Study on temporal-fuzzy fractional p-KdV equation with non-singular Mittag Leffler kernel
Abstract
This work discusses the solution of temporal-fuzzy fractional non-linear p-KdV equations employing a singular kernel and a non-singular Mittag Leffler kernel. A novel q-homotopy analysis approach with a generalised transform is proposed to study the fuzzy time-fractional model with two distinct fractional operators, and the behaviour of the solution is studied in both crisp and uncertain cases. Consequently, the efficiency and accuracy of the proposed method have been obtained by comparing the obtained numerical results with the available results under the assumption of crisp case for α = 1 that validate the obtained results. Finally, the efficiency of the proposed fractional orders is checked with distinct fractional operators.
Keyword : Atangana-Baleanu operator, Liouville-Caputo operator, fuzzy double parametric approach, fuzzy fractional differential equation, q-homotopy analysis Shehu transform method
How to Cite
Kumar, A., & Meher, R. (2024). Study on temporal-fuzzy fractional p-KdV equation with non-singular Mittag Leffler kernel. Mathematical Modelling and Analysis, 29(1), 57–76. https://doi.org/10.3846/mma.2024.17358
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
S. Ahmad, A. Ullah, A. Akgül and T. Abdeljawad. Semi-analytical solutions of the 3rd order fuzzy dispersive partial differential equations under fractional operators. Alexandria Engineering Journal, 60(6):5861–5878, 2021. https://doi.org/10.1016/j.aej.2021.04.065
I. Ahmed, P. Kumam, F. Jarad, P. Borisut, K. Sitthithakerngkiet and A. Ibrahim. Stability analysis for boundary value problems with generalized nonlocal condition via Hilfer–Katugampola fractional derivative. Advances in Difference Equations, 2020(225):1–18, 2020. https://doi.org/10.1186/s13662-020-02681-2
T. Ak and S. Dhawan. A practical and powerful approach to potential KdV and Benjamin equations. Beni-Suef University Journal of Basic and Applied Sciences, 6(4):383–390, 2017. https://doi.org/10.1016/j.bjbas.2017.07.008
T. Allahviranloo, Z. Gouyandeh, A. Armand and A. Hasanoglu. On fuzzy solutions for heat equation based on generalized Hukuhara differentiability. Fuzzy Sets and Systems, 265:1–23, 2015. https://doi.org/10.1016/j.fss.2014.11.009
T. Allahviranloo and N. Taheri. An analytic approximation to the solution of fuzzy heat equation by Adomian decomposition method. International Journal of Contemporary Mathematical Sciences, 4(3):105–114, 2009.
A. Atangana and D. Baleanu. New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model. Thermal Science, 20(2):763–769, 2016. https://doi.org/10.2298/TSCI160111018A
R. Belgacem, D. Baleanu and A. Bokhari. Shehu transform and applications to Caputo-fractional differential equations. International Journal of Analysis and Applications, 17(6):917–927, 2019.
A. Bokhari, D. Baleanu and R. Belgacem. Application of Shehu transform to Atangana-Baleanu derivatives. Journal of Mathematics and Computer Science, 20(2):101–107, 2019. https://doi.org/10.22436/jmcs.020.02.03
M. Caputo and M. Fabrizio. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2):73– 85, 2015.
S. Chakraverty and S. Perera. Recent Advances in Applications of Computational and Fuzzy Mathematics. Springer, Singapore, 2018. https://doi.org/10.1007/978-981-13-1153-6
M.W. Dingemans. Water wave propagation over uneven bottoms: Linear wave propagation, volume 13. World Scientific, 1997. https://doi.org/10.1142/1241-part2
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations, volume 204. Elsevier, 2006.
S. Kumar, A. Kumar and Z.M. Odibat. A nonlinear fractional model to describe the population dynamics of two interacting species. Mathematical Methods in the Applied Sciences, 40(11):4134–4148, 2017. https://doi.org/10.1002/mma.4293
S. Liao. Beyond perturbation: introduction to the homotopy analysis method. Chapman and Hall/CRC, New York, 2003. https://doi.org/10.1201/9780203491164
S. Liao. Notes on the homotopy analysis method: some definitions and theorems. Communications in Nonlinear Science and Numerical Simulation, 14(4):983– 997, 2009. https://doi.org/10.1016/j.cnsns.2008.04.013
S.-J. Liao. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University, 1992.
S. Maitama and W. Zhao. New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations. International Journal of Analysis and Applications, 17(2):167–190, 2019.
O. Martin. On the homotopy analysis method for solving a particle transport equation. Applied Mathematical Modelling, 37(6):3959–3967, 2013.
K.S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations. Wiley, 1993.
