Abstract
In this paper, we employ the certain theory of Lie symmetry analysis to discuss the time-fractional Gardner equation with time-dependent coefficients. The Lie point symmetry is applied to realize the symmetry reduction of the equation, and then the power series solutions in some specific cases are obtained. By virtue of the fractional conservation theorem, the conservation laws are constructed.
Highlights
The conversant Korteweg–de Vries(KdV) equation, as the prototype of an integrable nonlinear partial differential equation (NPDE), is an active subject in the area of mathematical physics
The solutions of nonlinear ordinary differential equations obtained by symmetry reduction of (6) cannot be expressed by elementary functions or their integral formulas, though it is feasible to provide this solutions by the power series method
We have shown feasible ways to determine the exact solutions and conservation laws of the time-fractional Gardner equation with time-dependent coefficients
Summary
The conversant Korteweg–de Vries(KdV) equation, as the prototype of an integrable nonlinear partial differential equation (NPDE), is an active subject in the area of mathematical physics. Taking m = β = p = δ = 0, we get its special case of the generalized cylindrical KdV type equation μ ut + αuu x + γu xxx + u = 0. They have produced the exact solutions of (4) and (5) by Painlevé analysis and Lie symmetry [12]. Conservation laws have important applications in the integrability of partial differential equations, stability and global behavior of solutions, reliability of numerical solutions, construction of nonlocal systems and extension of generalized symmetric methods.
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