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February  2022, 15(2): 315-338. doi: 10.3934/dcdss.2021011

## A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform

 1 Department of Mathematics, University of Malakand, Chakdara Dir (Lower), Khyber Pakhtunkhawa, 18000, Pakistan 2 Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey 3 Institute of IR 4.0, The National University of Malaysia, Bangi, Malaysia 4 Department of Law, Economics and Human Sciences & Decisions Lab, University Mediterranea of Reggio Calabria, Reggio Calabria, Italy

Received  June 2020 Revised  December 2020 Published  February 2022 Early access  January 2021

In this research article, the techniques for computing an analytical solution of 2D fuzzy wave equation with some affecting term of force has been provided. Such type of achievement for the aforesaid solution is obtained by applying the notions of a Caputo non-integer derivative in the vague or uncertainty form. At the first attempt the fuzzy natural transform is applied for obtaining the series solution. Secondly the homotopy perturbation (HPM) technique is used, for the analysis of the proposed result by comparing the co-efficient of homotopy parameter $q$ to get hierarchy of equation of different order for $q$. For this purpose, some new results about Natural transform of an arbitrary derivative under uncertainty are established, for the first time in the literature. The solution has been assumed in term of infinite series, which break the problem to a small number of equations, for the respective investigation. The required results are then determined in a series solution form which goes rapidly towards the analytical result. The solution has two parts or branches in fuzzy form, one is lower branch and the other is upper branch. To illustrate the ability of the considered approach, we have proved some test problems.

Citation: Muhammad Arfan, Kamal Shah, Aman Ullah, Soheil Salahshour, Ali Ahmadian, Massimiliano Ferrara. A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform. Discrete & Continuous Dynamical Systems - S, 2022, 15 (2) : 315-338. doi: 10.3934/dcdss.2021011
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##### References:
 [1] N. A. Abdul Rahman and M. Z. Ahmad, Solving fuzzy fractional differential equations using fuzzy Sumudu transform, J. Nonlinear Sci. Appl., 10 (2017), 2620-2632.  doi: 10.22436/jnsa.010.05.28.  Google Scholar [2] H. Ahmad, A. R. Seadawy, T. A. Khan and P. Thounthong, Analytic approximate solutions for some nonlinear parabolic dynamical wave equations, Taibah Uni. J. Sci., 14 (2020), 346-358.  doi: 10.1080/16583655.2020.1741943.  Google Scholar [3] S. Ahmed, A. Ullah, M. Arfan and K. Shah, On analysis of of the fractional mathematical model of rotravirous epidemic with the effect of breastfeeding and vaccination under Atangana-Baleanu (AB) derivative, Chaos Solitons Fractals, 140 (2020), 110233, 20 pp. doi: 10.1016/j.chaos.2020.110233.  Google Scholar [4] A. Ali, A. R. Seadawy and D. Lu, Computational methods and traveling wave solutions for the fourth-order nonlinear ablowitz-kaup-newell-segur water wave dynamical equation via two methods and its applications, Open Physics J., 16 (2018), 219-226.  doi: 10.1515/phys-2018-0032.  Google Scholar [5] M. Al-Refai and T. Abdeljawad, Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel, Adv. Difference Equ., 2017 (2017), Paper No. 315, 12 pp. doi: 10.1186/s13662-017-1356-2.  Google Scholar [6] M. Arshad, A. R. Seadawy and D. Lu, Modulation stability and optical soliton solutions of nonlinear schrödinger equation with higher order dispersion and nonlinear terms and its applications, Superlattice Microstr., 112 (2017), 422-434.  doi: 10.1016/j.spmi.2017.09.054.  Google Scholar [7] M. A. Asiru, Further properties of Sumudu transform and its applications, Internat. J. Math. Ed. Sci. Tech., 33 (2002), 441-449.  doi: 10.1080/002073902760047940.  Google Scholar [8] Z. Ayati and J. 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Math. Appl., 58 (2009), 2221-2230.  doi: 10.1016/j.camwa.2009.03.030.  Google Scholar [14] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470868279.  Google Scholar [15] S. S. Chang and L. A. Zadeh, On Fuzzy Mapping and Control: In Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems, World Scientific, Singapor, 1996. Google Scholar [16] M. Chen, Y. Fu, X. Xue and C. Wu, Two point boundary value problems of umdamped uncertain dynamical system, Fuzzy Sets and Systems, 159 (2008), 2077-2089.  doi: 10.1016/j.fss.2008.03.006.  Google Scholar [17] D. Dubois and H. Prade, Towards fuzzy differential calculus part 1: Integration of fuzzy mappings, Fuzzy Sets and Systems, 8 (1982), 1-17.  doi: 10.1016/0165-0114(82)90025-2.  Google Scholar [18] R. A. El-Nabulsi, Induced gravity from two occurrences of actions, The European Phy. J. Plus., 132 (2017), 295. doi: 10.1140/epjp/i2017-11560-3.  Google Scholar [19] H. Eltayeb and A. 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Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996.   Google Scholar [32] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7.  Google Scholar [33] Z. H. Khan and W. A. Khan, N-transform properties and applications, Nust. J. Eng. Sci., 1 (2008), 127-133.   Google Scholar [34] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar [35] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009. Google Scholar [36] B. D. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3-16.   Google Scholar [37] D. Loonker and P. K. Banerji, Natural transform for distribution and Boehmian spaces., Math. Engg. 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Representation of three dimensional(3D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5,$ of example 1. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of three dimensional(3D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5$ of example 1. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of two dimensional(2D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 1. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of two dimensional(2D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 1. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of comparison of fuzzy solution for upper and lower branches of example 1 by LADM and NTHPM in three dimension(3D). The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of comparison of fuzzy solution for upper and lower branches of example 1 by LADM and NTHPM in two dimension(2D). The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of three dimensional(3D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5$ of example 2. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of three dimensional(3D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5$ of example 2. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of two dimensional(2D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 2. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of two dimensional(2D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 2. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of comparison of fuzzy solution for upper and lower branches of example 2 by LADM and NTHPM in three dimension(3D). The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of comparison of fuzzy solution for upper and lower branches of example 2 by LADM and NTHPM in two dimension(2D). The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of three dimensional(3D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5,$ of example 3. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of three dimensional(3D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{y} = 0.5, \texttt{t} = 0.5$ of example 3. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of two dimensional(2D) graph of fuzzy solution at four different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 3. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of two dimensional(2D) graph of fuzzy solution at four other different fractional order of $\theta$ with $\texttt{x} = 0.5, \texttt{y} = 0.5, \texttt{t} = 0.5$ of example 3. The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of comparison of fuzzy solution for upper and lower branches of example 3 by LADM and NTHPM in three dimension(3D). The two similar color legends represents upper and lower portion of fuzzy solution respectively
Representation of comparison of fuzzy solution for upper and lower branches of example 3 by LADM and NTHPM in two dimension(2D). The two similar color legends represents upper and lower portion of fuzzy solution respectively
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