Symmetry Analysis,Conservation Laws And Exact Solutions For Several Classes Of Fractional Partial Differential Equations

Partial differential equation theory plays an important role in the research and development of nonlinear science,mathematical physics and any other related field.In order to describe the nonlocal time evolution or memory effect of spatial viscoelasticity in natural nonlinear phenomena more accurately,fractional calculus model is introduced.Since Gazizov proposed the continuous transformation group of fractional derivative in 2007,the symmetric structures,solutions and conservation laws of(1+1)-dimensional fractional real partial differential equations(systems)have been gradually concerned by scholars at home and abroad.However,there are only a few results on the reduction solution and conservation law of high-dimensional dispersion fractional order model with special structure,and the symmetry reduction problem of complex fractional order equation in quantum mechanics.On this basis,the paper mainly focus on the reductions and solution methods of high-dimensional fractional partial differential equations(systems),the construction of conservation laws and the Lie symmetry analysis of 1-D complex equations.Firstly,with the help of the fractional derivative,integral operation properties and Laplace transformation,by using our improved homogeneous balance method,we study the exact solutions of several kinds of high-dimensional nonlinear dispersion equations with time fractional derivatives.Secondly,we propose a generalized invariant subspace method to solve high-dimensional nonlinear dispersion equations with time fractional derivative by combining the properties of fractional calculus.These exact solutions have arbitrary functions or convolution integrals as the new form which is not easy to obtain via Lie symmetry reduction.Then,by using the classical Lie symmetry analysis method,Noether theorem and space composite transformation method,by constructing infinitesimal generator,Lagrange variation,conserved vector formula and invariant subspace of integral fractional mixed order derivative,we obtain the Lie symmetry,conservation law and exact solution of a class of high-dimensional fractional order long wave dispersion system.Finally,we study the symmetry reduction and solution of fractional two-component NLS systems and derivative NLS equations by using the non classical Lie group method of constructing extended systems.We find that the Lie symmetries obtained by this method are more extensive,and this idea can be extended to the study on symmetric reduction of other complex equations.