Optimality Conditions,Noether Theorem For Fractional Variational Problems And Exact Solutions Of Fractional Differential Equations

Fractional calculus has important applications in many fields and is a hot topic nowadays.It is found that earthquake intensity prediction systems and microscopic particle motion systems are represented by fractional logarithmic function models,which are more effective than integer order models.Multivariable fractional controller and multivariable fractional disturbance observer have higher precision and stronger anti-interference than integer order classics controllers.This paper mainly studies the optimality condition and Noether theorem of fractional single variable logarithmic function functional variational problems and multivariate fractional functional variational problems.At the same time,in order to obtain the exact solutions of fractional differential equations corresponding to the optimality conditions and Noether theorem,the invariant subspace method and the improved sub-equation method are studied.The exact solutions of some classical fractional differential equations are obtained as well.The details are as follows.1.For the logarithmic function Lagrange functional with integer classics derivative and Caputo fractional derivative,using the fractional variational principle,the Hamilton principle and Euler-Lagrange equation are obtained.The Noether symmetry of the fractional logarithmic function Lagrange functional is studied.The basic variational formula of the functional is given,and the criteria for Noether symmetry and Noether quasi-symmetry of the functional are obtained by using infinitesimal group transformation.Noether theorem and Noether inverse theorem of the functional are obtained,and the internal relationship between Noether symmetry and conserved quantity is established.2.Established the formula of integration by parts containing Riemann-Liouville fractional partial derivatives,Riemann-Liouville fractional partial integrals and Caputo fractional partial derivatives.We considered the functional containing Riemann-Liouville fractional partial derivatives,Riemann-Liouville fractional partial integrals and Caputo fractional partial derivatives.The first-order necessary condition Ostrogradsky equation of the functional to obtain extremum is given by using the fractional variational principle.The second-order necessary condition Legendre condition for the extreme value of the functional is obtained.At the same time,the necessary conditions for functional to obtain extremum under holonomic constraints and isoperimetric constraints are discussed.Finally,the necessary conditions for Noether symmetry of functional are studied.3.The invariant subspace method for solving fractional partial differential equations with fractional mixed partial derivatives in the sense of Caputo fractional partial derivatives is established.By constructing the power function and the Mittag-Leffler function as the invariant subspace of the fractional differential equations and combining the fractional Laplace transform,the exact solutions and initial value problems of some fractional differential equations are solved.These fractional differential equations are the fractional diffusion equation,the fractional wave differential equation with absorption term,the generalized fractional wave differential equation with absorption term,the fractional dispersion equation and the fractional nonlinear thermal equation.And the method is used to solve two second order differential equations with mixed partial derivatives,generalized hyperbolic heat conduction equation and Fokker-Planck equation.4.Using the improved sub-equation method,the exact solutions of the fractional differential equations in the sense of the modified Riemann-Liouville fractional derivative are solved.This method converts fractional differential equations into integer order ordinary differential equations through fractional complex transformations,and then uses the homogeneous equilibrium method and maple software to obtain the exact solutions of fractional differential equations.Using this method,the exact solutions of the generalized fractional-time biological population model,the generalized fractional-time composite KdV-Burgers equation,the fractional time-space regular long wave equation,and the(3+1)-dimensional fractional time-space Zakharov-Kuznetsov equation are solved.