Exact solutions for some time‐fractional evolution equations using Lie group theory

Abstract

In this paper, some classes of nonlinear partial fractional differential equations arising in some important physical phenomena are considered. Lie group method is applied to investigate the symmetry group of transformations under which the governing time-fractional partial differential equation remains invariant. The symmetry generators are used for constructing similarity variables, which leads to a reduced ordinary differential equation of Erdélyi-Kober fractional derivatives. Furthermore, a particular exact solution for each governing equation(s) is constructed. Moreover, the physical significance of the solution is investigated graphically based on numerical simulations in order to highlight the importance of the study.

Lie symmetry analysis, optimal system, and generalized group invariant solutions of the (2 + 1)‐dimensional Date–Jimbo–Kashiwara–Miwa equation

“…In the recent years, many direct methods have been developed to find the exact solutions of PDEs such as Backlund transformation method, 1 the exp-function method, 2,3 the inverse scattering method, 4 the auxiliary equation method, 5 the direct algebraic method, 6 the modified extended direct algebraic method, 7 the F-expansion method, 8 the Kudryashov method, 9 (G ′ ∕G)-expansion method, 10 Lax Pair, 11 novel test function method, 12 and Lie symmetry method. [13][14][15][16] Lie symmetry method was first introduced by Sophus Lie 17 in 1881. In this method, the number of independent variables is reduced to obtain Ordinary Differential Equations (ODEs).…”

Optimal classification, exact solutions, and wave interactions of Euler system with large friction

“…Partial invariance method, weak symmetry method, and conditional Lie‐Backlund symmetry method are some of the other branches of Lie symmetry. In order to solve fractional PDEs apart from usual methods, such as power series method and kernel Hilbert space method, one can also apply Lie symmetry technique . All these methods provide exact solutions for PDE systems as particular solutions.…”

Weak shock waves and its interaction with characteristic shocks in polyatomic gas

“…The purpose of this paper is to provide a contribution to the development and understanding of shocks and structured waves of polyatomic gases with any degree of freedom. It is motivated by our recent work on Lie symmetry group analysis to hyperbolic partial differential equations (PDEs) . It has been recognized that Lie symmetry group analysis provides a systematic and powerful technique to handle such equations.…”

“…It is motivated by our recent work on Lie symmetry group analysis to hyperbolic partial differential equations (PDEs). 17,18 It has been recognized that Lie symmetry group analysis provides a systematic and powerful technique to handle such equations. Lie symmetries in differential equations are discussed in several research papers.…”