一类时间分数阶偏微分方程的解

考虑一类时间分数阶偏微分方程,该方程包含几种特殊情况:时间分数阶扩散方程、时间分数阶反应-扩散方程、时间分数阶对流-扩散方程以及它们各自相对应的整数阶偏微分方程． 通过Laplace-Fourier变换及其逆变换,该方程在空间全平面和半平面内的基本解可以求出,但其表达式则是通过适当的变形来求．另外,对于有限域上的初边值问题,则可由Sine(Cosine)-Laplace变换导出该方程的一种级数形式的解,并通过两个数值例子来说明该方法的有效性．     

An inverse problem for a family of two parameters time fractional diffusion equations with nonlocal boundary conditions

The determination of a space‐dependent source term along with the solution for a 1‐dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter β>0 is considered. The fractional derivative is generalization of the Riemann‐Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over‐specified datum at 2 different time is given. The over‐specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative.     

Solvability of a nonlocal problem for a loaded parabolic-hyperbolic equation

For a mixed-type equation we study a problem with generalized fractional integrodifferentiation operators in the boundary condition. We prove its unique solvability under inequality-type conditions imposed on the known functions for various orders of fractional integrodifferentiation operators. We prove the existence of a solution to the problem by reducing the latter to a fractional differential equation.     

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.     

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 diffusion limit for a stochastic kinetic equation

We study the stochastic fractional diffusive limit of a kinetic equation involving a small parameter and perturbed by a smooth random term. Generalizing the method of perturbed test functions, under an appropriate scaling for the small parameter, and with the moment method used in the deterministic case, we show the convergence in law to a stochastic fluid limit involving a fractional Laplacian.     

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.     

Mixed Oblique Derivative Problem for the Helmholtz Equation in a Half-Disk

A mixed oblique derivative boundary value problem is considered for the Helmholtz equation in a half-disk. We prove the unique solvability of this problem for sufficiently large values of the parameter occurring in the equation, the leading part of the inverse operator being constructed explicitly.     

Regularity results for fractional diffusion equations involving fractional derivative with Mittag–Leffler kernel

This paper studies partial differential equation model with the new general fractional derivatives involving the kernels of the extended Mittag–Leffler type functions. An initial boundary value problem for the anomalous diffusion of fractional order is analyzed and considered. The fractional derivative with Mittag–Leffler kernel or also called Atangana and Baleanu fractional derivative in time is taken in the Caputo sense. We obtain results on the existence, uniqueness, and regularity of the solution.     

Numerical Solution for a Variable-Order Fractional Nonlinear Cable Equation via Chebyshev Cardinal Functions

In this paper, a variable-order fractional derivative nonlinear cable equation is considered. It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of class of fractional partial differential equation with variable coefficient of fractional differential equation in various continues functions of spatial and time orders. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Unique solvability of a non‐local problem for mixed‐type equation with fractional derivative

In this work, we investigate a boundary problem with non‐local conditions for mixed parabolic–hyperbolic‐type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.     

On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.     

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A biorthogonal system of functions consisting of two Riesz bases of the space L²[(0,1) × (0,1)], obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the biorthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data. Copyright © 2012 John Wiley & Sons, Ltd.     

On the solvability of a nonlocal problem for a hyperbolic equation of the second kind

In the characteristic triangle for a hyperbolic equation of the second kind we study a nonlocal problem, where the boundary value condition contains a linear combination of Riemann–Liouville fractional integro-differentiation operators. We establish variation intervals of orders of fractional integro-differentiation operators, taking into account parameters of the considered equation with which the mentioned problem has either a unique solution or more than one solution.     

Identification of the zeroth-order coefficient and fractional order in a time-fractional reaction-diffusion-wave equation

In this paper, we investigate an inverse problem of recovering the zeroth-order coefficient and fractional order simultaneously in a time-fractional reaction-diffusion-wave equation by using boundary measurement data from both of uniqueness and numerical method. We prove the uniqueness of the considered inverse problem and the Lipschitz continuity of the forward operator. Then the inverse problem is formulated into a variational problem by the Tikhonov-type regularization. Based on the continuity of the forward operator, we prove that the minimizer of the Tikhonov-type functional exists and converges to the exact solution under an a priori choice of regularization parameter. The steepest descent method combined with Nesterov acceleration is adopted to solve the variational problem. Three numerical examples are presented to support the efficiency and rationality of our proposed method.     

Reconstructing unknown nonlinear boundary conditions in a time‐fractional inverse reaction–diffusion–convection problem

This paper has focused on unknown functions identification in nonlinear boundary conditions of an inverse problem of a time‐fractional reaction–diffusion–convection equation. This inverse problem is generally ill‐posed in the sense of stability, that is, the solution of problem does not depend continuously on the input data. Thus, a combination of the mollification regularization method with Gauss kernel and a finite difference marching scheme will be introduced to solve this problem. The generalized cross‐validation choice rule is applied to find a suitable regularization parameter. The stability and convergence of the numerical method are investigated. Finally, two numerical examples are provided to test the effectiveness and validity of the proposed approach.     

On the solution of certain integral equations of mixed problems of plate bending theory

The problem of the bending of a Kirchhoff-Love plate in the shape of a strip under the impression of a thin linear rigid inclusion fastened at one of the edges of the plate when the other edge of the plate is rigidly clamped is considered. The problem is reduced by a Fourier integral transform to the solution of a convolution-type integral equation of the first kind in a finite segment with a regular kernel. The exact inversion of the principal part of the corresponding integral operator is constructed in the class of functions with non-integrable singularities on the segment edges. An effective asymptotic solution is given for the integral equation under investigation in this class of functions in the whole range of variation of the characteristic parameter λ. The results obtained are verified numerically. Analogous integral equations were examined in /1, 2/. The mode of investigation is similar to that proposed in /3/.     

Inverse initial-boundary value problem for a fractional diffusion-wave equation with a non-carleman shift

An inverse initial-boundary value problem is considered for a linear inhomogeneous second-order equation with a fractional time derivative and with delay in the spatial coordinate.     

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.     

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．