# Numerical Investigation of Time-Fractional Equivalent Width Equations That Describe Hydromagnetic Waves

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Concepts

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### Elzaki Transform Basic Concept

**Theorem**

**1.**

**Proof.**

## 3. Homotopy Perturbation Elzaki Transform Method

## 4. Implementation of the Technique

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Baleanu, D.; Guvenc, Z.B.; Machado, J.T. (Eds.) New Trends in Nanotechnology and Fractional Calculus Applications; Springer: New York, NY, USA, 2010. [Google Scholar]
- Shah, N.; Dassios, I.; Chung, J. A Decomposition Method for a Fractional-Order Multi-Dimensional Telegraph Equation via the Elzaki Transform. Symmetry
**2020**, 13, 8. [Google Scholar] [CrossRef] - Liu, Q.; Xu, Y.; Kurths, J. Active vibration suppression of a novel airfoil model with fractional order viscoelastic constitutive relationship. J. Sound Vib.
**2018**, 432, 50–64. [Google Scholar] [CrossRef] - Xu, Y.; Li, Y.; Liu, D. A method to stochastic dynamical systems with strong nonlinearity and fractional damping. Nonlinear Dyn.
**2016**, 83, 2311–2321. [Google Scholar] [CrossRef] - Xu, Y.; Li, Y.; Liu, D.; Jia, W.; Huang, H. Responses of Duffing oscillator with fractional damping and random phase. Nonlinear Dyn.
**2013**, 74, 745–753. [Google Scholar] [CrossRef] - Caputo, M. Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. Int.
**1967**, 13, 529–539. [Google Scholar] [CrossRef] - Ford, N.J.; Simpson, A.C. The numerical solution of fractional differential equations: Speed versus accuracy. Numer. Algorithms
**2001**, 26, 333–346. [Google Scholar] [CrossRef] - Oldham, K.B.; Spanier, J. The Fractional Calculus; Acadamic Press: New York, NY, USA, 1974. [Google Scholar]
- Ahmed, N.; Shah, N.; Vieru, D. Two-Dimensional Advection–Diffusion Process with Memory and Concentrated Source. Symmetry
**2019**, 11, 879. [Google Scholar] [CrossRef] [Green Version] - Ryzhkov, S.V.; Kuzenov, V.V. New realization method for calculating convective heat transfer near the hypersonic aircraft surface. Z. Fur Angew. Math. Und Phys.
**2019**, 70, 1–9. [Google Scholar] [CrossRef] - Saadeh, R.; Qazza, A.; Burqan, A. A New Integral Transform: ARA Transform and Its Properties and Applications. Symmetry
**2020**, 12, 925. [Google Scholar] [CrossRef] - Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl.
**2015**, 2, 731–785. [Google Scholar] - Losada, J.; Nieto, J.J. Properties of the new fractional derivative without singular kernel. Prog. Fract. Differ. Appl.
**2015**, 2, 87–92. [Google Scholar] - Shah, N.; Agarwal, P.; Chung, J.; El-Zahar, E.; Hamed, Y. Analysis of Optical Solitons for Nonlinear Schrödinger Equation with Detuning Term by Iterative Transform Method. Symmetry
**2020**, 12, 1850. [Google Scholar] [CrossRef] - Baleanu, D.; Mustafa, O.G. On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl.
**2010**, 59, 1835–1841. [Google Scholar] [CrossRef] [Green Version] - Yousef, F.; Alquran, M.; Jaradat, I.; Momani, S.; Baleanu, D. Ternary-fractional differential transform schema: Theory and application. Adv. Differ. Eqs.
**2019**, 2019, 197. [Google Scholar] [CrossRef] - Bokhari, A.; Baleanu, D.; Belgacem, R. Application of Shehu transform to Atangana-Baleanu derivatives. J. Math. Comput. Sci.
**2019**, 20, 101–107. [Google Scholar] [CrossRef] [Green Version] - He, J.H.; Ji, F.Y. Two-scale mathematics and fractional calculus for thermodynamics. Therm. Sci.
