarova [30], Evans and Bulut [31], and Steger and Warming [32] used several analytical and numerical approaches to solve various types of gas dynamics equations in physics. Aziz and Anderson [33] used a pocket computer to tackle several gas dynamics problems in 1985. Rasulov and Karaguler [34] used a different approach to solve nonlinear system of equations problems in gas dynamics for a category of discontinuous functions in 2003. Liu [35] published a paper in 1990 on nonlinear hyperbolic-parabolic partial differential equations in gas dynamics and mechanics. Biazar and Eslami [36], Das and Kumar [37], and Kumar et al. [38] have recently applied the homotopy perturbation transform method and differential transform to solve the homogeneous and nonhomogeneous time-fractional gas dynamics problem.

The rest of the article is divided as follows. In Section 2, we give basic definitions of fractional calculus, fractional fuzzy derivative, and fuzzy set. The general methodology of the present method is in Section 3. In Section 4, to detect the validity and effectiveness of the suggested algorithm, we present some numeric problems. Meanwhile, we present the results in figures to see the effect of the Atangana-Baleanu-Caputo operator to the considered model. Finally, the conclusion will be drawn in the last section.

2. Basic Notions of Fractional and Fuzzy Calculus

This portion shows clearly several important components related to the fuzzy set theory and fractional calculus, as well as specific discussion of results about the Shehu transform. For further information, we refer the reader to [39].

Definition 1. We suggest that φ : ℝ ↦ [0, 1] is a fuzzy set, then, it is known to be fuzzy number if it holds the subsequent assumptions [39–42]:

(1)
φ is normal (for some η₀ ∈ ℝ; φ(℘₀) = 1)
(2)
φ is upper semicontinuous
(3)
φ(℘₁ω + (1 − ω)℘₂) ≥ (φ(℘₁)∧φ(℘₂))∀ ω ∈ [0, 1], ℘₁, ℘₂ ∈ ℝ_,, i.e., φ is convex
(4)
cl{℘∈ℝ, φ(℘) > 0} is compact

Definition 2. We define that a fuzzy number φ is r-level set expressed as [39–42]

(1)

where r ∈ [0, 1] and Φ ∈ ℝ.

Definition 3. The parameterized version of a fuzzy number is denoted by such that r ∈ [0, 1] fulfills the following assumptions [39–42]:

(1)
φ(r) is left continuous, nondecreasing, bounded over (0, 1] and left continuous at 0
(2)
φ(r) is right continuous, nonincreasing, bounded over (0, 1] and right continuous at 0; 3.

Definition 4. For r ∈ [0, 1] and Y to be a scalar, assume that there are two fuzzy numbers [39–42] , then, the subtraction, addition, and multiplication, respectively, are stated as

(2)

(3)

(4)

Definition 5. Suppose a fuzzy mapping having two fuzzy numbers [39–42] , then, Θ-distance between and is represented as

(5)

Theorem 6. Consider a fuzzy valued function such that and r ∈ [0, 1]. Then [39–42]:

(1)
(η₀; r) and E(η₀; r) are differentiable, if E is a (1)-differentiable, and

(6)

(2)
and are differentiable, if E is a (2)-differentiable, and

(7)

Definition 7. Assume that a fuzzy mapping Then, the fuzzy -fractional Caputo differentiability of fuzzy-valued mapping Φ is represented as [39–42].

(8)

Therefore, the parameterized versions of

and

, and CFD in a fuzzy sense are stated as

(9)

where r = [r]:

(10)

Definition 8. Assume that a fuzzy mapping and ν ∈ [0, 1], then, the fuzzy -fractional Atangana-Baleanu differentiability of fuzzy-valued mapping is represented as

(11)

Thus, the parameterized formulation of

and

, and the fuzzy Atangana-Baleanu derivative in Caputo sense is stated as

(12)

where

(13)

(14)

where denotes the normalized function which is equal to 1 when ν is assumed to be 0 and 1. Furthermore, we suppose that type (i) exists. Thus, here, there is no need to consider (ii) differentiability.

Definition 9. Consider a continuous real-valued mapping Ψ and there is an improper fuzzy Riemann-integrable mapping on [0, +∞). Then, the integral is known to be the fuzzy Shehu transform, and it is stated over the set of mappings [39–42]:

(15)

(16)

Remark 10. In (50), fulfills the assumption of the decreasing diameter , tiameter of a fuzzy mapping Φ. If σ = 1, then fuzzy the Shehu transform is red ∗ Laplace transform [39–42].

