Abstract
In this work, we investigate invariance analysis, conservation laws, and exact power series solutions of time fractional generalized Drinfeld–Sokolov systems (GDSS) using Lie group analysis. Using Lie point symmetries and the Erdelyi–Kober (EK) fractional differential operator, the time fractional GDSS equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. Moreover, we have constructed conservation laws for time fractional GDSS and obtained explicit power series solutions of the reduced nonlinear ODEs that converge. Lastly, some figures are presented for explicit solutions.
Highlights
Because of the great importance of nonlinear fractional partial differential equations (NFPDEs) in physics, mechanics, hydrology, viscoelasticity, image processing, electromagnetics, and other fields, researchers have long been aware of the solutions and applications of fractional partial differential equations [1,2,3,4,5,6,7,8,9,10,11,12]
When we look through the literature, we realize Sophus Lie firstly put forward a methodology about symmetry analysis at the end of the nineteenth century [22]
Lie group methods were considered in order to obtain symmetries, symmetry groups, and symmetry reduction
Summary
Because of the great importance of nonlinear fractional partial differential equations (NFPDEs) in physics, mechanics, hydrology, viscoelasticity, image processing, electromagnetics, and other fields, researchers have long been aware of the solutions and applications of fractional partial differential equations [1,2,3,4,5,6,7,8,9,10,11,12]. Lie group methods were considered in order to obtain symmetries, symmetry groups, and symmetry reduction. The main role of Lie symmetry methods is to construct invariance properties having partial equations as invariant forms. With the aid of these properties, we can reduce an NFPDE into a nonlinear ODE of fractional order with the help of the Riemann-Liouville (RL) derivative. The link between Lie symmetry analysis and conservation laws of differential equations was revealed by Noether [27]. A generalized Noether theorem was used in [28] to construct conservation laws of NFPDEs with fractional Lagrangians. We deal with exact solutions of the time fractional GDSS by using Lie symmetry analysis and conservation laws. Lie symmetry methods have not been applied to the time fractional GDSS until now. E−t tz−1 dt, which converges in the complex plane when Re(z) > 0
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