Numerical Methods For Fractional Euler-Lagrange Equations And Space Fractional Schr(?)dinger Equations

In this thesis, we study three problems, i.e., the parametric symplectic parti-tioned Runge-Kutta(PRK) methods with energy conservation; the fractional vari-ational integrators for the fractional Euler-Lagrange(E-L) equations, and its gen-eralization for fractional Euler-Lagrange equations with holonomic constraints orwith indefinite integrals; the conservative implicit diference schemes for the cou-pled nonlinear Schr(?)dinger equations with space fractional derivative.In Chapter2, based on W-transformation, some parametric symplectic par-titioned Runge-Kutta(PRK) methods are constructed, the errors and the sim-plecticity of the parametric methods are discussed, and some examples are givenincluding the Lobatto IIIA-IIIB method and the Radau IA-IA method and soon. In particular, for the separable Hamiltonian systems, a parametric symplecticPRK method, which based on a explicit symplectic PRK method, is provided. Thecomputing speed of this scheme is much more quickly than the implicit methods,and can be used for long time computation. Then we study the energy conser-vation of the parametric symplectic PRK methods, and provide a conjecture onthe parameter range in the explicit symplectic PRK method. The numerical testsshow that the parametric symplectic PRK methods can conserve the Hamiltonianand the symplecticity at the same time. Finally, we give the chapter’s summary, inwhich we introduce the problems of symplectic and energy-conservation methodsfor the Hamiltonian system with holonomic constraints.In Chapter3, based on the spirit of classic variational integrators, the frac-tional variational integrators for fractional Euler-Lagrange equations are constructed.Firstly, we derive the fractional discrete E-L equations, provide some examplesof fractional variational integrators, and the fractional variational errors are dis-cussed. Some numerical examples are presented to illustrate these results andshow that the fractional variational integrators can deal with the fractional Euler-Lagrange equations very well. This method is generalized to solve the cases offractional Euler-Lagrange equations with holonomic constraints and indefinite in-tegrals. In the later two cases, we also derive the fractional discrete E-L equations, construct some example of fractional variational integrators, and discuss the frac-tional variational errors. Finally, we provide some open problems.In Chapter4, we construct two conservative implicit diference schemes for thespace fractional coupled nonlinear Schr(?)dinger equations, i.e., the linearly implicitdiference scheme(LIDS) and the Crank-Nicolson diference scheme(CNDS). Thetwo schemes can conserve the probability density in the discrete level. In thespatial direction, we use the fractional central diference scheme. The existenceand uniqueness of the diference solution are proved. The convergence of the twoschemes are discussed in discrete l~2norm, and the two schemes are shown to beconvergent of order O(τ~2+~h2) with the time step τ and mesh size h. Somenumerical examples are presented to illustrate these results. At the same time,the numerical experiments reveal the diferences between the fractional Schr(?)dingerequations and the classic Schr(?)dinger equations. In the summary of this chapter,we provide some unsolved problems.