Implementation of Yang residual power series method to solve fractional non-linear systems

Azzh Saad Alshehry; Roman Ullah; Nehad Ali Shah; Rasool Shah; Kamsing Nonlaopon; Azzh Saad Alshehry; Roman Ullah; Nehad Ali Shah; Rasool Shah; Kamsing Nonlaopon

doi:10.3934/math.2023418

AIMS Mathematics

2023, Volume 8, Issue 4: 8294-8309. doi: 10.3934/math.2023418

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Research article

Implementation of Yang residual power series method to solve fractional non-linear systems

1.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of General Studies, Higher Colleges of Technology, Dubai Women Campus, UAE
3.
Department of Mechanical Engineering, Sejong University, Seoul 05006, Korea
4.
Department of Mathematics, Abdul Wali khan University Mardan 23200, Pakistan
5.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
^† These authors contributed equally to this work and are co-first authors.

Received: 07 September 2022 Revised: 30 November 2022 Accepted: 12 December 2022 Published: 01 February 2023
MSC : 32B15, 34A34, 35A22, 35A24, 45A10

In this study, we implemented the Yang residual power series (YRPS) methodology, a unique analytical treatment method, to estimate the solutions of a non-linear system of fractional partial differential equations. The RPS approach and the Yang transform are togethered in the YRPS method. The suggested approach to handle fractional systems is explained along with its application. With fewer calculations and greater accuracy, the limit idea is used to solve it in Yang space to produce the YRPS solution for the proposed systems. The benefit of the new method is that it requires less computation to get a power series form solution, whose coefficients should be established in a series of algebraic steps. Two attractive initial value problems were used to test the technique's applicability and performance. The behaviour of the approximative solutions is numerically and visually discussed, along with the effect of fraction order $ \varsigma $. It was observed that the proposed method's approximations and exact solutions were completely in good agreement. The YRPS approach results highlight and show that the approach may be utilized to a variety of fractional models of physical processes easily and with analytical efficiency.
- Yang transform,
- Caputo operator,
- residual power series,
- systems of fractional differential equations
Citation: Azzh Saad Alshehry, Roman Ullah, Nehad Ali Shah, Rasool Shah, Kamsing Nonlaopon. Implementation of Yang residual power series method to solve fractional non-linear systems[J]. AIMS Mathematics, 2023, 8(4): 8294-8309. doi: 10.3934/math.2023418

