Analysis And Computation Of Riesz-type Fractional Differential Equations

Fractional calculus has increasing significance in various scientific fields[4,6,42,90,109,110],mainly because mathematical models using fractional differential operators are very suitable for descriptions of materials and processes with non-local properties and history-dependency.So far,the main research topics on fractional differential equations include:physical backgrounds of fractional models,properties of solutions to fractional differential equations,and numerical methods to fractional differential equations.Only a few of the fractional differential equations are analytically solvable.These few equations can be solved via integral transforms[47]and their closed form solutions can be expressed by special functions.Therefore,partial properties for the closed form solutions can be obtained through analyzing the relative special functions[13,58-60,121].However,most of the fractional differential equations are not analytically solvable.Therefore,numerical methods and scientific computations for these equations become more and more important.Similar to the case with integer order differential equations,numerical methods to fractional differential equations include the finite difference method,finite element method,and spectral method,etc.In[66],various numerical methods for fractional calculus and fractional differential equations are presented in details,as well as numerical stability analysis,convergence analysis,error estimates,and numerical computations.The key to constructing numerical schemes for fractional differential equations lies in discretizations of fractional operators.Numerical approximations to fractional differential/integral operators and properties of the coefficients in the numerical schemes are studied in more detail in[15,55]and references cited therein.Time fractional derivatives in fractional differential equations are often chosen as the Caputo type,the Caputo-Hadamard type,and so on.Space fractional derivatives are usually selected as Riemann-Liouville derivatives,Riesz derivatives,fractional Laplacians,and so on.This dissertation focuses on analyses and computations of Riesz fractional differential equations starting from Caputo derivatives and Riesz derivatives.The creative points of the dissertation are shown in the following aspects:(i)We construct numerical schemes for approximating Caputo derivatives.These approximations are of integer order convergence(some of)which are successfully applied to numerical solving the fractional differential equation.(ii)We prove the regularity of solutions to Riesz fractional differential equations defined on the whole real line and a finite interval where the definition conditions for the Riesz fractional differential equation on a finite interval are specifically determined.(iii)We establish numerical schemes for fractional partial differential equations with temporal Caputo derivative and spatial Riesz derivative in one-and two-dimensional spaces.The time derivative is approximated by the finite difference method while the space derivative is discretized by local discontinuous Galerkin finite element method.Numerical scheme of this type along with its analysis has not seen yet except this thesis.The main contents are introduced in more detail in the following four chapters(Chapters 2-5).Chapter 2 proposes numerical algorithms of integer order convergence to evaluate Caputo fractional derivatives.We derive numerical schemes for Caputo derivative via utilizing the relationship between Riemann-Liouville derivative and Caputo derivative,and adjusting Lubich's shifted formulae for Riemann-Liouville derivative.Chapter 3 discusses the property of Riesz derivative operator,together with the difference and relation with integer order differential operators and fractional Laplacians.Using Fourier transform for Riesz derivative on R shows that Riesz derivative is not consistent with the integer order differential operator.Moreover,the point-wise relation between Riesz derivative and fractional Laplacian in one dimension is proved.Finally,the relation between Riesz derivative and fractional Laplacian in higher dimension Rn(n?2)is discussed as well.Chapter 4 studies the regularity of the solution to the fractional differential equation with Riesz derivative.A closed form solution to the fractional differential equation on R is obtained via integral transform.Its regularity is then analyzed.What follows is the study of the fractional differential equation on the finite interval.To this end,the mapping properties and the null space of the Riesz derivative operator are introduced.Then the definition solution problems can be posed and the regularity of their solutions is studied in the weighted space.Chapter 5 numerically studies fractional advection equations with temporal Caputo derivative and spatial Riesz derivative in one-and two-dimensional spaces.The time derivative is approximated by the finite difference method.The space derivative is discretized by the local discontinuous Galerkin finite element method.We propose fully discrete schemes for one-dimensional and two-dimensional fractional advection equations.Choosing suitable numerical fluxes,we also show numerical stability,convergence analysis,and error estimates for the fully discrete schemes.The numerical experiment illustrates the feasibility of the proposed scheme.