Construction Of CQ/SCQ Difference Formulas With Applications In Numerically Solving Fractional Calculus Equations

Academic community has generally recognized the history of fractional derivatives is as long as the traditional integer ones.The nonlocality and possibly consisting of weak regular kernels of fractional calculus operators make fractional models rather applicable for anomalous diffusion process in various scientific fields,such as viscoelasticity,quantum mechanics,electromagnetism,non-Newtonian fluid mechanics,economics,biomedicine,and so forth.The importance of solving fractional calculus models therefore cannot be overemphasized in view of the successful application in the above fields,though generally analytical solutions are seldom derived or consist of special functions like Mittag-Leffler function,the H-function,and so on,which are not computationally friendly.There is no doubt that the construction of efficient numerical methods has become an important means to simulate the fractional calculus models.This thesis is devoted to analyzing numerical methods for fractional operators with singular kernels from three aspects:· In the second chapter,we design and prove two families of second-order fractional approximation formulas with a free parameter ? based on the theory of convolution quadrature(CQ),namely,fractional BT-? and BN-? methods.Meanwhile,by analyzing the dependence of the truncation error coefficient on the parameter ? and the A-stability of the two methods,we further point out the advantages of our proposed method compared with the traditional method,and verify it by numerical examples.In addition,we apply these two methods to the time fractional cable equation.By studying the properties of the discrete coefficients,we prove the unconditional stability of the discrete schemes,and then give the optimal error estimates under certain regularity conditions.· Considering the advantage of distributed order model in simulating ultro-slow diffusion problem,we apply the idea of discretizing fractional calculus in CQ to the distributed-order calculus in Chapter 3,and propose some approximation strategies which are different from those commonly used in literature.Under the assumption that the solution satisfies certain conditions,we derive the corresponding truncation error estimates,and develop correction technique in CQ theory and apply it to cases with distributed-order operators.Further,we consider the solution structure of a simple distributed-order differential equation,showing the difference from the solution of the traditional fractional-order problem.This result has a certain reference significance for the design and error analysis of the subsequent distributed-order approximation formulas.· Since the CQ theory reveals the basic characteristics of difference formulas at integer nodes,in Chapter 4 to Chapter 6,we introduce a shifted parameter ? and examine further the corresponding characteristics of difference formulas at shifted points.Specifically,in Chapter 4 we establish the shited convolution quadrature(SCQ)theory,and in Chapter 5 we design and analyze three second-order difference formulas with shifted parameter,applying them to fractional mobile/immobile transport equations,two-sided space-fractional advection-diffusion equations and multi-term timefractional reaction-diffusion-wave equations,respectively,with numerical analysis and simulations.In Chapter 6,we study further the shifted fractional trapezoidal rule(SFTR),constructing(a)efficient structure preserving finite difference methods for high dimension nonlinear space fractional Schr¨odinger equations and(b)fast algorithms for subdiffusion problems with nonsmooth solutions,and analyzing(c)the discrete energy dissipation law for time fractional Maxwell's equations.