A Comparative Study of the Fractional-Order System of Burgers Equations
Abstract
:1. Introduction
2. Preliminary Concepts
Basic Concept of Elzaki Transformation
3. The General Methodology of HPTM
4. The Methodology of EDM
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Sabatier, J.A.T.M.J.; Agrawal, O.P.; Machado, J.T. Advances in Fractional Calculus; Springer: Dordrecht, The Netherlands, 2007; Volume 4, No. 9. [Google Scholar]
- Atangana, A.; Baleanu, D. Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer. J. Eng. Mech. 2017, 143, D4016005. [Google Scholar] [CrossRef]
- Madani, M.; Fathizadeh, M.; Khan, Y.; Yildirim, A. On the coupling of the homotopy perturbation method and Laplace transformation. Math. Comput. Model. 2011, 53, 1937–1945. [Google Scholar] [CrossRef]
- Naeem, M.; Zidan, A.; Nonlaopon, K.; Syam, M.; Al-Zhour, Z.; Shah, R. A New Analysis of Fractional-Order Equal-Width Equations via Novel Techniques. Symmetry 2021, 13, 886. [Google Scholar] [CrossRef]
- Morales-Delgado, V.F.; Gómez-Aguilar, J.F.; Yépez-Martínez, H.; Baleanu, D.; Escobar-Jimenez, R.F.; Olivares-Peregrino, V.H. Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular. Adv. Differ. Equ. 2016, 2016, 164. [Google Scholar] [CrossRef] [Green Version]
- Li, Y.; Nohara, B.T.; Liao, S. Series solutions of coupled Van der Pol equation by means of homotopy analysis method. J. Math. Phys. 2010, 51, 063517. [Google Scholar] [CrossRef] [Green Version]
- Keskin, Y.; Oturanc, G. Reduced differential transform method for partial differential equations. Int. J. Nonlinear Sci. Numer. Simul. 2009, 10, 741–750. [Google Scholar] [CrossRef]
- Gupta, P.K. Approximate analytical solutions of fractional Benney—Lin equation by reduced differential transform method and the homotopy perturbation method. Comput. Math. Appl. 2011, 61, 2829–2842. [Google Scholar] [CrossRef] [Green Version]
- Huebner, K.H.; Dewhirst, D.L.; Smith, D.E.; Byrom, T.G. The Finite Element Method for Engineers; John Wiley & Sons: Hoboken, NJ, USA, 2001. [Google Scholar]
- Khan, H.; Shah, R.; Gomez-Aguilar, J.; Shoaib; Baleanu, D.; Kumam, P. Travelling waves solution for fractional-order biological population model. Math. Model. Nat. Phenom. 2021, 16, 32. [Google Scholar] [CrossRef]
- Bateman, H. Some recent researches on the motion of fluids. Mon. Weather Rev. 1915, 43, 163–170. [Google Scholar] [CrossRef]
- Burgers, J.M. A mathematical model illustrating the theory of turbulence. In Advances in Applied Mechanics; Elsevier: Amsterdam, The Netherlands, 1948; Volume 1, pp. 171–199. [Google Scholar]
- Al-Jawary, M.A.; Azeez, M.M.; Radhi, G.H. Analytical and numerical solutions for the nonlinear Burgers and advection—Diffusion equations by using a semi-analytical iterative method. Comput. Math. Appl. 2018, 76, 155–171. [Google Scholar] [CrossRef]
- Dehghan, M.; Hamidi, A.; Shakourifar, M. The solution of coupled Burgers, equations using Adomian—Pade technique. Appl. Math. Comput. 2007, 189, 1034–1047. [Google Scholar] [CrossRef]
- Abazari, R.; Borhanifar, A. Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method. Comput. Math. Appl. 2010, 59, 2711–2722. [Google Scholar] [CrossRef] [Green Version]
- Soliman, A.A. The modified extended tanh-function method for solving Burgers-type equations. Phys. A Stat. Mech. Its Appl. 2006, 361, 394–404. [Google Scholar] [CrossRef]
- Alomari, A.K.; Noorani, M.S.M.; Nazar, R. The homotopy analysis method for the exact solutions of the K (2, 2), Burgers and coupled Burgers equations. Appl. Math. Sci. 2008, 2, 1963–1977. [Google Scholar]
