In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. Three physical examples illustrate this point. The obtained results significantly facilitate finding Lie point symmetries for multi-dimensional time-fractional differential equations and their systems. Representations for the coordinates of corresponding infinitesimal group generators, as well as simplified determining equations are given in explicit form. It is proved that any Lie point symmetry group admitted by equations or systems belonging to considered class consists of only linearly-autonomous point symmetries. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied.
Through the results obtained in this paper, it can be found that the Lie group method is a very effective method, which can be used to deal with many models in natural phenomena.
Finally, for the general case, the symmetry of this equation is obtained, and based on the symmetry, the reduced equation is presented. And what is interesting is that the symmetry of the time fractional equation is obtained, and based on the symmetry, this equation is reduced to a fractional ordinary differential equation. The important thing is that for the special case of ϵ=3, the corresponding time fractional case are studied by Lie group method. Reciprocal Ba¨cklund transformations of conservation laws of momentum and energy are presented for the first time. In particular, for the particular equation, its conservation laws are obtained, including conservation of momentum and conservation of energy. Several special equations with PT symmetry are obtained by choosing different values, for which their symmetries are obtained simultaneously. In the present paper, PT-symmetric extension of the fifth-order Korteweg-de Vries-like equation are investigated. We show that the systems under consideration can be reduced to systems of ordinary differential equations with classical derivatives and with conformable fractional derivatives, respectively. For each system, all of the vector fields and the invariant variables are obtained. Thenceforward, the method is applied to time and time-space conformable fractional generalized Hirota-Satsuma coupled KdV systems, also, to time-space conformable fractional (2+1)-asymmetric Nizhnik-Novikov-Veselov system. In the case of time-fractional equations it turns out the reduced equations to differential equations with classical derivatives while in the case of time-space fractional equations, the reduced equations have conformable fractional derivatives.įurthermore, we propose prolongation formulas for systems of conformable time and time-space fractional partial differential equations with two independent variables. As an application of the symmetry reduction, we derived exact solutions of the considered equations in terms of solutions of ordinary or simpler partial differential equations. Subsequently, the corresponding vector fields and the symmetry reductions are derived. The efficiency of the method is illustrated by its applications to conformable time and time-space fractional Korteweg-de Vries, modified Korteweg-de Vries, Burgers, modified Burgers and Kadamtsev-Petviashvili equations. The chain rule is defined as the derivative of the composition of at least two different types of functions.In this study, Lie symmetry analysis is used to investigate invariance properties of some nonlinear time and time-space conformable fractional partial differential equations with two and three independent variables.
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