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.     

ON SOME BOUNDARY VALUE PROBLEMS FOR NONHOMOGENOUS POLYHARMONIC EQUATION WITH BOUNDARY OPERATORS OF FRACTIONAL ORDER

In the paper we study questions about solvability of some boundary value problems for a non-homogenous poly-harmonic equation.As a boundary operator we consider differentiation operator of fractional order in Miller-Ross sense.The considered problem is a generalization of well-known Dirichlet and Neumann problems.     

A LYAPUNOV-TYPE INEQUALITY FOR A PERIODIC BOUNDARY VALUE PROBLEM OF A FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we establish a Lyapunov-type inequality for fractional differential periodic boundary-value problems. As applications, a necessary condition is obtained to ensure the existence and uniqueness of nontrivial solutions to this problem.     

EXISTENCE OF SOLUTION FOR A BOUNDARY VALUE PROBLEM OF FRACTIONAL ORDER

This article considers the existence of solution for a boundary value problem of fractional order, involving Caputo's derivative     

两类分数阶微分方程的边值问题

本文研究了两类非线性项含有未知函数导数的分数阶微分方程的边值问题.利用分数阶微积分的性质及Banach不动点定理,获得了解的存在唯一性等有关结果,推广了已有文献的结论.     

SOLVABILITY FOR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION BOUNDARY VALUE PROBLEMS AT RESONANCE

The paper deals a fractional functional boundary value problems with integral boundary conditions. Besed on the coincidence degree theory, some existence criteria of solutions at resonance are established.     

一类核密度含高阶奇性Cauchy型积分的边值定理

本文推广「１」，「６」中的结果，讨论了一类开口弧核密度含高阶奇且情形更一般的Ｃａｕｃｈｙ型积分的边值定理，积分号下求导及Ｈ连续性。     

NONEXISTENCE OF POSITIVE SOLUTIONS FOR A FOUR-POINT BOUNDARY VALUE PROBLEM FOR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we investigate the nonexistence of positive solutions for a class of four-point boundary value problem of nonlinear differential equation with fractional order derivative. We give sufficient conditions on nonlinear term and the parameter such that the boundary value problem has no positive solutions. Some examples are presented to illustrate the main results.     

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO FRACTIONAL BOUNDARY VALUE PROBLEMS

In this paper, using fixed point theorems of general β-concave operators in ordered Banach space, we obtain the existence and uniqueness of positive solutions to a class of fractional boundary problem. In the end, an example is given to illustrate our main result.     

UNSTEADY BOUNDARY LAYER FLOW OF FRACTIONAL OLDROYD-B VISCOELASTIC FLUID PAST A MOVING PLATE IN A POROUS MEDIUM

This paper considers the unsteady boundary layer flow over a moving flat plate embedded in a porous medium with fractional Oldroyd-B viscoelastic fluid. The governing equations with mixed time-space fractional derivatives are solved numerically by using the finite difference method combined with an L1-algorithm. The effect of various physical parameters on the velocity and average skin friction are discussed and graphically illustrated in detail.Results show that the porosity € and fractional derivative α enhance the flow of Oldroyd-B viscoelastic fluid within porous medium, but fractional derivative βweakens the flow. Moreover, it is found that the average skin friction coefficient rises with the increase of fractional derivative β.     

A space-time finite element method for fractional wave problems

Numerical Algorithms - In this paper, we analyze a space-time finite element method for fractional wave problems involving the time fractional derivative of order γ (1 < γ...     

A class of degenerate fractional evolution systems in banach spaces

By using the notion of strongly (B, p)-sectorial operator and fractional differential calculus, we analyze the unique solvability of the Cauchy and Showalter problems for a class of degenerate fractional evolution systems. The results are used for the analysis of partial differential equations of fractional order with respect to the time variable.     

带有p-Laplace算子的分数阶边值问题的特征区间

本文研究了一类带有p-Laplace算子的分数阶微分方程两点边值问题.利用锥上的不动点定理,得到了这类边值问题的特征区间,推广了整数阶边值问题情形的结论.     

A reproducing kernel method for solving nonlocal fractional boundary value problems

In our previous works, we proposed a reproducing kernel method for solving singular and nonsingular boundary value problems of integer order based on the reproducing kernel theory. In this letter, we shall expand the application of reproducing kernel theory to fractional differential equations and present an algorithm for solving nonlocal fractional boundary value problems. The results from numerical examples show that the present method is simple and effective.     

A fractional partial differential equation based multiscale denoising model for texture image

Traditional integer‐order partial differential equation based image denoising approach can easily lead edge and complex texture detail blur, thus its denoising effect for texture image is always not well. To solve the problem, we propose to implement a fractional partial differential equation (FPDE) based denoising model for texture image by applying a novel mathematical method—fractional calculus to image processing from the view of system evolution. Previous studies show that fractional calculus has some unique properties that it can nonlinearly enhance complex texture detail in digital image processing, which is obvious different with integer‐order differential calculus. The goal of the modeling is to overcome the problems of the existed denoising approaches by utilizing the aforementioned properties of fractional differential calculus. Using classic definition and property of fractional differential calculus, we extend integer‐order steepest descent approach to fractional field to implement fractional steepest descent approach. Then, based on the earlier fractional formulas, a FPDE based multiscale denoising model for texture image is proposed and further analyze optimal parameters value for FPDE based denoising model. The experimental results prove that the ability for preserving high‐frequency edge and complex texture information of the proposed fractional denoising model are obviously superior to traditional integral based algorithms, as for texture detail rich images. Copyright © 2013 John Wiley & Sons, Ltd.     

A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.     

BOUNDARY VALUE METHODS FOR CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS

This paper deals with the numerical computation and analysis for Caputo fractional differential equations(CFDEs).By combining the p-order boundary value methods(B-VMs)and the m-th Lagrange interpolation,a type of extended BVMs for the CFDEs with y-order(0相似文献   

A NONLOCAL HYBRID BOUNDARY VALUE PROBLEM OF CAPUTO FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

In this paper,we discuss the existence of solutions for a nonlocal hybrid boundary value problem of Caputo fractional integro-differential equations.Our main result is based on a hybrid fixed point theorem for a sum of three operators due to Dhage,and is well illustrated with the aid of an example.     

实 Clifford 分析中的一个边值问题

考虑以 e_A=e_(α1)…e_(αh)(A={α_1,α_2,…,α_h)(?){1,2,3,…,n),1≤α_1<α_2<…<α_h≤n)为基底元素的实 Clifford 代数 A_n(R),其中 e_1=1,e_k~2=-1(k=2,3,4,…,n),e_ke_m e_me_k=0(k≠m,k,m=2,3,4,…,n).并用 V_n 表示由向量组 e_1,e_2,…,e_n 所张成的 A_n(R)的子空间,V_n 中元素为 x=(?)x_ke_k,A_n(R)中的     

A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation

In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann–Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracy of the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional order Van der Pol oscillator (VPO) equation.