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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 and Continuous Dynamical Systems - S, 2022, 15 (2) : 315-338. doi: 10.3934/dcdss.2021011
##### References:
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[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. [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. [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. [8] Z. Ayati and J. Biazar, On the convergence of Homotopy perturbation method, J. Egyptian Math. Soc., 23 (2015), 424-428.  doi: 10.1016/j.joems.2014.06.015. [9] V. A. Baǐdosov, Fuzzy differential inclusion, J. Appl. Math. Mech., 54 (1990), 8-13.  doi: 10.1016/0021-8928(90)90080-T. [10] F. B. M. Belgacem, A. A. Karaballi and S. L. 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Gustavo. I. Silva and U. M. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions, Chemical Eng. Sci., 117 (2014), 217-228. [27] A. Heinz, A boundary element procedure for transient wave propagations in two-dimensional isotropic elastic media, Finite Elements in Analysis and Design, 1 (1985), 313-322. [28] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747. [29] M. Iqbal, A. R. Seadawy, O. H. Khalil and D. Lu, Propagation of long internal waves in density stratified ocean for the $(2+1)$-dimensional nonlinear nizhnik-novikov-vesselov dynamical equation, Results in Physics, 16 (2020), 102838. doi: 10.1016/j.rinp.2019.102838. [30] M. Iqbal, A. R. Seadawy and D. Lu, Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods, Modern Phys. Lett. A, 33 (2018), 1850183, 13 pp. doi: 10.1142/S0217732318501833. [31] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996. [32] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7. [33] Z. H. Khan and W. A. Khan, N-transform properties and applications, Nust. J. Eng. Sci., 1 (2008), 127-133. [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. [35] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009. [36] B. D. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3-16. [37] D. Loonker and P. K. Banerji, Natural transform for distribution and Boehmian spaces., Math. Engg. Sci. Aerospace, 4 (2013), 69-76. [38] D. Lu, A. R. Seadawy and A. Ali, Applications of exact traveling wave solutions of modified liouville and the symmetric regularized long wave equations via two new techniques, Results in Physics, 9 (2018), 1403-1410.  doi: 10.1016/j.rinp.2018.04.039. [39] D. Lu, A. R. Seadawy and A. Ali, Dispersive traveling wave solutions of the equal-width and modified equal-width equations via mathematical methods and its applications, Results in Physics, 9 (2018), 313-320.  doi: 10.1016/j.rinp.2018.02.036. [40] Y. S. Ozkan, E. Yasar and A. R. Seadawy, A third-order nonlinear schrödinger equation; the exact solutions, group-invariant solutions and conservation laws, Taibah Uni.J. Sci., 14 (2020), 585-597. [41] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999. [42] Y. Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Thermal Stresses, 28 (2005), 83-102.  doi: 10.1080/014957390523741. [43] M. Ur. Rahman, M. Arfan, K. Shah and J. F. Gómez-Aguilar, Investigating a non-linear dynamical model of Covid-19 diesease under fuzzy caputo, randum an ABC fractional order derivative, Chaos-Solitons and Fractals, 140 (2020), 110232. doi: 10.1016/j.chaos.2020.110232. [44] H. Richard, Elementary Applied Partial Differential Equations, Englewood Cliffs, NJ: Prentice Hall, 1983. [45] F. J. Rizzo and D. J. Shippy, A method of solution for certain problems of transient heat conduction, AIAA J., 8 (1970), 2004-2009.  doi: 10.2514/3.6038. [46] Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997), 15-67.  doi: 10.1115/1.3101682. [47] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives (Theory and Applications), Gordon and Breach Science Publishers, Yverdon, 1993. [48] A. R. Seadawy and K. El-Rashidy, Dispersive solitary wave solutions of Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili dynamical equations in unmagnetized dust plasma, Results in Physics, 8 (2018), 1216-1222.  doi: 10.1016/j.rinp.2018.01.053. [49] A. R. Seadawy, M. Iqbal and D. Lu, Nonlinear wave solutions of the Kudryashov-Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity, Taibah Uni. J. Sci., 13 (2019), 1060-1072.  