(17)

Furthermore, considering the classical Shehu transform [39–42], we obtain

(18)

and

(19)

Then, the aforesaid expressions can be written as

(20)

Then, we will define the fuzzy Shehu transformation of the Caputo generalized Hukuhara derivative .

Definition 11. Suppose there is an integrable fuzzy-valued mapping , and is the primitive of on [0, +∞), then, the CFD of order ν is presented as [39–42]

(21)

(22)

Bokhari et al. defined the fractional derivative of ABC operator in the sense of Shehu transform. Furthermore, in the context of a fuzzy Shehu transform, we extend the concept of fuzzy ABC fractional derivative as follows.

Definition 12. Consider such that , r ∈ [0, 1]; then, the Shehu transformation of the fuzzy ABC of order ν ∈ [0, 1] is defined as follows:

(23)

Furthermore, using the fact of Allahviranloo [39], we have

(24)

3. Main Work

In this section, we investigate our proposed model for a semianalytical solution. For this, we use the Shehu transform of the Atangana-Baleanu fractional differential operator along with iterative transform method as

(25)

where ν ∈ (0, 1]; therefore, the Shehu transform of (25) is

(26)

On using the initial condition, we obtain

(27)

Decompose the solution as

; then, (27) implies

(28)

Take parts of the solution by choice of comparison as

(29)

Taking the inverse Shehu transform, we obtain

(30)

Thus, the solution becomes

(31)

(32)

Equation (31) is the solution in series form.

4. Numerical Examples

Example 1. The first example is of a fuzzy fractional gas dynamics equation along with a fuzzy initial condition, as follows:

(33)

(34)

Using the scheme of Equation (30), we obtain

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

In a similar fashion, we can obtain the higher terms. The series solution is obtained using Equation (31); therefore, we write

(43)

while, in upper and lower portion form, it is

(44)

(45)

(46)

(47)

The exact solution of equation (33) is given as

(48)

Figure 1 indicates how the efficiency of various (upper and lower bound accuracy) surface plots for example 1 connect with the fuzzy Atangana-Baleanu operator and Shehu transformation are being showed in this research. Figure 2 shows that the lower and upper bounded graph of two dimensional. The pattern identifies the fluctuation in the mapping on the space coordinate ξ with the considering of and the uncertainty parameter rε[0, 1]. The figure shows that, as time passes, the mapping will become more intricate.

Example 2. The second example is of a fuzzy fractional gas dynamics equation along with a fuzzy initial condition, as follows

(49)

(50)

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

First graph shows that three dimensional fuzzy lower and upper branch graph of analytical series solution and second graph the different fractional-order of ν.

Using the scheme of Equation (50), we obtain

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

In a similar fashion, we can obtain the higher terms. The series solution is obtained using Equation (67); therefore, we write

(59)

While, in upper and lower portion form, it is

(60)

(61)

(62)

(63)

The exact solution of equation (50) is given as

(64)

Figure 3 indicates how the efficiency of various (upper and lower bound accuracy) surface plots for example 1 connect with the fuzzy Atangana-Baleanu operator and Shehu transformation is being shown in this research. Figure 4 shows that the lower and upper bounded graph of two dimensional. The pattern identifies the fluctuation in the mapping on the space coordinate ξ with the considering of and the uncertainty parameter rε[0, 1]. The figure shows that, as time passes, the mapping will become more intricate.

Example 3. The third example is of a fuzzy fractional nonlinear homogenous gas dynamics equation along with a fuzzy initial condition, as follows

(65)

(66)

Using the scheme of Equation (65), we obtain

(67)

(68)

(69)

(70)

(71)

(72)

In a similar fashion, we can obtain the higher terms. The series solution is obtained using Equation (71); therefore, we write

(73)

While, in upper and lower portion form, it is

(74)

(75)

(76)

(77)

The exact solution of equation (65) is given as

(78)

Figure 5 indicates how the efficiency of various (upper and lower bound accuracy) surface plots for example 1 connect with the fuzzy Atangana-Baleanu operator and Shehu transformation is being shown in this research. Figure 6 shows that the lower and upper bounded graph of two dimensional. The pattern identifies the fluctuation in the mapping on the space coordinate ξ with the considering of and the uncertainty parameter rε[0, 1]. The figure shows that, as time passes, the mapping will become more intricate.

