Numerical Methods And Analysis For Some Fractional Or Dispersive Partial Differential Equations

In recent,years,with the rapid development of science and technology,many practical problems in science and engineering can be converted to the definite solution problems for partial differential equations(PDEs).But in general,it is difficult to directly obtain the analytic solutions to these definite solution prob-lems.Therefore,the research and development of the numerical methods for PDEs are particularly important.This work aims to study the numerical methods and analysis for some frac-tional or dispersive PDEs.Firstly,we present a finite difference/spectral method for the two-dimensional generalized time fractional Cable equation,and analyze the stability and convergence of the discrete scheme.Then,based on the numeri-cal solutions obtained by the implicit finite difference method,we study the prob-lem of parameters identification for the fractional derivative,relaxation and retar-dation time parameters in the unsteady helical flows of a generalized Oldroyd-B fluid model with time fractional derivative.Thirdly,we utilize a Jacobi-Galerkin spectral method for computing the ground and first excited states of the non-linear fractional Schrodinger equation,prove the energy diminishing property of the time discrete scheme under the modified energy,and further derive the lower bound of the fundamental gap.Finally,by using a uniformly and optimally ac-curate multiscale time integrator Fourier pseudospectral method,we investigate the Klein-Gordon-Zakharov system in the subsonic limit,regime.Specifically:In chapter 1,we first briefly introduce the origin of the fractional calculus,and give the definitions and relations of several fractional differential operators used in this thesis,then briefly present the main content of this dissertation.In chapter 2,we consider the two-dimensional generalized time fractional Cable equation(?)with the boundary and initial conditions u(x,t)= 0,(x,t)?(?)?×I,u(x,0)= u0(x),x ??.We apply a finite difference/spectral method to theoretical analysis and numerical computation for the two-dimensional generalized time fractional Cable equation.Firstly,we combine the second-order backward difference method in time and Galerkin spectral method in space with the Legendre polynomials.Secondly,a detailed theoretical analysis demonstrates that the scheme is unconditionally stable.This method is proved to be min{2-?,2-?}-order convergence in time and spectral accuracy in space for smooth solutions,where ?,? are two exponents of fractional derivatives.Finally,the numerical results are reported to confirm our error bounds,which demonstrate the effectiveness of the proposed method.This research provides an efficient numerical method that can be applied to diffusion models and viscoelastic non-Newtonian fluid flow models.In chapter 3.we study the unsteady helical flows of a generalized Oldroyd-B fluid model with time fractional derivative(?)(?)with initial and boundary conditions v(r,0)=(?)tv(r,0)= 0,w(r,0)=(?)tw(r,0)= 0,a<r<b,v(a,t)=?(t),v(b,t)=?(t),?(a,t)=?(t),?(b,t)=?(t),0?t?T.Based on the finite difference method,we investigate the problem of parameter-s identification for the unsteady helical flows of a.generalized Oldroyd-B fluid between two infinitely long coaxial circular cylinders.Firstly,we employ the implicit finite difference method to obtain the numerical solution for the direct problem.By means of the Levenberg-Marquardt method,the numerical inversion for identifying optimal three parameters of the model is implemented simultane-ously,that is the Riemann-Liouville time-fractional derivative a,relaxation time? and retardation time ?r.Then,in order to testify the validity of the proposed numerical method,varied initial guesses and the observed data whether or not involving random error have been used to verify the reliable of estimation.This work provides an efficient method to obtain the estimated value of the unknown parameters for the generalized non-Newtonian fluids model.In chapter 4,we consider the time-independent nonlinear space fractional Schrodinger equation as following:find u(x)and ??R(?(?)Rd),such that(?)u(x)= 0,x ??c=Rd/?,with the normalization condition?u?2:= ?? u(x)|2dx=1.We study the ground and first excit ed states of the nonlinear fractional Schrodinger equation numerically by a Jacobi-Galerkin spectral method.Firstly,we introduce a normalized gradient flow with discrete normalization in order to treat the non-liinear term of the problem efficiently,then we discretize it by the semi-implicit backward Euler method in time and the Jacobi-Galerkin spectral method in s-pace.Secondly,the energy diminishing property of the time discrete scheme is proved under the modified energy,and the lower bound of the fundamental gap is further derived.Finally.some numerical examples are carried out for confirming the accuracy,and computing the ground and first excited states in one and two dimensional cases respectively.We find that the ground and first excited states become more peaked and narrower as the fractional derivative order becomes bigger or the local nonlinear interaction becomes smaller.In addition,the funda-mental gap is also numerically investigated which verifies the theoretical estimate.This study provides an efficient numerical method,which can be generalized to solve the fractional PDEs with Riesz space fractional derivatives for both linear and nonlinear cases.In chapter 5,we consider the Klein-Gordon-Zakharov(KGZ)system in the subsonic limit regime(?)with initial data?(x,0)=?0(x),(?)t?(x,0)=?1(x),?(x,0)=?0?(x),(?)t?(x,0)=?1?(x).By using a uniformly and optimally accurate multiscale time integrator Fourier pseudospectral method,we study the KGZ system with a dimensionless param-eter 0<?<1 and inversely proportional to the acoustic speed.In the subsonic limit regime,i.e.,0<?<<1,the solution of KGZ system propagates waves with O(?)-and O(1)-wavelength in time and space,respectively,and rapid outspread-ing initial layers with speed O(1/?)in space due to the singular perturbation of the wave operator in KGZ and/or the incompatibility of the initial data.First.of all,based on a multiscale decomposition by frequency and amplitude,we pro-pose a multiscale time integrator Fourier pseudospectral method by applying the Fourier spectral discretization for spatial derivatives followed by using the expo-nential wave integrator in phase space for integrating the decomposed system at each time step.Then,the method is explicit and easy to be implemented,exten-sive numerical results show that the MTI-FP method converges optimally in both space and time,with exponential and quadratic convergence rate,respectively,which is uniform for ??(0,1].This research provides a numerical method that can be applied to study the convergence rates of the KGZ system to its limit--ing models in the subsonic limit and wave dynamics and interactions of the two dimensional KGZ system.In chapter 6,we summarize this dissertation,and exhibit the prospect for the future research.