Fundamental Solution And Numerical Solution Of Fractional Partial Differential Equation With Riesz(-Feller) Potential Operator

In the end of 17th century the integral calculus were on the seedtime,Leibniz and L'Hospital discussed the fractional calculus and simple fractional differential equations by letters.In the earlist time,since the fractional calculus were short of physical and mechanical background and were contradiction with the classical Newtonian intergral system,the development of fractional calculus was very slow. Till the 20th century,many academicians find that the fractional calculus are used widely to simulate some special phenominon in fractal kinetics,diffusion and trans-portation,spread and multiply of population biology,chaos and overfall,random walks,finance,viscoelastic materials and non-Newtonian fluid mechanics,etc.Since then fractional calculus obtain rapid development.Now the fractional theories and applications become a pop reseach subject in the world.Podlubny concludes that "the complete theory of fractional differential equa-tions,especially the theory of boundary value problems for fractional differential euqtions,can be developed only with the use of both left and right derivatives." So the spatial derivatives discussed in the paper are all Riesz potential operator or Riesz-Feller potential operator,which include the left and right Riemann-Liouville fractional derivatives.In Chapter 1,the developmental history of fractional calculus and the previous works about fractional calculs are introduced,then we present the definiens and properties of some common fractional operators and their relationship.In Chapter 2,The fundamental solutions of the space Riesz fractional partial differential equation(SRFPDE)and the time-space Riesz fractional partial differen-tial equation(TSRFPDE)are discussed,respectively.Under the condition of peri-odicity of function about the spatial variable,we solve the Cauchy problems of the SRFPDE and TSRFPDE by the expansion of Fourier series and Laplace integral transform.The obtained fundamental solutions can be expressed in the form of series,then computed easily.The space fractional Levy-Feller diffusion equations(SFLFDE)are obtained from the standard diffusion equation by replacing the second-order space deriva-tive with a Riesz-Feller derivative D_θ~αof orderα∈(0,2](α≠1)and skewnessθ(|θ|≤min{α,2-α}).In Chapter 3 we consider the Cauchy problem of SFLFDE in the sense of probability and numerical approximate computation,respectively. Firstly,two difference discrete schemes are constructed using numerical integral technique for the case 0＜α＜1 and 1＜α≤2 in infinite interval,which can be simulated the random walk's models.Furthermore,we discuss the domain of attraction of the stable Lévy distribution corresponding to Lévy-Feller diffusion. Secondly,in view of practical significance,numerical approximation of the initial and boundary values problem of SFLFDE is discussed for 1＜α≤2 in a finite interval.A conditionally stable and convergent explicit difference discrete scheme is presented.Finally,a numerical example is given to confirm our theoretical analysis.In Chapter 4,space Riesz fractional diffusion equation with nonlinear source term is discussed.Due to the equivalence of R-L fractional derivative and G-L fractional derivative,we discretize the Riesz potential operator by the shifted G-L technique.Moreover,discretizing the time derivative by the backward difference quotient deduces a implicit finite difference discrete scheme(IFDDS).Furthermore, if the nonlinear source term satisfies the Lipschitz's condition,the IFDDS is uncon-ditionally stable and convergent.To evaluate the efficiency of the above obtained IFDDS,a comparison with Method of Lines(MOL)is used.Finally,two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.Discretizing the fractional partial differential equations by finite difference tech-nique(i.e,numerical integral technique or the shifted G-L technique),the conver-gence order of the numerical approximation of R-L fractional derivative is no more than one order,so it is necessary to find other numerical discrete scheme with higher convergence order.In Chapter 5,we use two numerical methods:finite difference method and Galerkin finite element method to approximate the space Riesz frac-tional advection-diffusion equation.Firstly,discretizing the Riesz potential operator by numerical integral technique and time derivative by the backward difference quo-tient deduce to a implicit finite difference discrete scheme,which is unconditionally stable and convergent.But it is pity that the convergence order of the numeri-cal scheme about spatial variable is less than one order.In Order to obtain higher convergence order we moreover approximate the space Riesz fractional advection-diffusion equation by Galerkin finite element method.We transform the equation into a equavilent weak form,which is proven that the solution is existent and unique. Approximating time derivative by the backward difference quotient and the space Riesz derivative by Galerkin finite element deduce a implicit Galerkin finite element fully discrete scheme,which is also unconditionally stable and convergent.Further-more,if the solution of the equation satisfies some regularity,the convergence order about spatial variable can reach high order.At the end of Chapter 5 we compare the convergence order of numerical examples computed by the above two numerical approximate schemes,which are good agreement with the theoretical analysis.In Chapter 6 we conclude the works in this thesis.