5. Conclusion

This investigation is aimed at providing a semianalytical result to the fuzzy fractional third-order KdV equations by considering the Atangana-Baleanu fractional derivatives. An important example has confirmed the result obtained. Also, we have provided graphs of the numerical solution at numerous fractional orders. As seen in the figures, the plots will close with the curve at classical order 1 when the fractional-order ν approaches its integer value. Hence, we noted that fractional calculus also identifies the global nature of the dynamics of the equations concerning the fuzzy idea. We will use this approach to more dynamics of the models that involve fuzzy with future study. In future research, this technique can be implemented to obtain analytical and approximate results of perturbed fractional differential equations under the uncertainty equipped with nonclassical and integral boundary conditions in the sense of the Atangana-Baleanu operator.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia, for funding this work through research groups program under grant number R.G.P-1/192/42.

Open Research

Data Availability

The numerical data used to support the findings of this study are included within the article.

References

1 Rossikhin Y. A. and Shitikova M. V., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Applied Mechanics Reviews. (1997) 50, no. 1, 15–67, https://doi.org/10.1115/1.3101682, 2-s2.0-0030867045.
10.1115/1.3101682
Google Scholar
2 Lakshmikantham V., Leela S., and Vasundhara J., Theory of Fractional Dynamic Systems, 2009, Cambridge Academic Publishers, Cambridge, UK.
Google Scholar
3 Podlubny I., Fractional Differential Equations, Mathematics in Science and Engineering, 1999, Academic Press, New York.
Google Scholar
4 Naeem M., Zidan A. M., Nonlaopon K., Syam M. I., Al-Zhour Z., and Shah R., A new analysis of fractional-order equal-width equations via novel techniques, Symmetry. (2021) 13, no. 5, https://doi.org/10.3390/sym13050886.
10.3390/sym13050886
Web of Science® Google Scholar
5 Sunthrayuth P., Zidan A. M., Yao S. W., Shah R., and Inc M., The comparative study for solving fractional-order Fornberg–Whitham equation via ρ-laplace transform, Symmetry. (2021) 13, no. 5, https://doi.org/10.3390/sym13050784.
10.3390/sym13050784
Web of Science® Google Scholar
6 Niazi A. U., Iqbal N., Shah R., Wannalookkhee F., and Nonlaopon K., Controllability for fuzzy fractional evolution equations in credibility space, Fractal and Fractional. (2021) 5, no. 3, https://doi.org/10.3390/fractalfract5030112.
10.3390/fractalfract5030112
PubMed Web of Science® Google Scholar
7 Zadeh L. A., Fuzzy sets, Information and Control. (1965) 8, no. 3, 338–353, https://doi.org/10.1016/S0019-9958(65)90241-X, 2-s2.0-34248666540.
10.1016/S0019-9958(65)90241-X
CAS Web of Science® Google Scholar
8 Zadeh L. A., The role of fuzzy logic in modeling, identification and control, Fuzzy Sets, Fuzzy Logic, And Fuzzy Systems: Selected Papers by Lotfi A Zadeh, 1996, World Scientific.
10.1142/9789814261302_0041
Google Scholar
9 Dubois D. and Prade H., Towards fuzzy differential calculus part 3: differentiation, Fuzzy Sets and Systems. (1982) 8, no. 3, 225–233, https://doi.org/10.1016/S0165-0114(82)80001-8, 2-s2.0-0020184062.
10.1016/S0165-0114(82)80001-8
Web of Science® Google Scholar
10 Kaleva O., Fuzzy differential equations, Fuzzy Sets and Systems. (1987) 24, no. 3, 301–317, https://doi.org/10.1016/0165-0114(87)90029-7, 2-s2.0-0023573517.
10.1016/0165-0114(87)90029-7
Web of Science® Google Scholar
11 Aljahdaly N. H., Agarwal R. P., Shah R., and Botmart T., Analysis of the time fractional-order coupled burgers equations with non-singular kernel operators, Mathematics. (2021) 9, no. 18, https://doi.org/10.3390/math9182326.
10.3390/math9182326
Google Scholar
12 Buckley J. J. and Feuring T., Fuzzy differential equations, Fuzzy Sets and Systems. (2000) 110, no. 1, 43–54, https://doi.org/10.1016/S0165-0114(98)00141-9, 2-s2.0-0012838528.
10.1016/S0165-0114(98)00141-9
Google Scholar
13 Liu R., Wang J. R., and O'Regan D., On the solutions of first-order linear impulsive fuzzy differential equations, Fuzzy Sets and Systems. (2020) 400, 1–33, https://doi.org/10.1016/j.fss.2019.11.001.