doi: 10.1080/16583655.2019.1680170. [50] K. Shah, M. A. Alqudah, F. Jarad and T. Abdeljawad, Semi-analytical study of Pine Wilt disease model with convex rate under Caputo-Fabrizio fractional order derivative, Chaos Solitons Fractals, 135 (2020), 109754, 9 pp. doi: 10.1016/j.chaos.2020.109754. [51] H. Shatha, A. El-Ajou, S. Hadid, M. Al-Smadi and S. Momani, Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system, Chaos Solitons Fractals, 133 (2020), 109624, 10 pp. doi: 10.1016/j.chaos.2020.109624. [52] M. R. Spiegel, Theory and Problems of Laplace Transforms, Schaum Publishing Co., New York, 1965. [53] L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis, 71 (2009), 1311-1328.  doi: 10.1016/j.na.2008.12.005. [54] R. Toledo-Hernandez, V. Rico-Ramirez, A. Gustavo. Iglesias-Silva and M. Urmila, A fractional calculus approach to the dynamic optimization of biological reactive systems. 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show all references

##### 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. [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. [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. [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. [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. [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. [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. [8] Z. Ayati and J. Biazar, On the convergence of Homotopy perturbation method, J. Egyptian Math. Soc., 23 (2015), 424-428.  doi: 10.1016/j.joems.2014.06.015. [9] V. A. Baǐdosov, Fuzzy differential inclusion, J. Appl. Math. Mech., 54 (1990), 8-13.  doi: 10.1016/0021-8928(90)90080-T. [10] F. B. M. Belgacem, A. A. Karaballi and S. L. Kalla., Analytical, Investigations of the Sumudu transform and applications to integral production equations, Math. Prob. Eng., 2003 (2003), 103-118.  doi: 10.1155/S1024123X03207018. [11] F. B. M. Belgacem and R. Silambarasan, Theory of Natural Transform:, Math. Eng. Sci. Aero. J., 3 (2012), 99-124. [12] F. B. M. Belgacem and R. Silambarasan, Maxwells equations solutions by means of the Natural transform, I. J. Math. Eng. Sci. Aero., 3 (2012), 313-323. [13] J. Biazar and H. Aminikhahb, Study of convergence of homotopy perturbation method for systems of partial differential equations, Comput. Math. Appl., 58 (2009), 2221-2230.  doi: 10.1016/j.camwa.2009.03.030. [14] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470868279. [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. [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. [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. [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. [19] H. Eltayeb and A. Kiliçman, A note on solutions of wave, Laplace's and heat equations with convolution terms by using a double Laplace transform, Appl. Math. Lett., 21 (2008), 1324-1329.  doi: 10.1016/j.aml.2007.12.028. [20] B. Esmail, H. Sadeghi Goghary and S. Abbasbandy, Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method, Appl. Math. Comput., 161 (2005), 733-744.  doi: 10.1016/j.amc.2003.12.071. [21] N. Farah, A. R. Seadawy, S. Ahmad, S. T. R. Rizvi and M. Younis, Interaction properties of soliton molecules and Painleve analysis for nano bioelectronics transmission model, Optical and Quantum Electronics, 52 (2020), 1-15. [22] R. Goetschel, Jr . and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43.  doi: 10.1016/0165-0114(86)90026-6. [23] F. Haq, K. Shah, G. Rahman and M. Shahzad, Numerical solution of fractional order smoking model via Laplace Adomian decomposition method, Alex. Eng. J., 57 (2018), 1061-1069.  doi: 10.1016/j.aej.2017.02.015. [24] F. Haq, K. Shah, G. Rahman and M. Shahzad, Numerical analysis of fractional order model of HIV-1 infection of CD4+ T-cells, Comput. Method. Diff. Equs., 5 (2017), 1-11. [25] M. A. Helal, A. R. Seadawy and M. H. Zekry, Stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation, Appl. Math. Comput., 232 (2014), 1094-1103.  doi: 10.1016/j.amc.2014.01.066. [26] T. Hernandez, Ra siel, V. R. Ramirez, A. Gustavo. I. Silva and U. M. