Study Of Efficient Spectral Methods For Fractional PDEs

Fractional PDEs appear in modeling transport dynamics in complex systems.It has been demonstrated that many phenomena in various fields of science,engineering,bio-engineering,and economics are more accurately described by using fractional derivatives.Due to the non-local nature of the fractional derivatives,local methods such as finite difference method and finite element method loss the advantage that they enjoy for usu-al PDEs.Since the spectral method is fully capable of solving problems with non-local operator,they have become very helpful for numerical FPDEs.Another advantage of Spectral method to FPDEs is that it can naturally deal with the singular kernel involved in fractional calculus.The main purpose of this thesis is to develop some efficient and accurate spectral algorithms to solve a class of FPDEs.The details are given as follows:In Chapter 1,we give a brief review of the recent progress on the numerical method for fractional differential equation,present the motivations and main contributions of the thesis,and list some relevant preliminaries.In Chapter 2,we develop spectral/space-time spectral methods for time-fractional diffusion equations(TFDEs)involving either a Caputo or Riemann-Liouville derivative.An essential difficulty arises as the time fractional operator is not self-adjoint which makes the diagonalization process very ill-conditioned.We shall propose a novel approach to overcome this difficulty.This new approach is very efficient,and the computational cost is almost the same as the diagonalization method[163].Moreover,we also use the Enriched Petrov-Galerkin method to solve TFDEs in time.Error estimates of the proposed scheme are also obtained.Extensive numerical results are presented to verify the theoretical analysis and the robustness of the numerical scheme.In Chapter 3,the Fourier-like basis functions are constructed as the discrete eigen-functions of fractional Laplacian with Dirichlet,the Neumann,and the mixed boundary conditions.Then,we develop a novel space-time spectral method based on Fourier-like basis function for nonlinear PDEs involving fractional Laplacian on bounded domains.A detailed error analysis for the model problem with fractional Laplacian is established,which play important role in numerical analysis for Laplacian operator.We also provide ample numerical results to show high-order accuracy and efficiency of our method.In Chapter 4,we introduce an orthogonal family of new generalized Hermite functions(GHFs),with the weight function |x|2?,?>—1/2,and develop a novel spectral scheme for fractional PDEs on unbounded domain.The main ingredient of our approach is to define a new class of GHFs,which is intrinsically related to Fourier transform of fractional Laplacian and can serve as natural basis functions for properly designed spectral methods for fractional PDEs in unbounded domain.We also establish spectral approximation results for these GHFs.As examples of applications,the spectral schemes are provided for two model problems.Numerical results demonstrate the efficiency and accuracy of our method.In Chapter 5,we derive the Caffarelli-Silvestre extension problem of the FPDEs on unbounded domains Rd,namely,the Caffarelli and Silvestre extended the complex d dimensional FPDEs to the simple d + 1 dimensional integer order problem.Due to the Caffarelli-Silvestre extension problem caused the singularity in the extended direction,we use the Enriched spectral method to efficiently solve the extension problem.Due to the slow algebraic decay of the solution,we shall use mapped Chebyshev functions for this extension problem in original direction.We further presented ample numerical results to show that this method is very effective in dealing with fractional Laplacian problems and outperforms the existing methods by a wide margin.In Chapter 6,we study asymptotically and numerically the fundamental gap-the difference between the first two smallest(and distinct)eigenvalues-of the fractional Schr(?)dinger operator(FSO)and formulate a gap conjecture on the fundamental gap of the FSO.We begin with an introduction of the FSO on bounded domains with homo-geneous Dirichlet boundary conditions,while the fractional Laplacian operator defined either via the local fractional Laplacian(i.e.via the eigenfunctions decomposition of the Laplacian operator)or via the classical fractional Laplacian(i.e.via the Fourier trans-form).For the FSO on bounded domains with either the local fractional Laplacian or the classical fractional Laplacian,we obtain the fundamental gap of the FSO analytically on simple geometry without potential and numerically on complicated geometries and/or with different convex potentials.Based on the asymptotic and extensive numerical result-s,a gap conjecture on the fundamental gap of the FSO is formulated.Surprisingly,for two and higher dimensions,the lower bound of the fundamental gap depends not only on the diameter of the domain,but also the diameter of the largest inscribed ball of the do*main,which is completely different from the case of the Schr(?)dinger operator.Extensions of these results for the FSO in the whole space and on bounded domains with periodic boundary conditions are presented.