Lie symmetry analysis, power series solutions and conservation laws of the time-fractional breaking soliton equation. Issue 6 (2nd November 2022) - Zhang, Zhi-Yong; Zhu, Hui-Min; Zheng, Jia
Abstract : The main attention of this work focus on extending the Lie symmetry and conservation law theories to the fractional partial differential equations involving the mixed derivative of the Riemann-Liouville time-fractional and first-order x -derivatives. More specifically, we first present a new prolongation formula of the infinitesimal generators of Lie symmetries for the time-fractional breaking soliton equation since the equation involves the mixed derivative, then perform Lie symmetry analysis for the equation. Furthermore, we construct an optimal system of one-dimensional Lie subalgebras and use them to reduce the equation to lower-dimensional fractional partial differential equations involving the Erdélyi-Kober operator. In order to construct the power series solution of the equation, we introduce the Hadamard's finite-part integral to deal with the divergence of the integrals. The convergence and error estimate of the power series solution are proved. Finally, a new conservation law formula for the equation is given by means of the nonlinear self-adjointness method and nontrivial conservation laws are found.