**2019**, 21, 2131–2133. [Google Scholar] [CrossRef] - Wang, K.L.; Yao, S.W.; Yang, H.W. A fractal derivative model for snow’s thermal insulation property. Therm. Sci.
**2019**, 23, 2351–2354. [Google Scholar] [CrossRef] - Kakutani, T.; Ono, H. Weak non-linear hydromagnetic waves in a cold collision-free plasma. J. Phys. Soc. Jpn.
**1969**, 26, 1305–1318. [Google Scholar] [CrossRef] - Yang, X.J.; Srivastava, H.M.; Machado, J.A. A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow. Therm. Sci.
**2016**, 20, 753–756. [Google Scholar] [CrossRef] - Yang, X.J. Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm. Sci.
**2017**, 21, 1161–1171. [Google Scholar] [CrossRef] [Green Version] - Singh, J.; Kumar, D.; Kumar, S. A new fractional model of nonlinear shock wave equation arising in flow of gases. Nonlinear Eng.
**2014**, 3, 43–50. [Google Scholar] [CrossRef] - Yang, X.J.; Machado, J.T. A new fractional operator of variable order: Application in the description of anomalous diffusion. Phys. A Stat. Mech. Appl.
**2017**, 481, 276–283. [Google Scholar] [CrossRef] [Green Version] - Di Barba, P.; Fattorusso, L.; Versaci, M. Electrostatic field in terms of geometric curvature in membrane MEMS devices. Commun. Appl. Ind. Math.
**2017**, 8, 165–184. [Google Scholar] - Lei, Y.; Wang, H.; Chen, X.; Yang, X.; You, Z.; Dong, S.; Gao, J. Shear property, high-temperature rheological performance and low-temperature flexibility of asphalt mastics modified with bio-oil. Constr. Build. Mater.
**2018**, 174, 30–37. [Google Scholar] [CrossRef] - Meerschaert, M.M.; Tadjeran, C. Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math.
**2006**, 56, 80–90. [Google Scholar] [CrossRef] - Ray, S.S.; Bera, R.K. Analytical solution of the Bagley Torvik equation by Adomian decomposition method. Appl. Math. Comput.
**2005**, 168, 398–410. [Google Scholar] [CrossRef] - Jiang, Y.; Ma, J. High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math.
**2011**, 235, 3285–3290. [Google Scholar] [CrossRef] [Green Version] - Odibat, Z.; Momani, S.; Erturk, V.S. Generalized differential transform method: Application to differential equations of fractional order. Appl. Math. Comput.
**2008**, 197, 467–477. [Google Scholar] [CrossRef] - Arikoglu, A.; Ozkol, I. Solution of fractional differential equations by using differential transform method. Chaos Solitons Fractals
**2007**, 34, 1473–1481. [Google Scholar] [CrossRef] - Zhang, X.; Zhao, J.; Liu, J.; Tang, B. Homotopy perturbation method for two dimensional time-fractional wave equation. Appl. Math. Model.
**2014**, 38, 5545–5552. [Google Scholar] [CrossRef] - Prakash, A. Analytical method for space-fractional telegraph equation by homotopy perturbation transform method. Nonlinear Eng.
**2016**, 5, 123–128. [Google Scholar] [CrossRef] - Dhaigude, C.; Nikam, V. Solution of fractional partial differential equations using iterative method. Fract. Calc. Appl. Anal.
**2012**, 15, 684–699. [Google Scholar] [CrossRef] - Safari, M.; Ganji, D.D.; Moslemi, M. Application of He’s variational iteration method and Adomian’s decomposition method to the fractional KdV-Burgers-Kuramoto equation. Comput. Math. Appl.
**2009**, 58, 2091–2097. [Google Scholar] [CrossRef] [Green Version] - Liao, S.J. The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, China, 1992. [Google Scholar]
- Liao, S. Homotopy analysis method: A new analytical technique for nonlinear problems. Commun. Nonlinear Sci. Numer. Simulation