K. Oldham and J. Spanier. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.
I. Podlubny. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego, CA, 1998.
M.L. Puri and D.A. Ralescu. Differentials of fuzzy functions. Journal of Mathematical Analysis and Applications, 91(2):552–558, 1983. https://doi.org/10.1016/0022-247X(83)90169-5
S. Rashid, R. Ashraf, A.O. Akdemir, M.A. Alqudah, T. Abdeljawad and M.S. Mohamed. Analytic fuzzy formulation of a time-fractional Fornberg–Whitham model with power and Mittag–Leffler kernels. Fractal and Fractional, 5(3):113, 2021. https://doi.org/10.3390/fractalfract5030113
S. Rashid, R. Ashraf and Z. Hammouch. New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography. Journal of Ocean Engineering and Science, 8(1):55–78, 2023. https://doi.org/10.1016/j.joes.2021.11.004
P.P. Sartanpara and R. Meher. A robust computational approach for ZakharovKuznetsov equations of ion-acoustic waves in a magnetized plasma via the Shehu transform. Journal of Ocean Engineering and Science, 8(1):79–90, 2023. https://doi.org/10.1016/j.joes.2021.11.006
P.P. Sartanpara and R. Meher. A robust fuzzy-fractional approach for the atmospheric internal wave model. Journal of Ocean Engineering and Science, 8(3):308–322, 2023. https://doi.org/10.1016/j.joes.2022.02.001
P.P. Sartanpara, R. Meher and S.K. Meher. The generalized timefractional Fornberg–Whitham equation: An analytic approach. Partial Differential Equations in Applied Mathematics, 5:100350, 2022. https://doi.org/10.1016/j.padiff.2022.100350
L. Verma and R. Meher. Effect of heat transfer on Jeffery–Hamel Cu/Ag–water nanofluid flow with uncertain volume fraction using the double parametric fuzzy homotopy analysis method. The European Physical Journal Plus, 137(3):1–20, 2022. https://doi.org/10.1140/epjp/s13360-022-02586-x
L. Verma and R. Meher. Solution for generalized fuzzy time-fractional Fisher’s equation using a robust fuzzy analytical approach. Journal of Ocean Engineering and Science, 2022. https://doi.org/10.1016/j.joes.2022.03.019
L. Verma, R. Meher, Z. Avazzadeh and O. Nikan. Solution for generalized fuzzy fractional Kortewege-de Varies equation using a robust fuzzy double parametric approach. Journal of Ocean Engineering and Science, 2022. https://doi.org/10.1016/j.joes.2022.04.026
G.-W. Wang, T.-Z. Xu, G. Ebadi, S. Johnson, A. J. Strong and A. Biswas. Singular solitons, shock waves, and other solutions to potential Kdv equation. Nonlinear Dynamics, 76(2):1059–1068, 2014. https://doi.org/10.1007/s11071-013-1189-9
X.-J. Yang. General fractional derivatives: theory, methods and applications. Chapman and Hall/CRC, New York, 2019. https://doi.org/10.1201/9780429284083
Z. Yang and S. Liao. A HAM-based wavelet approach for nonlinear partial differential equations: Two dimensional Bratu problem as an application. Communications in Nonlinear Science and Numerical Simulation, 53:249–262, 2017. https://doi.org/10.1016/j.cnsns.2017.05.005
L.A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. ISSN 0019-9958. https://doi.org/10.1016/S0019-9958(65)90241-X
L.A. Zadeh. Linguistic variables, approximate reasoning and dispositions. Medical Informatics, 8(3):173–186, 1983. https://doi.org/10.3109/14639238309016081
H.-J. Zimmermann. Fuzzy set theory—and its applications. Springer Science & Business Media, New York, 2011. https://doi.org/10.1007/978-94-010-0646-0
I. Ahmed, P. Kumam, F. Jarad, P. Borisut, K. Sitthithakerngkiet and A. Ibrahim. Stability analysis for boundary value problems with generalized nonlocal condition via Hilfer–Katugampola fractional derivative. Advances in Difference Equations, 2020(225):1–18, 2020. https://doi.org/10.1186/s13662-020-02681-2
T. Ak and S. Dhawan. A practical and powerful approach to potential KdV and Benjamin equations. Beni-Suef University Journal of Basic and Applied Sciences, 6(4):383–390, 2017. https://doi.org/10.1016/j.bjbas.2017.07.008
T. Allahviranloo, Z. Gouyandeh, A. Armand and A. Hasanoglu. On fuzzy solutions for heat equation based on generalized Hukuhara differentiability. Fuzzy Sets and Systems, 265:1–23, 2015. https://doi.org/10.1016/j.fss.2014.11.009
T. Allahviranloo and N. Taheri. An analytic approximation to the solution of fuzzy heat equation by Adomian decomposition method. International Journal of Contemporary Mathematical Sciences, 4(3):105–114, 2009.
A. Atangana and D. Baleanu. New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model. Thermal Science, 20(2):763–769, 2016. https://doi.org/10.2298/TSCI160111018A
R. Belgacem, D. Baleanu and A. Bokhari. Shehu transform and applications to Caputo-fractional differential equations. International Journal of Analysis and Applications, 17(6):917–927, 2019.
A. Bokhari, D. Baleanu and R. Belgacem. Application of Shehu transform to Atangana-Baleanu derivatives. Journal of Mathematics and Computer Science, 20(2):101–107, 2019. https://doi.org/10.22436/jmcs.020.02.03
M. Caputo and M. Fabrizio. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2):73– 85, 2015.
S. Chakraverty and S. Perera. Recent Advances in Applications of Computational and Fuzzy Mathematics. Springer, Singapore, 2018. https://doi.org/10.1007/978-981-13-1153-6
M.W. Dingemans. Water wave propagation over uneven bottoms: Linear wave propagation, volume 13. World Scientific, 1997. https://doi.org/10.1142/1241-part2
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations, volume 204. Elsevier, 2006.