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Abstract

In this study, we implemented the Yang residual power series (YRPS) methodology, a unique analytical treatment method, to estimate the solutions of a non-linear system of fractional partial differential equations. The RPS approach and the Yang transform are togethered in the YRPS method. The suggested approach to handle fractional systems is explained along with its application. With fewer calculations and greater accuracy, the limit idea is used to solve it in Yang space to produce the YRPS solution for the proposed systems. The benefit of the new method is that it requires less computation to get a power series form solution, whose coefficients should be established in a series of algebraic steps. Two attractive initial value problems were used to test the technique's applicability and performance. The behaviour of the approximative solutions is numerically and visually discussed, along with the effect of fraction order $ \varsigma $. It was observed that the proposed method's approximations and exact solutions were completely in good agreement. The YRPS approach results highlight and show that the approach may be utilized to a variety of fractional models of physical processes easily and with analytical efficiency.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Preface, North-Holl. Math. Stud., 204 (2006). https://doi.org/10.1016/s0304-0208(06)80001-0 doi: 10.1016/s0304-0208(06)80001-0
[2]	H. Dutta, A. O. Akdemir, A. Atangana, Fractional Order Analysis: Theory, Methods and Applications, Hoboken: Wiley, 2020.
[3]	B. Bira, T. Raja Sekhar, D. Zeidan, Exact solutions for some time-fractional evolution equations using lie group theory, Math. Meth. Appl. Sci., 41 (2018), 6717–6725. https://doi.org/10.1002/mma.5186 doi: 10.1002/mma.5186
[4]	M. Alabedalhadi, M. Al-Smadi, S. Al-Omari, D. Baleanu, S. Momani, Structure of optical soliton solution for nonlinear resonant space-time Schrödinger equation in conformable sense with full nonlinearity term, Phys. Scr., 95 (2020), 105215. https://doi.org/10.1088/1402-4896/abb739 doi: 10.1088/1402-4896/abb739
[5]	M. Al-Smadi, O. Abu Arqub, Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates, Appl. Math. Comput., 342 (2019), 280–294. https://doi.org/10.1016/j.amc.2018.09.020 doi: 10.1016/j.amc.2018.09.020
[6]	D. Baleanu, A. K. Golmankhaneh, A. K. Golmankhaneh, M. C. Baleanu, Fractional electromagnetic equations using fractional forms, Int. J. Theor. Phys., 48 (2009), 3114–3123. https://doi.org/10.1007/s10773-009-0109-8 doi: 10.1007/s10773-009-0109-8
[7]	M. H. Al-Smadi, G. N. Gumah, On the homotopy analysis method for fractional SEIR epidemic model, Res. J. Appl. Sci. Eng. Technol., 7 (2014), 3809–3820. https://doi.org/10.19026/rjaset.7.738 doi: 10.19026/rjaset.7.738
[8]	S. Kumar, A new fractional modeling arising in engineering sciences and its analytical approximate solution, Alex. Eng. J., 52 (2013), 813–819. https://doi.org/10.1016/j.aej.2013.09.005 doi: 10.1016/j.aej.2013.09.005
[9]	M. H. Al-Smadi, A. Freihat, O. Abu Arqub, N. Shawagfeh, A novel multistep generalized differential transform method for solving fractional-order Lü chaotic and hyperchaotic systems, J. Comput. Anal. Appl., 19 (2015), 713–724.
[10]	K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, New York: Academic Press, 1974.
[11]	V. N. Kovalnogov, R. V. Fedorov, A. V. Chukalin, T. E. Simos, C. Tsitouras, Eighth order two-step methods trained to perform better on Keplerian-type orbits, Mathematics, 9 (2021), 3071. https://doi.org/10.3390/math9233071 doi: 10.3390/math9233071
[12]	L. J. Sun, J. Hou, C. J. Xing, Z. W. Fang, A robust Hammerstein-Wiener model identification method for highly nonlinear systems, Processes, 10 (2022), 2664. https://doi.org/10.3390/pr10122664 doi: 10.3390/pr10122664
[13]	F. W. Meng, A. P. Pang, X. F. Dong, C. Han, X. P. Sha, H-infinity optimal performance design of an unstable plant under bode integral constraint, Complexity, 2018 (2018), 4942906. https://doi.org/10.1155/2018/4942906 doi: 10.1155/2018/4942906
[14]	S. M. Ali, W. Shatanawi, M. D. Kassim, M. S. Abdo, S. Saleh, Investigating a class of generalized Caputo-type fractional integro-differential equations, J. Funct. Spaces, 2022 (2022), 8103046. https://doi.org/10.1155/2022/8103046 doi: 10.1155/2022/8103046
[15]	S. Etemad, M. M. Matar, M. A. Ragusa, S. Rezapour, Tripled fixed points and existence study to a tripled impulsive fractional differential system via measures of noncompactness, Mathematics, 10 (2022), 25. https://doi.org/10.3390/math10010025 doi: 10.3390/math10010025
