EXPLICIT SOLUTION OF TIME FRACTIONAL MODIFIED EQUAL WIDTH WAVE EQUATION BY LIE SYMMETRY ANALYSIS

Copyright © 2021 the author(s). This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: In present article, the time fractional modified equal width wave equation has been examined by Lie symmetry reduction technique. This schemed methodology with generalized Erdelyi-Kober (E-K) integral and differential operator have been used to transformed the partial differential equations (PDEs) of generalized (non integer) order into ordinary differential equations (ODEs) of fractional order with insertion of some independent variable. At last, explicit solution obtained by power series method.


INTRODUCTION
The generalized calculus literature is as ancient as classical calculus. The concepts of fractional differential equations (FDEs) are utilized in modeling distinct phenomenon of mechanics, dynamics and drug therapy in biological systems. It is also used to study new age advance problems in neurons network, image processing, geology and hydrology. Podlubny [1], Oldham 5032 HARISH KUMAR, DIMPLE SINGH, AMIT TOMAR [2] and Debnath [3] illustrated the content on fractional order calculus. They provided Grunwald, Caputo and Riemann-Liuovili (RL) fractional derivatives and integrals definition along with their physical and geometrical interpretations in real modeling.
The schemed study of Lie symmetry and their application has been derived by Olver [4].
Bakkayaraj and Sehdaven [5] provoked about the group formalism of geometrical transforms in this technique. Biswas et al. [6,7] suggested the dual dispersion and non-linearity laws with the exclusive use of infinitesimals in symmetry reduction. Invariance criterion of some fractional PDEs, Hirota nonlinear, Hirota-Satsoma systems has been studied by Singla et al. [8]. Sneddon [9] used the concept of E-K operators and remarked that the system of FPDEs can be reduced to FODEs with the efficient use of these operators. The symmetry properties and exact solution of real time fractional KdV of third, fourth, fifth and generalized order compiled by Zhang [10], Wang et al. [17] and Gandhi [25,26]. Kaur et al. [11][12][13][14][15] has been implemented Painlike and Lie symmetry to Einstein vacuum field equation, Complex Hirota forms in multiple real and complex solutions. Huang [16] provided the total solutions of time fractional Harry-Dym equation with R-L fractional derivatives approach. Garrido et al. [19] prompted on travelling wave generalized solution of Driffield-Sokolov system and Arora et al. [20] found solitary wave solutions of modified equal width wave equations by Lie infinitesimals. The unremarkable criterion of solitary waves of equal width and regularized long wave equation has been solved by Gardner et al. [21,22] in late 20 th century. The physical phenomenon, scattering of regular long solitary wave has been studied by Morison et al. [23]. Rudin [24] attempted the implicit function theorem in principal of mathematical analysis, which has been used for convergence of power series solution by [10,17,[25][26].
In recent times, mathematicians have devoted lot of efforts to analyze the explicit and exact solutions of linear and nonlinear PDEs. It's difficult to obtain the exact solution of nonlinear differential equation as compared to linear differential equation. Therefore, some researchers have used numerical methods. It is always challenging task to find the exact or analytic solutions for nonlinear equations, and finding such solutions is even more difficult for fractional nonlinear equations. Hence, we are accepting the challenge and in this paper, we will 5033 TIME FRACTIONAL MODIFIED EQUAL WIDTH WAVE EQUATION try to find the explicit solution of a nonlinear time fractional equation.
The MEWWE occurring from the nonlinear media with dispersion process has been paid special concentration in the past decades. Our motivation is to generate mathematical formulation of the infinitesimals and investigate the symmetry reduction with power series solution of generalized TFMEWW equation with one parameter '  ' (1) and (x,t) is space-time coordinate and u(x, t) is amplitude of wave for the one dimensional wave- We proposed definition and terminology in section 2; in section 3, Lie symmetry approach has been discussed. The application of series solution with its convergence to MEWW model described in sections 4 and 5. Finally, the remarks and conclusions established.

PRELIMINARIES
In this section, we would like to present the needful definitions and terminology related to fractional calculus.
In the first two definitions, let ) (t h be an integrable function on ( ) t , 0 and for 0 0 except for a set of measure zero.

R-L PARTIAL DERIVATIVE OF GENERALIZED ORDER
This R-L definition holds for the function of two variables and  is order of fractional derivative.

THE LEIBNITZ RULE FOR R-L FRACTIONAL DERIVATIVES:
Leibnitz rule is defined for the product of two functions. Hence, below is the definition of Leibnitz rule for fractional derivative of the product of two functions. of two variable such that they are differentiable and integrable.  is order of fractional derivative.

DEFINITION OF A E-K FRACTIONAL DIFFERENTIAL AND INTEGRAL OPERATOR
5035 TIME FRACTIONAL MODIFIED EQUAL WIDTH WAVE EQUATION

LIE SYMMETRY ANALYSIS
There are several semi-analytic and analytic techniques to obtain exact and approximate solutions of FPDEs but we imposed Lie approach to address the infinitesimal symmetries of generalized differential equations; as conversion of FPDEs into FODEs is major task and it is feasible after the prolongation technique explained under: Suppose a general fractional PDE with space-time variables and ). , Lie group of transformations with parameter  is taken as Apply prolongation of rd 3 order on ) By preserving the operator Here, we use essential terms only which are usable in this paper and ) .

EXPLICIT POWER SERIES SOLUTION
Comparing the powers of 'z' on both sides,

CONVERGENCE OF THE SERIES SOLUTION OF TFMEWW EQUATION
We can take 0 a and 1 a as two arbitrary constants from equation (52). So let us assume We now consider the new power series   In last, convergence of the power series solution is proved. Overall, we conclude that the mathematical solution with use of dual techniques Lie symmetry analysis along with power series solution is reliable and efficient tool in obtaining the solution of such kind of linear and nonlinear time fractional wave models.