Current issue
Archive
About the Journal
About
Editorial Board
Abstract & Indexing
Contact
Editorial Policies
Authorship & COI
Ethical principles
Principles of Transparent Checklist
Open access
For Authors
Instructions to Authors
Detailing Guidelines for Authors
Review process
Review sheet
How to submit
Open Access license
ORIGINAL PAPER
New Iterative Method of Solving Nonlinear Equations in Fluid Mechanics
1
Department of Information Technology in the Agro-Industrial Complex, Russian State Agrarian University - Moscow Timiryazev Agricultural Academy, Moscow, Russia, E-mail: paliivetsmax@rambler.ru
2
Department of Agricultural Construction and Real Estate Expertise, Russian State Agrarian University - Moscow Timiryazev Agricultural Academy, Moscow, Russia
3
Department of Integrated Water Management and Hydraulics, Russian State Agrarian University - Moscow, Timiryazev Agricultural Academy, Moscow, Russia
4
Department of Information Technology in the Agro-Industrial Complex, Russian State Agrarian University - Moscow Timiryazev Agricultural Academy, Moscow, Russia
Online publication date: 2021-08-26
Publication date: 2021-09-01
International Journal of Applied Mechanics and Engineering 2021;26(3):163-176
KEYWORDS
Adomian decomposition method (ADM)earthquakefiltrationfluid mechanicsiterative methodnonlinear fractional partial differential equationsporous medium
ABSTRACT
This paper presents the results of applying a new iterative method to linear and nonlinear fractional partial differential equations in fluid mechanics. A numerical analysis was performed to find an exact solution of the fractional wave equation and fractional Burgers’ equation, as well as an approximate solution of fractional KdV equation and fractional Boussinesq equation. Fractional derivatives of the order α are described using Caputo's definition with 0 < α ≤ 1 or 1 < α ≤ 2. A comparative analysis of the results obtained using a new iterative method with those obtained by the Adomian decomposition method showed the first method to be more efficient and simple, providing accurate results in fewer computational operations. Given its flexibility and ability to solve nonlinear equations, the iterative method can be used to solve more complex linear and nonlinear fractional partial differential equations.
REFERENCES (33)
1.
Singh H., Kumar D. and Baleanu D. (2019): Methods of Mathematical Modelling: Fractional Differential Equations.– Boca Raton: CRC Press.
2.
Shishkina E. and Sitnik S. (2020): Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics.– Cambridge: Academic Press.
3.
Milici C., Drăgănescu G. and Machado J.T. (2018): Introduction to Fractional Differential Equations.– Cham: Springer, vol.25.
4.
Kaplan M., Bekir A., Akbulut A. and Aksoy E. (2016): The modified simple equation method for solving some fractional-order nonlinear equations.– Pramana, vol.87, No.1, pp.1-5. https://doi.org/10.1007/s12043....
5.
Brociek R., Słota D., Król M., Matula G. and Kwaśny W. (2017): Modeling of heat distribution in porous aluminum using fractional differential equation.– Fractal Fract., vol.1, No.1, pp.17. doi.org/10.3390/fractalfract1010017.
6.
Bekir A., Aksoy E. and Cevikel A.C. (2015): Exact solutions of nonlinear time fractional partial differential equations by sub-equation method.– Math. Methods Appl. Sci., vol.38, No.13, pp.2779-2784. https://doi.org/10.1002/mma.32....
7.
Sonmezoglu A. (2015): Exact solutions for some fractional differential equations.– Adv. Math. Phys., vol.2015, pp.567842. https://doi.org/10.1155/2015/5....
8.
Gulian M., Raissi M., Perdikaris P., Karniadakis G. and Karniadakis G. (2019): Machine learning of space-fractional differential equations.– SIAM J. Sci. Comp., vol.41, No.4, pp.2485-2509. https://doi.org/10.1137/18M120....
9.
Esmailzadeh E., Younesian D. and Askari H. (2018): Analytical Methods in Nonlinear Oscillations.– Amsterdam: Springer.
10.
Arqub O.A., El-Ajou A. and Momani S. (2015): Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations.– J. Comput. Phys., vol.293, pp.385-399. https://doi.org/10.1016/j.jcp.....
11.
Eltayeb H., Abdalla Y.T., Bachar I. and Khabir M.H. (2019): Fractional telegraph equation and its solution by natural transform decomposition method.– Symmetry, vol.11, No.3, pp.334. https://doi.org/10.3390/sym110....
12.
El-Ajou A., Arqub O.A. and Momani S. (2015): Approximate analytical solution of the nonlinear fractional KdVBurgers equation: a new iterative algorithm.– J. Comput. Phys., vol.293, pp.81-95. https://doi.org/10.1016/j.jcp.....
13.
Wang K.L., Wang K.J. and He C.H. (2019): Physical insight of local fractional calculus and its application to fractional Kdv-Burgers-Kuramoto equation.– Fractals, vol.27, No.7, pp.1950122. https://doi.org/10.1142/S02183....
14.
Goodrich C. and Peterson A.C. (2015): Discrete Fractional Calculus.– Berlin: Springer.
15.