S. Kumar, A. Kumar and Z.M. Odibat. A nonlinear fractional model to describe the population dynamics of two interacting species. Mathematical Methods in the Applied Sciences, 40(11):4134–4148, 2017. https://doi.org/10.1002/mma.4293
S. Liao. Beyond perturbation: introduction to the homotopy analysis method. Chapman and Hall/CRC, New York, 2003. https://doi.org/10.1201/9780203491164
S. Liao. Notes on the homotopy analysis method: some definitions and theorems. Communications in Nonlinear Science and Numerical Simulation, 14(4):983– 997, 2009. https://doi.org/10.1016/j.cnsns.2008.04.013
S.-J. Liao. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University, 1992.
S. Maitama and W. Zhao. New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations. International Journal of Analysis and Applications, 17(2):167–190, 2019.
O. Martin. On the homotopy analysis method for solving a particle transport equation. Applied Mathematical Modelling, 37(6):3959–3967, 2013.
K.S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations. Wiley, 1993.
K. Oldham and J. Spanier. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.
I. Podlubny. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego, CA, 1998.
M.L. Puri and D.A. Ralescu. Differentials of fuzzy functions. Journal of Mathematical Analysis and Applications, 91(2):552–558, 1983. https://doi.org/10.1016/0022-247X(83)90169-5
S. Rashid, R. Ashraf, A.O. Akdemir, M.A. Alqudah, T. Abdeljawad and M.S. Mohamed. Analytic fuzzy formulation of a time-fractional Fornberg–Whitham model with power and Mittag–Leffler kernels. Fractal and Fractional, 5(3):113, 2021. https://doi.org/10.3390/fractalfract5030113
S. Rashid, R. Ashraf and Z. Hammouch. New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography. Journal of Ocean Engineering and Science, 8(1):55–78, 2023. https://doi.org/10.1016/j.joes.2021.11.004
P.P. Sartanpara and R. Meher. A robust computational approach for ZakharovKuznetsov equations of ion-acoustic waves in a magnetized plasma via the Shehu transform. Journal of Ocean Engineering and Science, 8(1):79–90, 2023. https://doi.org/10.1016/j.joes.2021.11.006
P.P. Sartanpara and R. Meher. A robust fuzzy-fractional approach for the atmospheric internal wave model. Journal of Ocean Engineering and Science, 8(3):308–322, 2023. https://doi.org/10.1016/j.joes.2022.02.001
P.P. Sartanpara, R. Meher and S.K. Meher. The generalized timefractional Fornberg–Whitham equation: An analytic approach. Partial Differential Equations in Applied Mathematics, 5:100350, 2022. https://doi.org/10.1016/j.padiff.2022.100350
L. Verma and R. Meher. Effect of heat transfer on Jeffery–Hamel Cu/Ag–water nanofluid flow with uncertain volume fraction using the double parametric fuzzy homotopy analysis method. The European Physical Journal Plus, 137(3):1–20, 2022. https://doi.org/10.1140/epjp/s13360-022-02586-x
L. Verma and R. Meher. Solution for generalized fuzzy time-fractional Fisher’s equation using a robust fuzzy analytical approach. Journal of Ocean Engineering and Science, 2022. https://doi.org/10.1016/j.joes.2022.03.019
L. Verma, R. Meher, Z. Avazzadeh and O. Nikan. Solution for generalized fuzzy fractional Kortewege-de Varies equation using a robust fuzzy double parametric approach. Journal of Ocean Engineering and Science, 2022. https://doi.org/10.1016/j.joes.2022.04.026
G.-W. Wang, T.-Z. Xu, G. Ebadi, S. Johnson, A. J. Strong and A. Biswas. Singular solitons, shock waves, and other solutions to potential Kdv equation. Nonlinear Dynamics, 76(2):1059–1068, 2014. https://doi.org/10.1007/s11071-013-1189-9
X.-J. Yang. General fractional derivatives: theory, methods and applications. Chapman and Hall/CRC, New York, 2019. https://doi.org/10.1201/9780429284083
Z. Yang and S. Liao. A HAM-based wavelet approach for nonlinear partial differential equations: Two dimensional Bratu problem as an application. Communications in Nonlinear Science and Numerical Simulation, 53:249–262, 2017. https://doi.org/10.1016/j.cnsns.2017.05.005
L.A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. ISSN 0019-9958. https://doi.org/10.1016/S0019-9958(65)90241-X
L.A. Zadeh. Linguistic variables, approximate reasoning and dispositions. Medical Informatics, 8(3):173–186, 1983. https://doi.org/10.3109/14639238309016081
H.-J. Zimmermann. Fuzzy set theory—and its applications. Springer Science & Business Media, New York, 2011. https://doi.org/10.1007/978-94-010-0646-0