Generalized Fractional Theory And Symmetry Group Analysis And Applications Of Fractional Partial Differential Equations

The theory of fractional calculus is not only one of the important branches in the field of modern mathematical analysis,but also has a wide range of applications in the field of engineering technology.When describing complex nonlinear phenomena,fractional nonlinear partial differential equation(system)plays an important role.This thesis mainly studies the generalized fractional theory,including constructing Ψ-Hilfertype fractional derivative,generalizing the classic Maxwell model,defining two types of weighted Caputo-type differential operators,and generalizing the symmetry group analysis to the time-,time-space fractional differential equation(system).The specific research contents are expanded the following aspects:Chapter 1 first introduces the research background and significance,and then separately discusses the research status and main results of fractional order theory and symmetry group analysis theory,and finally summarizes the main work and innovations of this thesis.Chapter 2 studies several types of generalized fractional differential operators under the framework of generalized fractional theory.First of all,based on the idea ofΨ-Hilfer fractional derivative with respect to another function,the Sigmoid function and Sigmoid-type function are introduced as the kernel function to construct a new class of generalized fractional differential operators,and some basic properties and applications are given.Then,the k-Hilfer-Prabhakar fractional derivative is used to simulate the classic Maxwell model.As a result,we obtained the general form of the material function expression,and strictly prove its convergence and show the influence of the parameter k on the material function.Next,based on two types of generalized Caputo-type differential operators containing non-singular kernel,we defined their corresponding weighted Caputo-type differential operators.Finally,by solving related linear differential equations,we defined their weighted Caputo-type integral operators.In addition,the solution of nonlinear differential equations of the existence of uniqueness,the commutability of differential operators and the infinite series representation of solutions,have also been studied in detail.Chapter 3 addresses the applications of the symmetry group analysis technology to the time fractional partial differential equations with physical background,namely the(1+1)-dimensional nonlinear wave equation,the(2+1)-dimensional extended ZK equation and the(3+1)-dimensional KdV-type equation.In the first place,using the fractional symmetry group analysis can get their symmetries,one-parameter Lie transformation group,group invariant solution and similarity transformation.In the next place,with the help of the Olver method and the commutation table method,we can separately construct the one-dimensional subalgebra optimal system of the time fractional extended(2+1)-dimensional ZK equation and the time fractional(3+1)-dimensional KdV-type equation.Next,by considering the two-parameter left Erdelyi-Kober fractional operator,they can be reduced to the low-dimensional fractional differential equations and some special forms of exact solutions are also constructed.In the end,by using the new fractional conservation theorem,we can obtain their conservation laws.Chapter 4 further extends the applications of the symmetry group analysis method to the shallow water wave type time-space fractional partial differential equations,namely the(1+1)-dimensional KdV-type equation,the(2+1)-dimensional linear Burgers equation and the(3+1)-dimensional dissipative Burgers equation.Firstly,based on the analysis technique of fractional symmetry group,we can get their symmetries.Secondly,according to the above obtained symmetries,the one-parameter Lie transformation group and the several special forms of exact solutions are constructed.Subsequently,with the help of the multi-parameter left and right Erdelyi-Kober fractional operators,they can be reduced to the lower one-dimensional fractional differential equations,respectively.Moreover,for the time-space fractional(2+1)-dimensional linear Burgers equation,we have also constructed its one-dimensional sub-algebraic optimal system.Lastly,based on the nonlinear self-adjointness of these considered models,we can accurately construct their conservation laws.Chapter 5 investigations the applications of the symmetry group analysis scheme to the time fractional coupled partial differential equations,namely the(1+1)dimensional coupled Burgers-type equations,the(2+1)-dimensional integrable coupled KdV equations and the(3+1)-dimensional coupled unsteady Euler equations.In the first instance,we use the variational technique to derive the time fractional(1+1)-dimensional coupled Burgers-type equations.In the next place,by applying the fractional symmetry group analysis scheme,we can get its symmetries and oneparameter Lie transformation group.Subsequently,with the help of the one-parameter left Erdelyi-Kober fractional operator,the systems are reduced to a fractional ordinary differential equations.At the same time,we derive the power series solution of the systems and give the convergence analysis.Furthermore,the stability of the solitary wave of the model is also discussed.In the end,by using the new conservation theorem and its nonlinear self-adjointness,we construct the conservation law of the time fractional(1+1)-dimensional coupled Burgers-type equations.For two high-dimensional time fractional coupled partial differential equations,namely the time fractional(2+1)-dimensional integrable coupled KdV equations and the time fractional(3+1)-dimensional coupled unsteady Euler equations.We can also discuss similarly.Chapter 6 further extends the idea of the symmetry group analysis to several coupled time-space fractional partial differential equations,namely the(1+1)-dimensional coupled KdV-type equations,the(2+1)-dimensional coupled Burgers equations and the(3+1)-dimensional coupled unsteady Euler equations.Above all,the symmetries of these considered time-space coupled fractional models by using the fractional symmetry group analysis technique can be obtained.Then,based on their symmetries,these time-space fractional coupled systems can be reduced to the low-dimensional fractional coupled system containing multi-parameter left and right Erdelyi-Kober fractional operators.Besides,for the time-space fractional(2+1)-dimensional coupled Burgers equations,its explicit power series solution is constructed and the convergence analysis is also given.Finally,the conservation laws of the time-space fractional(1+1)dimensional coupled KdV-type equations through the new conservation theorem and the generalized fractional Noether operator,are found.Chapter 7 provides a comprehensive summary of the main work of this thesis and prospects for future research directions and problems that need to be further resolved.There are 8 figures,7 tables and 218 references in this thesis.