**1997**, 2, 95–100. [Google Scholar] [CrossRef] - Liao, S. On the homotopy analysis method for nonlinear problems. Appl. Math. Comput.
**2004**, 147, 499–513. [Google Scholar] [CrossRef] - Abbasbandy, S.; Hashemi, M.S.; Hashim, I. On convergence of homotopy analysis method and its application to fractional integro-differential equations. Quaest. Math.
**2013**, 36, 93–105. [Google Scholar] [CrossRef] - Kumar, D.; Singh, J.; Baleanu, D. A fractional model of convective radial fins with temperature-dependent thermal conductivity. Rom. Rep. Phys.
**2017**, 69, 103. [Google Scholar] - Kumar, D.; Agarwal, R.P.; Singh, J. A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation. J. Comput. Appl. Math.
**2018**, 339, 405–413. [Google Scholar] [CrossRef] - Elzaki, T.M. The new integral transform ‘Elzaki transform’. Glob. J. Pure Appl. Math.
**2011**, 7, 57–64. [Google Scholar] - Elzaki, T.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.; Hilal, E.M.; Arabia, J.S.; Arabia, J.S. Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations. Math. Theory Model.
**2012**, 2, 33–42. [Google Scholar] - Shakeri, F.; Dehghan, M. Solution of delay differential equations via a homotopy perturbation method. Math. Comput. Model.
**2008**, 48, 486–498. [Google Scholar] [CrossRef] - Sakar, M.G.; Uludag, F.; Erdogan, F. Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method. Appl. Math. Model.
**2016**, 40, 6639–6649. [Google Scholar] [CrossRef] - Goswami, A.; Singh, J.; Kumar, D. An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Phys. A Stat. Mech. Appl.
**2019**, 524, 563–575. [Google Scholar] [CrossRef] - Kumar, D.; Singh, J.; Purohit, S.D.; Swroop, R. A hybrid analytical algorithm for nonlinear fractional wave-like equations. Math. Model. Nat. Phenomena
**2019**, 14, 304. [Google Scholar] [CrossRef] [Green Version] - Abdel-Aty, A.H.; Khater, M.M.; Baleanu, D.; Abo-Dahab, S.M.; Bouslimi, J.; Omri, M. Oblique explicit wave solutions of the fractional biological population (BP) and equal width (EW) models. Adv. Differ. Eqs.
**2020**, 2020, 1–17. [Google Scholar] - Li, X.; Haq, A.U.; Zhang, X. Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature. AIMS Math.
**2020**, 5, 5287. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The exact and analytical solutions graph and (

**b**) different fractional-order $\alpha $ graph of Example 1.

**Figure 2.**(

**a**) Exact and analytical solutions graph and (

**b**) different fractional-order $\alpha $ graph of Example 2.

**Figure 3.**(

**a**) Exact and analytical solutions graph and (

**b**) different fractional-order $\alpha $ graph of Example 3.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shah, N.A.; Dassios, I.; Chung, J.D.
Numerical Investigation of Time-Fractional Equivalent Width Equations That Describe Hydromagnetic Waves. *Symmetry* **2021**, *13*, 418.
https://doi.org/10.3390/sym13030418

**AMA Style**

Shah NA, Dassios I, Chung JD.
Numerical Investigation of Time-Fractional Equivalent Width Equations That Describe Hydromagnetic Waves. *Symmetry*. 2021; 13(3):418.
https://doi.org/10.3390/sym13030418

**Chicago/Turabian Style**

Shah, Nehad Ali, Ioannis Dassios, and Jae Dong Chung.
2021. "Numerical Investigation of Time-Fractional Equivalent Width Equations That Describe Hydromagnetic Waves" *Symmetry* 13, no. 3: 418.
https://doi.org/10.3390/sym13030418