10.1016/j.fss.2019.11.001
Web of Science® Google Scholar
14 Khastan A., Nieto J. J., and Rodriguez-Lopez R., Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets and Systems. (2011) 177, no. 1, 20–33, https://doi.org/10.1016/j.fss.2011.02.020, 2-s2.0-79958159192.
10.1016/j.fss.2011.02.020
Web of Science® Google Scholar
15 Nieto J. J. and Rodriguez-Lopez R., Some results on boundary value problems for fuzzy differential equations with functional dependence, Fuzzy Sets and Systems. (2013) 230, 92–118, https://doi.org/10.1016/j.fss.2013.05.010, 2-s2.0-84884676070.
10.1016/j.fss.2013.05.010
Web of Science® Google Scholar
16 Atangana A. and Koca I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons & Fractals. (2016) 89, 447–454, https://doi.org/10.1016/j.chaos.2016.02.012, 2-s2.0-84960970311.
10.1016/j.chaos.2016.02.012
Web of Science® Google Scholar
17 Baleanu D., Mousalou A., and Rezapour S., A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative, Adv. Difference Equ.(2017) 2017, no. 1, 1–12, https://doi.org/10.1186/s13662-017-1088-3, 2-s2.0-85012033986.
10.1186/s13662-017-1088-3
Web of Science® Google Scholar
18 Korpinar Z., Inc M., Baleanu D., and Bayram M., Theory and application for the time fractional Gardner equation with Mittag-Leffler kernel, Journal of Taibah University for Science. (2019) 13, no. 1, 813–819, https://doi.org/10.1080/16583655.2019.1640446.
10.1080/16583655.2019.1640446
Web of Science® Google Scholar
19 Atangana A. and Gómez-Aguilar J. F., Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu, Numerical Methods for Partial Differential Equations. (2018) 34, no. 5, 1502–1523, https://doi.org/10.1002/num.22195, 2-s2.0-85050740100.
10.1002/num.22195
Web of Science® Google Scholar
20 J. F. Gomez, L. Torres, and R. F. Escobar, Fractional Derivatives with Mittag-Leffler Kernel, 2019, Springer International Publishing.
10.1007/978-3-030-11662-0
Google Scholar
21 Allahviranloo T., Salahshour S., and Abbasbandy S., Explicit solutions of fractional differential equations with uncertainty, Soft Computing. (2012) 16, no. 2, 297–302, https://doi.org/10.1007/s00500-011-0743-y, 2-s2.0-84855651248.
10.1007/s00500-011-0743-y
Web of Science® Google Scholar
22 Rahaman M., Mondal S. P., El Allaoui A., Alam S., Ahmadian A., and Salahshour S., Solution strategy for fuzzy fractional order linear homogeneous differential equation by Caputo-H differentiability and its application in fuzzy EOQ model, Advances in Fuzzy Integral and Differential Equations, 2022, Springer, Cham, 143–157.
10.1007/978-3-030-73711-5_5
Google Scholar
23 Choi H., Sin K., Pak S., Sok K., and So S., Representation of solution of initial value problem for fuzzy linear multi-term fractional differential equation with continuous variable coefficient, AIMS Mathematics. (2019) 4, no. 3, 613–625, https://doi.org/10.3934/math.2019.3.613, 2-s2.0-85073387078.
10.3934/math.2019.3.613
Web of Science® Google Scholar
24 Salahshour S., Ahmadian A., Senu N., Baleanu D., and Agarwal P., On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem, Entropy. (2015) 17, no. 2, 885–902, https://doi.org/10.3390/e17020885, 2-s2.0-84923359923.
10.3390/e17020885
Web of Science® Google Scholar
25 Anastassiou G. A., Fuzzy fractional calculus and the Ostrowski integral inequality, Intelligent Mathematics: Computational Analysis, 2011, Springer, Berlin, Heidelberg, 553–574.
10.1007/978-3-642-17098-0_34
Google Scholar
26 Caputo M. and Fabrizio M., The kernel of the distributed order fractional derivatives with an application to complex materials, Fractal and Fractional. (2017) 1, no. 1, https://doi.org/10.3390/fractalfract1010013.
10.3390/fractalfract1010013
Web of Science® Google Scholar
27 Khan I., New idea of Atangana and Baleanu fractional derivatives to human blood flow in nanofluids, Chaos: An Interdisciplinary Journal of Nonlinear Science. (2019) 29, no. 1, article 013121, https://doi.org/10.1063/1.5078738, 2-s2.0-85060490529.
10.1063/1.5078738
Google Scholar
28 Jafari H., Chun C., Seifi S., and Saeidy M., Analytical solution for nonlinear gas dynamics equation by homotopy analysis method, Applications and Applied Mathematics. (2009) 4, no. 1, 149–154.
Google Scholar