- Veeresha, P.; Prakasha, D.G. A novel technique for (2 + 1)-dimensional time-fractional coupled Burgers equations. Math. Comput. Simul. 2019, 166, 324–345. [Google Scholar] [CrossRef]
- Oruç, O.; Bulut, F.; Esen, A. Chebyshev Wavelet Method for Numerical Solutions of Coupled Burgers’ Equation. Hacet. J. Math. Stat. 2019, 48, 1–16. [Google Scholar] [CrossRef] [Green Version]
- He, J.H. Homotopy perturbation method: A new nonlinear analytical technique. Appl. Math. Comput. 2003, 135, 73–79. [Google Scholar] [CrossRef]
- Elzaki, T.M. The new integral transform ‘Elzaki transform’. Glob. J. Pure Appl. Math. 2011, 7, 57–64. [Google Scholar]
- Alshikh, A.A. A Comparative Study between Laplace Transform and Two New Integrals “ELzaki” Transform and “Aboodh” Transform. Pure Appl. Math. J. 2016, 5, 145. [Google Scholar] [CrossRef] [Green Version]
- Elzaki, T.; Alkhateeb, S. Modification of Sumudu transform “Elzaki transform” and adomian decomposition method. Appl. Math. Sci. 2015, 9, 603–611. [Google Scholar] [CrossRef]
- Jena, R.; Chakraverty, S. Solving time-fractional Navier-Stokes equations using homotopy perturbation Elzaki transform. SN Appl. Sci. 2018, 1, 16. [Google Scholar] [CrossRef] [Green Version]
- Mahgoub, M.; Sedeeg, A. A Comparative Study for Solving Nonlinear Fractional Heat -Like Equations via Elzaki Transform. Br. J. Math. Comput. Sci. 2016, 19, 1–12. [Google Scholar] [CrossRef]
- Das, S.; Gupta, P. An Approximate Analytical Solution of the Fractional Diffusion Equation with Absorbent Term and External Force by Homotopy Perturbation Method. Zeitschrift Fur Naturforschung A 2010, 65, 182–190. [Google Scholar] [CrossRef]
- Singh, P.; Sharma, D. Comparative study of homotopy perturbation transformation with homotopy perturbation Elzaki transform method for solving nonlinear fractional PDE. Nonlinear Eng. 2019, 9, 60–71. [Google Scholar] [CrossRef]
- Nonlaopon, K.; Alsharif, A.; Zidan, A.; Khan, A.; Hamed, Y.; Shah, R. Numerical Investigation of Fractional-Order Swift–Hohenberg Equations via a Novel Transform. Symmetry 2021, 13, 1263. [Google Scholar] [CrossRef]
- Adomian, G. Solution of physical problems by decomposition. Comput. Math. Appl. 1994, 27, 145–154. [Google Scholar] [CrossRef] [Green Version]
- Adomian, G. A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 1988, 135, 501544. [Google Scholar] [CrossRef] [Green Version]
- Sunthrayuth, P.; Zidan, A.; Yao, S.; Shah, R.; Inc, M. The Comparative Study for Solving Fractional-Order Fornberg–Whitham Equation via ρ-Laplace Transform. Symmetry 2021, 13, 784. [Google Scholar] [CrossRef]
- Elzaki, T.M.; Ezaki, S.M. Applications of new transform ”Elzaki Transform” to partial differential equations. Glob. J. Pure Appl. Math. 2011, 7, 65–70. [Google Scholar]
- Elzaki, T.M.; Ezaki, S.M. On the connections between Laplace and ELzaki transforms. Adv. Theo. Appl. Math. 2011, 6, 1–10. [Google Scholar]
- Elzaki, T.M.; Ezaki, S.M. On the ELzaki transform and ordinary differential equation with variable coefficients. Adv. Theor. Appl. Math. 2011, 6, 41–46. [Google Scholar]
- He, J.H. Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul. 2005, 6, 207–208. [Google Scholar] [CrossRef]
- He, J.H. Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals 2005, 26, 695–700. [Google Scholar] [CrossRef]
ℑ | AE () | AE () | AE ( | AE ( | |
---|---|---|---|---|---|
1 | 5.1298036 | 8.5968820 | 1.1526960 | 1.4693 | |
2 | 1.0533254900 | 1.7251291 | 2.3064850 | 2.1224 | |
0.1 | 3 | 1.5936706200 | 2.5905700 | 3.4602750 | 2.7754 |
4 | 2.1340157400 | 3.4560108 | 4.6140640 | 3.4285 | |