Li C. and Zeng F. (2015): Numerical Methods for Fractional Calculus.– Boca Raton: CRC Press, Vol. 24.
16.
Sun H., Zhang Y., Baleanu D., Chen W. and Chen Y. (2018): A new collection of real world applications of fractional calculus in science and engineering.– Comm. Nonlinear Sci. Numer. Simulat., vol.64, pp.213-231. https://doi.org/10.1016/j.cnsn....
17.
Hu Z. and Du X. (2015): First order reliability method for time-variant problems using series expansions.– Struct. Multidiscip. Optim., vol.51, No.1, pp.1-21. https://doi.org/10.1007/s00158....
18.
Turkyilmazoglu M. (2019): Accelerating the convergence of Adomian decomposition method (ADM).– J. Comput. Sci., vol.31, pp.54-59. https://doi.org/10.1016/j.jocs....
19.
Jajarmi A. and Baleanu D. (2020): A new iterative method for the numerical solution of high- order non-linear fractional boundary value problems.– Front. Phys., vol.8, pp.220. https://doi.org/10.3389/fphy.2....
20.
Bhalekar S. and Daftardar-Gejji V. (2010): Solving evolution equations using a new iterative method.– Numer. Methods Partial Differ. Equ., vol.26, No.4, pp.906-916. https://doi.org/10.1002/num.20....
21.
Bhalekar S. and Daftardar-Gejji V. (2008): New iterative method: application to partial differential equations.– Appl. Math. Comput., vol.203, No.2, pp.778-783. https://doi.org/10.1016/j.amc.....
22.
Awawdeh F. (2010): On new iterative method for solving systems of nonlinear equations.– Numer. Algorithms, vol.54, No.3, pp.395-409. https://doi.org/10.1007/s11075....
23.
Bocanegra S.Y., Gil-González W. and Montoya O.D. (2020): A new iterative power flow method for ac distribution grids with radial and mesh topologies.– In: 2020 IEEE International Autumn Meeting on Power, Electronics and Computing (ROPEC) (Vol. 4). New York: IEEE. https://doi.org/10.1109/ROPEC5....
24.
Qureshi S. and Yusuf A. (2019): Modeling chickenpox disease with fractional derivatives: From Caputo to Atangana-Baleanu.– Chaos Solitons Fractals, vol.122, pp.111-118. https://doi.org/10.1016/j.chao....
25.
Giusti A. (2020): General fractional calculus and Prabhakar's theory.– Comm. Nonlinear Sci. Numer. Simulat., vol.83, pp.105114. https://doi.org/10.1016/j.cnsn....
26.
Shen J.M., Rashid S., Noor M.A., Ashraf R. and Chu Y.M. (2020): Certain novel estimates within fractional calculus theory on time scales.– AIMS Math., vol.5, No.6, pp.6073-6086. https://doi.org/10.3934/math.2....
27.
Qureshi S. and Yusuf A. (2019): Fractional derivatives applied to MSEIR problems: Comparative study with real world data.– Eur. Phys. J. Plus, vol.134, No.4, pp.171. https://doi.org/10.1140/epjp/i....
28.
Luchko Y. and Gorenflo R. (1998): The initial value problem for some fractional differential equations with the Caputo derivatives.– Preprint No. A-98-08.
29.
Gómez-Aguilar J.F., Yépez-Martínez H., Escobar-Jiménez R.F., Astorga-Zaragoza C.M. and Reyes-Reyes J. (2016): Analytical and numerical solutions of electrical circuits described by fractional derivatives.– Appl. Math. Model., vol.40, No.21-22, pp.9079-9094. https://doi.org/10.1016/j.apm.....
30.
El-Ajou A., Arqub O.A., Momani S., Baleanu D. and Alsaedi A. (2015): A novel expansion iterative method for solving linear partial differential equations of fractional order.– Appl. Math. Comput., vol.257, pp.119-133. https://doi.org/10.1016/j.amc.....
31.
Rasheed M., Shihab S., Rashid T. and Enneffati M. (2021): Some step iterative method for finding roots of a nonlinear equation.– JQCM, vol.13, No.1, pp.95-102. https://doi.org/10.29304/jqcm.....
32.
Atangana A. (2016): On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation.– Appl. Math. Comput., vol.273, pp.948-956. https://doi.org/10.1016/j.amc.....
33.
Amat S., Busquier S. and Gutiérrez J.M. (2003): Geometric constructions of iterative functions to solve nonlinear equations.– J. Comput. Appl. Math., vol.157, No.1, pp.197-205. https://doi.org/10.1016/S0377-....
Share
RELATED ARTICLE
| eISSN: | 2353-9003 |
| ISSN: | 1734-4492 |
Task: edition of a scientific journal, its internationalization and ensuring open access to the journal via the Internet
© 2006-2024 Journal hosting platform by Bentus
We process personal data collected when visiting the website. The function of obtaining information about users and their behavior is carried out by voluntarily entered information in forms and saving cookies in end devices. Data, including cookies, are used to provide services, improve the user experience and to analyze the traffic in accordance with the Privacy policy. Data are also collected and processed by Google Analytics tool (more).
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