[16]	P. Kumar, V. S. Erturk, M. Vellappandi, H. Trinh, V. Govindaraja, A study on the maize streak virus epidemic model by using optimized linearization-based predictor-corrector method in caputo sense, Chaos Solitons Fractals, 158 (2022), 112067. https://doi.org/10.1016/j.chaos.2022.112067 doi: 10.1016/j.chaos.2022.112067
[17]	Z. Odibat, V. S. Erturk, P. Kumar, A. B. Makhlouf, V. Govindaraj, An implementation of the generalized differential transform scheme for simulating impulsive fractional differential equations, Math. Probl. Eng., 2022 (2022), 8280203. https://doi.org/10.1155/2022/8280203 doi: 10.1155/2022/8280203
[18]	L. Akinyemi, H. Rezazadeh, Q. H. Shi, M. Inc, M. M. A. Khater, H. Ahmad, et al., New optical solitons of perturbed nonlinear Schrodinger-Hirota equation with spatio-temporal dispersion, Results Phys., 29 (2021), 104656. https://doi.org/10.1016/j.rinp.2021.104656 doi: 10.1016/j.rinp.2021.104656
[19]	D. Ntiamoah, W. Ofori-Atta, L. Akinyemi, The higher-order modified Korteweg-de Vries equation: Its soliton, breather and approximate solutions, J. Ocean Eng. Sci., 2022. https://doi.org/10.1016/j.joes.2022.06.042 doi: 10.1016/j.joes.2022.06.042
[20]	P. Liu, J. P. Shi, Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Continuous Dyn. Syst. B, 18 (2013), 2597–2625. https://doi.org/10.3934/dcdsb.2013.18.2597 doi: 10.3934/dcdsb.2013.18.2597
[21]	S. G. Li, Z. M. Liu, Scheduling uniform machines with restricted assignment, Math. Biosci. Eng., 19 (2022), 9697–9708. https://doi.org/10.3934/mbe.2022450 doi: 10.3934/mbe.2022450
[22]	J. S. Duan, R. Rach, D. Baleanu, A. M. Wazwaz, A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Fract. Calc., 3 (2012), 73–99.
[23]	A. Kumar, S. Kumar, S. P. Yan, Residual power series method for fractional diffusion equations, Fundam. Inform., 151 (2017), 213–230. https://doi.org/10.3233/fi-2017-1488 doi: 10.3233/fi-2017-1488
[24]	Y. Qin, A. Khan, I. Ali, M. Al Qurashi, H. Khan, R. Shah, et al., An efficient analytical approach for the solution of certain fractional-order dynamical systems, Energies, 13 (2020), 2725. https://doi.org/10.3390/en13112725 doi: 10.3390/en13112725
[25]	A. A. Alderremy, S. Aly, R. Fayyaz, A. Khan, R. Shah, N. Wyal, The analysis of fractional-order nonlinear systems of third order kdv and burgers equations via a novel transform, Complexity, 2022 (2022), 4935809. https://doi.org/10.1155/2022/4935809 doi: 10.1155/2022/4935809
[26]	P. Sunthrayuth, H. A. Alyousef, S. A. El-Tantawy, A. Khan, N. Wyal, Solving fractional-order diffusion equations in a plasma and fluids via a novel transform, J. Funct. Spaces, 2022 (2022), 1899130. https://doi.org/10.1155/2022/1899130 doi: 10.1155/2022/1899130
[27]	H. Khan, A. Khan, M. Al-Qurashi, R. Shah, D. Baleanu, Modified modelling for heat like equations within Caputo operator, Energies, 13 (2020), 2002. https://doi.org/10.3390/en13082002 doi: 10.3390/en13082002
[28]	K. Nonlaopon, A. M. Alsharif, A. M. Zidan, A. Khan, Y. S. Hamed, R. Shah, Numerical investigation of fractional-order Swift-Hohenberg equations via a novel transform, Symmetry, 13 (2021), 1263. https://doi.org/10.3390/sym13071263 doi: 10.3390/sym13071263
[29]	Hajira, H. Khan, A. Khan, P. Kumam, D. Baleanu, M. Arif, An approximate analytical solution of the Navier-Stokes equations within Caputo operator and Elzaki transform decomposition method, Adv. Differ. Equ., 2020 (2020), 622. https://doi.org/10.1186/s13662-020-03058-1 doi: 10.1186/s13662-020-03058-1
[30]	M. K. Alaoui, K. Nonlaopon, A. M. Zidan, A. Khan, R. Shah, Analytical investigation of fractional-order Cahn-Hilliard and Gardner equations using two novel techniques, Mathematics, 10 (2022), 1643. https://doi.org/10.3390/math10101643 doi: 10.3390/math10101643
[31]	T. Botmart, R. P. Agarwal, M. Naeem, A. Khan, R. Shah, On the solution of fractional modified Boussinesq and approximate long wave equations with non-singular kernel operators, AIMS Mathematics, 7 (2022), 12483–12513. https://doi.org/10.3934/math.2022693 doi: 10.3934/math.2022693
[32]	A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 31–52. https://doi.org/10.5373/jaram.1447.051912 doi: 10.5373/jaram.1447.051912
[33]	E. A. Az-Zobi, A. Yildirim, W. A. AlZoubi, The residual power series method for the one-dimensional unsteady flow of a van der waals gas, Physica A, 517 (2019), 188–196. https://doi.org/10.1016/j.physa.2018.11.030 doi: 10.1016/j.physa.2018.11.030
