分数阶微分方程的一些新结果

第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性（次线性）条件下C310,11正解存在的充分必要条件．最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．     

The dependence on fractional orders of mild solutions to the fractional diffusion equation with memory

In this paper we investigate the Cauchy problem for a fractional diffusion equation and the time-fractional derivative is taken in the Caputo type sense. We give a representation of solutions under Fourier series and analyze initial value problems for the semi-linear fractional diffusion equation with a memory term. We also discuss the stability of the fractional derivative order for the time under some assumptions on the input data. Our key idea is to use Mittag-Leffler functions, the Banach fixed point theorem, and some Sobolev embeddings.     

非线性分数阶微分方程三点边值问题解的存在性

讨论了一类非线性分数阶微分方程三点边值问题解的存在性.微分算子是Riemann-Liouville导算子并且非线性项依赖于低阶分数阶导数.通过将所考虑的问题转化为等价的Fredholm型积分方程,利用Schauder不动点定理获得该三点边值问题至少存在一个解.     

Fractional variational problems with the Riesz-Caputo derivative

In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz-Caputo derivative. First we prove a generalized Euler-Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional derivative. Next we consider integral dynamic constraints on the problem, for several different cases. Finally, we determine optimality conditions for functionals depending not only on the admissible functions, but on time also, and we present a necessary condition for a pair function-time to be an optimal solution to the problem.     

On a Problem for a Mixed-Type Equation With Fractional Derivative

For a mixed-type equation with the Riemann–Liouville partial fractional derivative we study a problem where the boundary condition contains a linear combination of generalized fractional operators with the Gauss hypergeometric function. We find a solution to the considered problem explicitly by solving an equation with fractional derivatives of various orders and prove the uniqueness of the solution for various values of parameters of the mentioned operators.     

A new reliable algorithm based on the sinc function for the time fractional diffusion equation

This paper deals with the numerical solution of time fractional diffusion equation. In this work, we consider the fractional derivative in the sense of Riemann-Liouville. At first, the time fractional derivative is discretized by integrating both sides of the equation with respect to the time variable and we arrive at a semi–discrete scheme. The stability and convergence of time discretized scheme are proven by using the energy method. Also we show that the convergence order of this scheme is O(τ2?α). Then we use the sinc collocation method to approximate the solution of semi–discrete scheme and show that the problem is reduced to a Sylvester matrix equation. Besides by performing some theorems, the exponential convergence rate of sinc method is illustrated. The numerical experiments are presented to show the excellent behavior and high accuracy of the proposed hybrid method in comparison with some other well known methods.     

On a nonclassical fractional boundary-value problem for the Laplace operator

In this paper we consider a boundary-value problem for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides dependent on time. We prove the one-valued solvability of this problem, and provide the coercive estimates of the solution.     

On a boundary value problem for an equation of mixed type with a Riemann–Liouville fractional partial derivative

For an equation of mixed type with a Riemann–Liouville fractional partial derivative, we prove the uniqueness and existence of a solution of a nonlocal problem whose boundary condition contains a linear combination of generalized fractional integro-differentiation operators with the Gauss hypergeometric function in the kernel. A closed-form solution of the problem is presented.     

A nonlocal problem with fractional derivatives for the mixed type equation

For an equation of the mixed elliptic-hyperbolic type we investigate the boundary-value problem, when on the elliptic part of domain boundary a co-normal derivative of solution is given, and in the hyperbolic part the generalized fractional derivatives of solution value on characteristics are pointwise connected with the solution value and its derivative on the line of parabolic degeneration. The unique resolvability of problem is proven.     

Nonlocal problem for an equation of mixed type in a domain whose elliptic part is a half-strip

For the Gellerstedt equation, we study a problem with shift in a domain whose elliptic part is an infinite half-strip. By using generalized fractional differentiation operators, we specify a linear combination that relates the value of the solution on characteristics of the equation with the value of the solution and its derivative on the parabolic degeneration line. We prove the unique solvability of this problem.     

The Fundamental Solutions of the Space, Space-Time Riesz Fractional Partial Differential Equations with Periodic Conditions

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (SRFPDE) and the space-time Riesz fractional partial differential equation (STRFPDE) are discussed, respectively. Using methods of Fourier series expansion and Laplace transform, we derive the explicit expressions of the fundamental solutions for the SRFPDE and the STRFPDE, respectively.     

The space-time fractional diffusion equation with Caputo derivatives

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order β∈(0, 2] and the first-order time derivative with Caputo derivative of order α∈(0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.     

非线性分数微分方程边值问题正解的存在性(英文)

In this paper,we study a Dirichlet-type boundary value problem（BVP） of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D^α_o⁺is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel’skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.     

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann problems with operators of a fractional order.     

Continuation theorem of fractional order evolutionary integral equations

The fractional order evolutionary integral equations have been considered by the first author in [6], the existence, uniqueness and some other properties of the solution have been proved. Here we study the continuation of the solution and its fractional order derivative. Also we study the generality of this problem and prove that the fractional order diffusion problem, the fractional order wave problem and the initial value problem of the equation of evolution are special cases of it. The abstract diffusion-wave problem will be given also as an application.     

Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative

In this paper, we investigate the existence of solutions of the periodic boundary value problem for nonlinear impulsive fractional differential equation involving Riemann-Liouville sequential fractional derivative by using monotone iterative method. An example is presented to illustrate our main result.     

Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative by using the monotone iterative method.     

Inverse problem for a space‐time fractional diffusion equation: Application of fractional Sturm‐Liouville operator

An inverse problem of determining a time‐dependent source term from the total energy measurement of the system (the over‐specified condition) for a space‐time fractional diffusion equation is considered. The space‐time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional‐order time derivative and Sturm‐Liouville operator by fractional‐order Sturm‐Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.     

Symmetry analysis, conservation laws of a time fractional fifth-order Sawada-Kotera equation

In this paper, we intend to study the symmetry properties and conservation laws of a time fractional fifth-order Sawada-Kotera (S-K) equation with Riemann-Liouville derivative. Applying the well-known Lie symmetry method, we analysis the symmetry properties of the equation. Based on this, we find that the S-K equation can be reduced to a fractional ordinary differential equation with Erdelyi-Kober derivative by the similarity variable and transformation. Furthermore, we construct some conservation laws for the S-K equation using the idea in the Ibragimov theorem on conservation laws and the fractional generalization of the Noether operators.     

Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations

We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.