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions, Chemical Eng. Sci., 117 (2014), 217-228. [27] A. Heinz, A boundary element procedure for transient wave propagations in two-dimensional isotropic elastic media, Finite Elements in Analysis and Design, 1 (1985), 313-322. [28] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747. [29] M. Iqbal, A. R. Seadawy, O. H. Khalil and D. Lu, Propagation of long internal waves in density stratified ocean for the $(2+1)$-dimensional nonlinear nizhnik-novikov-vesselov dynamical equation, Results in Physics, 16 (2020), 102838. doi: 10.1016/j.rinp.2019.102838. [30] M. Iqbal, A. R. Seadawy and D. Lu, Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods, Modern Phys. Lett. A, 33 (2018), 1850183, 13 pp. doi: 10.1142/S0217732318501833. [31] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996. [32] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7. [33] Z. H. Khan and W. A. Khan, N-transform properties and applications, Nust. J. Eng. Sci., 1 (2008), 127-133. [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. [35] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009. [36] B. D. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3-16. [37] D. Loonker and P. K. Banerji, Natural transform for distribution and Boehmian spaces., Math. Engg. Sci. Aerospace, 4 (2013), 69-76. [38] D. Lu, A. R. Seadawy and A. Ali, Applications of exact traveling wave solutions of modified liouville and the symmetric regularized long wave equations via two new techniques, Results in Physics, 9 (2018), 1403-1410.  doi: 10.1016/j.rinp.2018.04.039. [39] D. Lu, A. R. Seadawy and A. Ali, Dispersive traveling wave solutions of the equal-width and modified equal-width equations via mathematical methods and its applications, Results in Physics, 9 (2018), 313-320.  doi: 10.1016/j.rinp.2018.02.036. [40] Y. S. Ozkan, E. Yasar and A. R. Seadawy, A third-order nonlinear schrödinger equation; the exact solutions, group-invariant solutions and conservation laws, Taibah Uni.J. Sci., 14 (2020), 585-597. [41] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999. [42] Y. Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Thermal Stresses, 28 (2005), 83-102.  doi: 10.1080/014957390523741. [43] M. Ur. Rahman, M. Arfan, K. Shah and J. F. Gómez-Aguilar, Investigating a non-linear dynamical model of Covid-19 diesease under fuzzy caputo, randum an ABC fractional order derivative, Chaos-Solitons and Fractals, 140 (2020), 110232. doi: 10.1016/j.chaos.2020.110232. [44] H. Richard, Elementary Applied Partial Differential Equations, Englewood Cliffs, NJ: Prentice Hall, 1983. [45] F. J. Rizzo and D. J. Shippy, A method of solution for certain problems of transient heat conduction, AIAA J., 8 (1970), 2004-2009.  doi: 10.2514/3.6038. [46] Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997), 15-67.  doi: 10.1115/1.3101682. [47] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives (Theory and Applications), Gordon and Breach Science Publishers, Yverdon, 1993. [48] A. R. Seadawy and K. El-Rashidy, Dispersive solitary wave solutions of Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili dynamical equations in unmagnetized dust plasma, Results in Physics, 8 (2018), 1216-1222.  doi: 10.1016/j.rinp.2018.01.053. [49] A. R. Seadawy, M. Iqbal and D. Lu, Nonlinear wave solutions of the Kudryashov-Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity, Taibah Uni. J. Sci., 13 (2019), 1060-1072.  doi: 10.1080/16583655.2019.1680170. [50] K. Shah, M. A. Alqudah, F. Jarad and T. Abdeljawad, Semi-analytical study of Pine Wilt disease model with convex rate under Caputo-Fabrizio fractional order derivative, Chaos Solitons Fractals, 135 (2020), 109754, 9 pp. doi: 10.1016/j.chaos.2020.109754. [51] H. Shatha, A. El-Ajou, S. Hadid, M. Al-Smadi and S. Momani, Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system, Chaos Solitons Fractals, 133 (2020), 109624, 10 pp. doi: 10.1016/j.chaos.2020.109624. [52] M. R. Spiegel, Theory and Problems of Laplace Transforms, Schaum Publishing Co., New York, 1965. [53] L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis, 71 (2009), 1311-1328.  doi: 10.1016/j.na.2008.12.005. [54] R. Toledo-Hernandez, V. Rico-Ramirez, A. Gustavo. Iglesias-Silva and M. Urmila, A fractional calculus approach to the dynamic optimization of biological reactive systems. 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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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