29 Mohamad-Jawad A. J.’., Petković M. D., and Biswas A., Applications of He's principles to partial differential equations, Applied Mathematics and Computation. (2011) 217, no. 16, 7039–7047, https://doi.org/10.1016/j.amc.2011.02.013, 2-s2.0-79952452222.
10.1016/j.amc.2011.02.013
Web of Science® Google Scholar
30 Elizarova T. G., Quasi-Gas Dynamic Equations (Computational Fluid and Solid Mechanics), 2009, Springer Verlag.
10.1007/978-3-642-00292-2
Google Scholar
31 Evans D. J. and Bulut H., A new approach to the gas dynamics equation: an application of the decomposition method, International Journal of Computer Mathematics. (2002) 79, no. 7, 817–822, https://doi.org/10.1080/00207160211297, 2-s2.0-0042793363.
10.1080/00207160211297
Web of Science® Google Scholar
32 Steger J. L. and Warming R. F., Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods, Journal of Computational Physics. (1981) 40, no. 2, 263–293, https://doi.org/10.1016/0021-9991(81)90210-2, 2-s2.0-49049151571.
10.1016/0021-9991(81)90210-2
Web of Science® Google Scholar
33 Aziz A. and Anderson D., The use of a pocket computer in gas dynamics, Computers & Education. (1985) 9, no. 1, 41–56, https://doi.org/10.1016/0360-1315(85)90025-9, 2-s2.0-50849152956.
10.1016/0360-1315(85)90025-9
Web of Science® Google Scholar
34 Rasulov M. and Karaguler T., Finite difference schemes for solving system equations of gas dynamic in a class of discontinuous functions, Applied Mathematics and Computation. (2003) 143, no. 1, 145–164, https://doi.org/10.1016/S0096-3003(02)00353-3, 2-s2.0-0038486012.
10.1016/S0096-3003(02)00353-3
Web of Science® Google Scholar
35 Liu T. P., Nonlinear waves in mechanics and gas dynamics, 1990, Defense Technical Information Center.
10.21236/ADA238340
Google Scholar
36 Biazar J. and Eslami M., Differential transform method for nonlinear fractional gas dynamics equation, International Journal of Physical Sciences. (2011) 6, no. 5.
Google Scholar
37 Das S. and Kumar R., Approximate analytical solutions of fractional gas dynamic equations, Applied Mathematics and Computation. (2011) 217, no. 24, 9905–9915, https://doi.org/10.1016/j.amc.2011.03.144, 2-s2.0-79959205476.
10.1016/j.amc.2011.03.144
Web of Science® Google Scholar
38 Kumar S., Kocak H., and Yildirim A., A fractional model of gas dynamics equations and its analytical approximate solution using Laplace transform, Zeitschrift für Naturforschung. (2012) 67, no. 6-7, 389–396, https://doi.org/10.5560/zna.2012-0038, 2-s2.0-84867725131.
10.5560/zna.2012-0038
CAS Google Scholar
39 Allahviranloo T., Fuzzy Fractional Differential Operators and Equation Studies in Fuzziness and Soft Computing, 2021, Springer, Berlin, Germany.
10.1007/978-3-030-51272-9
Google Scholar
40 Gottwald S., Fuzzy Set Theory and Its Applications, 1992, Kluwer Academic Publishers, Dordrecht.
Google Scholar
41 Allahviranloo T. and Ahmadi M. B., Fuzzy Laplace transforms, Soft Computing. (2010) 14, no. 3, 235–243, https://doi.org/10.1007/s00500-008-0397-6, 2-s2.0-70350649223.
10.1007/s00500-008-0397-6
Web of Science® Google Scholar
42 Maitama S. and Zhao W., Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives, Computational and Applied Mathematics. (2021) 40, no. 3, 1–30, https://doi.org/10.1007/s40314-021-01476-9.
10.1007/s40314-021-01476-9
Web of Science® Google Scholar

Citing Literature

All articles

[Retracted] Novel Analysis of Fuzzy Physical Models by Generalized Fractional Fuzzy Operators

Retraction(s) for this article

Retracted: Novel Analysis of Fuzzy Physical Models by Generalized Fractional Fuzzy Operators

Abstract

1. Introduction

2. Basic Notions of Fractional and Fuzzy Calculus

3. Main Work

4. Numerical Examples

5. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

Citing Literature

Figures

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

[Retracted] Novel Analysis of Fuzzy Physical Models by Generalized Fractional Fuzzy Operators

Retraction(s) for this article

Retracted: Novel Analysis of Fuzzy Physical Models by Generalized Fractional Fuzzy Operators

Abstract

1. Introduction

2. Basic Notions of Fractional and Fuzzy Calculus

3. Main Work

4. Numerical Examples

5. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

Citing Literature

Figures

References

Related

Information