5 | 2.6743608600 | 4.3214516 | 5.7678540 | 4.0815 | |
1 | 6.5526269 | 1.2845443 | 1.9524030 | 8.9043500 | |
2 | 1.3585147 | 2.5823037 | 3.9081030 | 1.2243480 | |
0.2 | 3 | 2.0617667100 | 3.8800631 | 5.8638020 | 1.5582620 |
4 | 2.7650187200 | 5.1778225 | 7.8195020 | 1.8921750 | |
5 | 3.4682707200 | 6.4755819 | 9.7752020 | 2.2260880 | |
1 | 7.5239217 | 1.6203247 | 2.6503270 | 9.9570730 | |
2 | 1.5715901300 | 3.2621458 | 5.3069340 | 1.2801951 | |
0.3 | 3 | 2.3907880800 | 4.9039669 | 7.9635420 | 1.5646829 |
4 | 3.2099860300 | 6.5457880 | 1.0620149 | 1.8491707 | |
5 | 4.0291839800 | 8.1876092 | 1.3276756 | 2.1336585 | |
1 | 8.2762123 | 1.9075950 | 3.2874570 | 5.7825882 | |
2 | 1.7398405800 | 3.8455493 | 6.5848290 | 6.7463529 | |
0.4 | 3 | 2.6520599300 | 5.7835036 | 9.8822010 | 7.7101176 |
4 | 3.5642792800 | 7.7214580 | 1.3179574 | 8.6738823 | |
5 | 4.4764986200 | 9.6594122 | 1.6476946 | 9.6376470 | |
1 | 8.8947364 | 2.1627817 | 3.8817930 | 2.3900 | |
2 | 1.8806520800 | 4.3652454 | 7.7777130 | 3.5300 | |
0.5 | 3 | 2.8718305200 | 6.5677091 | 1.1673633 | 4.6600 |
4 | 3.8630089800 | 8.7701729 | 1.5569554 | 5.8 | |
5 | 4.8541874400 | 1.0972636700 | 1.9465475 | 6.9300000 |
ℑ | AE () | AE () | AE ( | AE ( | |
---|---|---|---|---|---|
1 | 5.6770988 | 8.7119340 | 1.1548840 | 1.6326 | |
2 | 5.3834512 | 8.4544080 | 9.5379000 | 2.0407510200 | |
0.1 | 3 | 5.0898036 | 8.1968820 | 7.5269600 | 4.0814857200 |
4 | 4.7961560 | 7.9393560 | 5.5160200 | 6.1222204100 | |
5 | 4.5025084 | 7.6818300 | 3.5050800 | 8.1629551100 | |
1 | 7.5124132 | 1.3109744 | 1.9589970 | 2.2260880 | |
2 | 6.9925200 | 1.2577593 | 1.5557000 | 4.3444869600 | |
0.2 | 3 | 6.4726268 | 1.2045442 | 1.1524030 | 8.6867478200 |
4 | 5.9527336 | 1.1513291 | 7.4910600 | 1.3029008690 | |
5 | 5.4328404 | 1.0981140 | 3.4580900 | 1.7371269560 | |
1 | 8.8600372900 | 1.6633175900 | 2.6628869 | 4.2673170 | |
2 | 8.1319795 | 1.5818212 | 2.0566070 | 7.2886243900 | |
0.3 | 3 | 7.4039217 | 1.5003248 | 1.4503270 | 1.4534575610 |
4 | 6.6758639 | 1.4188284 | 8.4404700 | 2.1780526830 | |
5 | 5.9478061 | 1.3373320 | 2.3776700 | 2.9026478050 | |
1 | 9.9681746700 | 1.9683136700 | 3.3072887 | 3.8550588 | |
2 | 9.0421934 | 1.8579543 | 2.4973720 | 1.1668329410 | |
0.4 | 3 | 8.1162122 | 1.7475950 | 1.6874560 | 2.2951152940 |
4 | 7.1902310 | 1.6372357 | 8.7754000 | 3.4233976470 | |
5 | 6.2642497 | 1.5268763 | 6.7623000 | 4.5516800 | |
1 | 1.0928832650 | 2.2421458500 | 3.9100485 | 1.2600000 | |
2 | 9.8117844 | 2.1024637 | 2.8959200 | 1.0050239900 | |
0.5 | 3 | 8.6947361 | 1.9627815 | 1.8817910 | 2.0100478600 |
4 | 7.5776879 | 1.8230994 | 8.6766300 | 3.0150717200 | |
5 | 6.4606397 | 1.6834173 | 1.4646500 | 4.0200955900 |
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Cui, Y.; Shah, N.A.; Shi, K.; Saleem, S.; Chung, J.D. A Comparative Study of the Fractional-Order System of Burgers Equations. Symmetry 2021, 13, 1786. https://doi.org/10.3390/sym13101786
Cui Y, Shah NA, Shi K, Saleem S, Chung JD. A Comparative Study of the Fractional-Order System of Burgers Equations. Symmetry. 2021; 13(10):1786. https://doi.org/10.3390/sym13101786
Chicago/Turabian StyleCui, Yanmei, Nehad Ali Shah, Kunju Shi, Salman Saleem, and Jae Dong Chung. 2021. "A Comparative Study of the Fractional-Order System of Burgers Equations" Symmetry 13, no. 10: 1786. https://doi.org/10.3390/sym13101786
APA StyleCui, Y., Shah, N. A., Shi, K., Saleem, S., & Chung, J. D. (2021). A Comparative Study of the Fractional-Order System of Burgers Equations. Symmetry, 13(10), 1786. https://doi.org/10.3390/sym13101786