[34]	M. Khader, M. H. DarAssi, Residual power series method for solving nonlinear reaction-diffusion-convection problems, Bol. da Soc. Parana. de Mat., 39 (2021), 177–188. https://doi.org/10.5269/bspm.41741 doi: 10.5269/bspm.41741
[35]	R. Saadeh, M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. Salma Din, Application of fractional residual power series algorithm to solve Newell-Whitehead-Segel equation of fractional order, Symmetry, 11 (2019), 1431. https://doi.org/10.3390/sym11121431 doi: 10.3390/sym11121431
[36]	M. Al-Smadi, O. Abu Arqub, S. Momani, Numerical computations of coupled fractional resonant schrodinger equations arising in quantum mechanics under conformable fractional derivative sense, Phys. Scr., 95 (2020), 075218. https://doi.org/10.1088/1402-4896/ab96e0 doi: 10.1088/1402-4896/ab96e0
[37]	M. Al-Smadi, O. Abu Arqub, S. Hadid, Approximate solutions of nonlinear fractional Kundu-Eckhaus and coupled fractional massive Thirring equations emerging in quantum field theory using conformable residual power series method, Phys. Scr., 95 (2020), 105205. https://doi.org/10.1088/1402-4896/abb420 doi: 10.1088/1402-4896/abb420
[38]	R. Saadeh, M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. S. Din, Application of fractional residual power series algorithm to solve Newell-Whitehead-segel equation of fractional order, Symmetry, 11 (2019), 1431. https://doi.org/10.3390/sym11121431 doi: 10.3390/sym11121431
[39]	M. Al-Smadi, O. Abu Arqub, S. Hadid, An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative, Commun. Theor. Phys., 72 (2020), 085001. https://doi.org/10.1088/1572-9494/ab8a29 doi: 10.1088/1572-9494/ab8a29
[40]	I. Komashynska, M. Al-Smadi, O. A. Arqub, S. Momani, An efficient analytical method for solving singular initial value problems of nonlinear systems, Appl. Math. Inform. Sci., 10 (2016), 647–656. https://doi.org/10.18576/amis/100224 doi: 10.18576/amis/100224
[41]	M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. S. Din, Numerical computation of fractional Fredholm integrodifferential equation of order 2b arising in natural sciences, J. Phys. Conf. Ser., 1212 (2019), 012022. https://doi.org/10.1088/1742-6596/1212/1/012022 doi: 10.1088/1742-6596/1212/1/012022
[42]	M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. S. Din, An analytical numerical method for solving fuzzy fractional Volterra Integro-differential equations, Symmetry, 11 (2019), 205. https://doi.org/10.3390/sym11020205 doi: 10.3390/sym11020205
[43]	M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. S. Din, Computational optimization of residual power series algorithm for certain classes of fuzzy fractional differential equations, Int. J. Differ. Equ., 2018 (2018), 8686502. https://doi.org/10.1155/2018/8686502 doi: 10.1155/2018/8686502
[44]	A. Khan, M. Junaid, I. Khan, F. Ali, K. Shah, D. Khan, Application of Homotopy analysis natural transform method to the solution of nonlinear partial differential equations, Sci. Int., 29 (2017), 297–303.
[45]	M. F. Zhang, Y. Q. Liu, X. S. Zhou, Efficient homotopy perturbation method for fractional non-linear equations using Sumudu transform, Therm. Sci., 19 (2015), 1167–1171. https://doi.org/10.2298/tsci1504167z doi: 10.2298/tsci1504167z
[46]	A. M. Zidan, A. Khan, R. Shah, M. K. Alaoui, W, Weera, Evaluation of time-fractional Fisher's equations with the help of analytical methods, AIMS Mathematics, 7 (2022), 18746–18766. https://doi.org/10.3934/math.20221031 doi: 10.3934/math.20221031
[47]	M. Alquran, M. Ali, M. Alsukhour, I. Jaradat, Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics, Results Phys., 19 (2020), 103667. https://doi.org/10.1016/j.rinp.2020.103667 doi: 10.1016/j.rinp.2020.103667
[48]	A. El-Ajou, Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach, Eur. Phys. J. Plus, 136 (2021), 229. https://doi.org/10.1140/epjp/s13360-020-01061-9 doi: 10.1140/epjp/s13360-020-01061-9
[49]	M. Areshi, A. Khan, R. Shah, K. Nonlaopon, Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform, AIMS Mathematics, 7 (2022), 6936–6958. https://doi.org/10.3934/math.2022385 doi: 10.3934/math.2022385
[50]	M. K. Alaoui, R. Fayyaz, A. Khan, R. Shah, M. S. Abdo, Analytical investigation of Noyes-Field model for time-fractional Belousov-Zhabotinsky reaction, Complexity, 2021 (2021), 3248376. https://doi.org/10.1155/2021/3248376 doi: 10.1155/2021/3248376

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