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Research Papers

# Time Fractional Third-Order Evolution Equation: Symmetry Analysis, Explicit Solutions, and Conservation Laws

[+] Author and Article Information
Dumitru Baleanu

Department of Mathematics,
Cankaya University,
Ankara 6400, Turkey;
Institute of Space Sciences,
Magurele, Bucharest 77125, Romania
e-mail: dumitru@cankaya.edu.tr

Mustafa Inc

Department of Mathematics,
Firat University,
Elazič 23119, Turkey
e-mail: minc@firat.edu.tr

Abdullahi Yusuf

Department of Mathematics,
Firat University,
Elazič 23119, Turkey;
Department of Mathematics,
Federal University Dutse,
Jigawa 7156, Nigeria
e-mail: yusufabdullahi@fud.edu.ng

Aliyu Isa Aliyu

Department of Mathematics,
Firat University,
Elazič 23119, Turkey;
Department of Mathematics,
Federal University Dutse,
Jigawa 7156, Nigeria
e-mail: aliyu.isa@fud.edu.ng

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 16, 2017; final manuscript received August 14, 2017; published online November 20, 2017. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 13(2), 021011 (Nov 20, 2017) (8 pages) Paper No: CND-17-1216; doi: 10.1115/1.4037765 History: Received May 16, 2017; Revised August 14, 2017

## Abstract

In this work, Lie symmetry analysis for the time fractional third-order evolution (TOE) equation with Riemann–Liouville (RL) derivative is analyzed. We transform the time fractional TOE equation to nonlinear ordinary differential equation (ODE) of fractional order using its Lie point symmetries with a new dependent variable. In the reduced equation, the derivative is in Erdelyi–Kober (EK) sense. We obtain a kind of an explicit power series solution for the governing equation based on the power series theory. Using Ibragimov's nonlocal conservation method to time fractional partial differential equations (FPDEs), we compute conservation laws (CLs) for the TOE equation. Two dimensional (2D), three-dimensional (3D), and contour plots for the explicit power series solution are presented.

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## Figures

Fig. 1

3D plot of Eq. (38) with a0 = 1, a1 = 1, a2 = 0.5, c = 1, α = 0.75, k = 2, and Γ = 1

Fig. 2

Contour plot of Eq. (38) with a0 = 1, a1 = 1, a2 = 0.5, c = 1, α = 0.75, k = 2, and Γ = 1

Fig. 3

3D plot of Eq. (38) with a0 = 0.5, a1 = 1, a2 = 0.5, a3 = –12, a4 = 3, c = 1, α = 0.75, k = 2, and Γ = 1

Fig. 4

Contour plot of Eq. (38) with a0 = 0.5, a1 = 1, a2 = 0.5, a3 = –12, a4 = 3, c = 1, α = 0.75, k = 2, and Γ = 1

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