Solution of model equation of completely passive natural convection by improved differential transform method

Ekici, Mustafa; Ayaz, Fatma

doi:10.17515/resm2015.10me0818

Cited by 4 publications

(3 citation statements)

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“…Many scientists have explored exact solutions for nonlinear fractional differential equations using various methodologies. Some of these methods include the Adomian decomposition method [1,2], the differential transformation method [3,4], the finite difference method [5], the homogeneous balance method [6], the (G /G)-expansion method [7][8][9], the trial function method [10], Jacobi elliptic function expansion [11], the sub-ODE method [12,13], the homotopy analysis method [14], the tanhfunction expansion method [15], the sinc-collocation method [16], the exponential function method [17,18], the fractional sub-equation method [19], the generalized Kudryashov method [20], etc.…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions to Some Nonlinear Time-Fractional Evolution Equations Using the Generalized Kudryashov Method in Mathematical Physics

Ekici

2023

Symmetry

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In this study, we utilize the potent generalized Kudryashov method to address the intricate obstacles presented by fractional differential equations in the field of mathematical physics. Specifically, our focus centers on obtaining novel exact solutions for three pivotal equations: the time-fractional seventh-order Sawada-Kotera-Ito equation, the time-fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation, and the time-fractional seventh-order Kaup–Kupershmidt equation. The generalized Kudryashov method, celebrated for its versatility and efficacy in addressing intricate nonlinear problems, plays a central role in our research. This method not only simplifies the equations but also unveils their inner dynamics, rendering them amenable to meticulous analysis. It is worth noting that our fractional derivatives are defined in the context of the conformable fractional derivative, providing a solid foundation for our mathematical investigations. One notable aspect of our study is the visual representation of our findings. Graphical representations of the yielded solutions enliven intricate mathematical structures, providing a concrete insight into the dynamics and behaviors of said equations. This paper highlights the proficiency of the generalized Kudryashov method in resolving complex issues presented by fractional differential equations. Our study not only broadens the range of mathematical methods but also enhances our comprehension of the intriguing realm of nonlinear physical phenomena.

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Section: Introductionmentioning

confidence: 99%

Exact Solutions to Some Nonlinear Time-Fractional Evolution Equations Using the Generalized Kudryashov Method in Mathematical Physics

Ekici

2023

Symmetry

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“…Many researchers have been proposed various different methods to find solutions for nonlinear partial differential equations and nonlinear fractional differential equations [36][37][38][39][40]. Such as the inverse scattering transform method [1], the Hirota's bilinear method [2], truncated Painlevé expansion method [3], the tanh-function expansion method [4], the Jacobi elliptic function expansion method [5], the homogeneous balance method [6][7][8], the trial function method [9], the exp-function method [10,34], differential transform method [33], the Bäcklund transform method [11], the generalized Riccati equation method [12][13][14][15], the sub-ODE method [17][18][19][20], the original (G ′ /G)-expansion method [16,29], the double (G ′ /G,1/G)-expansion method [35] etc.. Since there is not a common method that can be used to solve all types of nonlinear evolution equations.…”

Section: Introductionmentioning

confidence: 99%

Application of the rational (G' /G)-expansion method for solving some coupled and combined wave equations

Ekici¹,

Ünal²

2022

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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In this paper, we explore the travelling wave solutions for some nonlinear partial differential equations by using the recently established rational (G ′ /G)-expansion method. We apply this method to the combined KdV-mKdV equation, the reaction-diffusion equation and the coupled Hirota-Satsuma KdV equations. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are also derived from the travelling waves. We have also given some figures for the solutions. 2020 Mathematics Subject Classification. Primary 35C07; Secondary 35C08. Keywords. The rational (G ′ /G)-expansion method, travelling wave solution, the combined KdV-mKdV equation, the reaction-diffusion equation, the coupled Hirota-Satsuma KdV equations.

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“…Many researchers have been proposed various different methods to find solutions for nonlinear fractional differential equations, and they discovered many useful methods in order to find exact solutions of FDEs. Namely, the sine-cosine method (Taşcan and Bekir, 2009), the homogeneous balance method (En-Gui and Hong-Qing, 1998;Wang et al, 1996), the hyperbolic tangent expansion method (Yang et al, 2001), the tanh-function expansion method (Fan, 2000), the exponential function method (He and Wu, 2006), fractional subequation method (Guo et al, 2012;Lu, 2012), the double ( ′ / , / )-expansion method (Li et al, 2010), the sub-ODE method (Zhang et al, 2006;Wang et al, 2007), the theta function method (Fan,2002), the differential transformation method (Odibat and Momani, 2008;Ekici and Ayaz, 2017), F-expansion method (Wang and Zhou, 2003;Wang and Li, 2005), the homotopy anlysis method (Arafa et al,2011), and so on.…”

Section: Introductionmentioning

confidence: 99%

The Double (G^( ^' )/G,1/G)-Expansion Method and Its Applications for Some Nonlinear Partial Differential Equations

Ünal

Ekici

2021

Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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The double (′ / , /)-expansion method is used to find exact travelling wave solutions to the fractional differantial equations in the sense of Jumarie's modified Riemann-Liouville derivative. We exploit this method for the combined KdV-negative-order KdV equation (KdV-nKdV) and the Calogero-Bogoyavlinskii-Schiff equation (CBS) of fractional order. We see that these solutions are concise and easy to understand the physical phenomena of the nonlinear partial differential equations. These solutions can be shown in terms of trigonometric, hyperbolic and rational functions.

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Solution of model equation of completely passive natural convection by improved differential transform method

Cited by 4 publications

References 0 publications

Exact Solutions to Some Nonlinear Time-Fractional Evolution Equations Using the Generalized Kudryashov Method in Mathematical Physics

Exact Solutions to Some Nonlinear Time-Fractional Evolution Equations Using the Generalized Kudryashov Method in Mathematical Physics

Application of the rational (G' /G)-expansion method for solving some coupled and combined wave equations

The Double (G^( ^' )/G,1/G)-Expansion Method and Its Applications for Some Nonlinear Partial Differential Equations

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