General methods for converting impulsive fractional differential equations to integral equations and applications

In this paper, we propose the concepts of Caputo fractional derivatives and Caputo type Hadamard fractional derivatives for piecewise continuous functions. We obtain general solutions of four classes of impulsive fractional differential equations (Theorem 3.1–Theorem 3.4) respectively. These results are applied to converting boundary value problems for impulsive fractional differential equations to integral equations. Some comments are made on recently published papers (see Section 4).     

On the concept and existence of solution for impulsive fractional differential equations

This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results.     

Multiquadric quasi-interpolation method for fractional integral-differential equations

In this paper, Multiquadric quasi-interpolation method is used to approximate fractional integral equations and fractional differential equations. Firstly, we construct two operators for approximating the Hadamard integral-differential equation based on quasi interpolators, and verify their properties and order of convergence. Secondly, we obtain that the approximation order of the integral scheme is 3, and the approximation order of the differential scheme is $3-\mu$ for $\mu(0<\mu<1)$ order fractional Hadamard derivative. Finally, The results of numerical experiments show that the numerical results are in greement with the theoretical analysis.     

The finite element method for fractional diffusion with spectral fractional Laplacian

This paper focuses on the finite element method for Caputo-type parabolic equation with spectral fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0,1) and the spatial derivative is the spectral fractional Laplacian. The time discretization is based on the Hadamard finite-part integral (or the finite-part integral in the sense of Hadamard), where the piecewise linear interpolation polynomials are used. The spatial fractional Laplacian is lifted to the local spacial derivative by using the Caffarelli–Silvestre extension, where the finite element method is used. Full-discretization scheme is constructed. The convergence and error estimates are obtained. Finally, numerical experiments are presented which support the theoretical results.     

Maximum principle for Hadamard fractional differential equations involving fractional Laplace operator

The purpose of the current study is to investigate IBVP for spatial-time fractional differential equation with Hadamard fractional derivative and fractional Laplace operator(−Δ)^β. A new Hadamard fractional extremum principle is established. Based on the new result, a Hadamard fractional maximum principle is also proposed. Furthermore, the maximum principle is applied to linear and nonlinear Hadamard fractional equations to obtain the uniqueness and continuous dependence of the solution of the IBVP at hand.     

Oscillatory solutions of fractional integro-differential equations

We give necessary conditions to get oscillatory solutions of a class of fractional order neutral differential equations with continuously distributed delay by means of the fractional derivative with respect to a given function. In particular, oscillatory solutions of the considered fractional equations with Caputo and Hadamard type of fractional derivatives are established. Some explicit examples are given to illustrate the main results.     

Initial Value Problems of Fractional Order with Fractional Impulsive Conditions

In this paper we intend to accomplish two tasks firstly, we address some basic errors in several recent results involving impulsive fractional equations with the Caputo derivative, and, secondly, we study initial value problems for nonlinear differential equations with the Riemann–Liouville derivative of order 0 < α ≤ 1 and the Caputo derivatives of order 1 < δ < 2. In both cases, the corresponding fractional derivative of lower order is involved in the formulation of impulsive conditions.     

Direct and inverse problems for an abstract differential equation containing Hadamard fractional derivatives

In a Banach space, we consider a Cauchy type problem with a left Hadamard fractional derivative of order α ∈ (0, 1) and a Cauchy problem with a regularized Hadamard fractional derivative. We prove the well-posed solvability of such problems with a bounded operator as well as with the generator of a strongly continuous semigroup. For inverse coefficient problems, we indicate sufficient conditions for their unique solvability.     

Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces

In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic PC-mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic PC-mild solution of a two-dimensional impulsive fractional predator-prey system with diffusion are investigated.     

Oscillation behavior of solution of impulsive fractional differential equation

In this paper, we study the oscillation of impulsive Caputo fractional differential equation. Sufficient conditions for the asymptotic and oscillation of the equation are obtained by using the inequality principle and Bihari Lemma. An example is given to illustrate the results. This is the first time to study the oscillation of impulsive fractional differential equation with Caputo derivative.     

A derivative concept with respect to an arbitrary kernel and applications to fractional calculus

In this paper, we propose a new concept of derivative with respect to an arbitrary kernel function. Several properties related to this new operator, like inversion rules and integration by parts, are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann‐Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.     

Dynamics of a Rod Made of Generalized Kelvin-Voigt Visco-elastic Material

In this paper we develop a new mathematical model for the lateral vibration of an axially compressed visco-elastic rod. As the basis for this model we use a fractional derivative type of stress-strain relation. We show that the dynamics of the lateral vibration is governed by two coupled linear differential equations with fractional derivatives. For a special case of the generalized Kelvin-Voigt body, this system is reduced to a single fractional derivative differential equation (Eq. (19)). For a class of problems to which (19) belongs the questions of the existence of a solution and its regularity are analyzed. Both continuous and impulsive loading are treated.     

Expansion Formulas in Terms of Integer-Order Derivatives for the Hadamard Fractional Integral and Derivative

We obtain series expansion formulas for the Hadamard fractional integral and fractional derivative of a smooth function. When considering finite sums only, an upper bound for the error is given. Numerical simulations show the efficiency of the approximation method.     

Caputo–Hadamard Fractional Derivatives of Variable Order

In this article, we present three types of Caputo–Hadamard derivatives of variable fractional order and study the relations between them. An approximation formula for each fractional operator, using integer-order derivatives only, is obtained and an estimation for the error is given. At the end, we compare the exact fractional derivative of a concrete example with some numerical approximations.     

非线性分数阶常微分方程的分段线性插值多项式方法

通过分段线性插值多项式方法构造了一类含有Hadamard有限部分积分的非线性常微分方程的数值离散格式.在时间方向上, 利用分段线性插值多项式方法对分数阶导数项进行近似, 并通过二阶向后差分格式来离散整数阶导数项.经过详细的证明, 得到了收敛精度为O(τmin{1+α,1+β})的误差估计结果.最后,通过数值算例和理论结果的对比直观地说明了理论分析的正确性.     

Variational Problems Involving a Caputo-Type Fractional Derivative

The aim of this paper is to study certain problems of calculus of variations that are dependent upon a Lagrange function on a Caputo-type fractional derivative. This type of fractional operator is a generalization of the Caputo and the Caputo–Hadamard fractional derivatives that are dependent on a real parameter \(\rho \). Sufficient and necessary conditions of the first and second order are presented. The cases of integral and holonomic constraints are also considered.     

Study on some qualitative properties for solutions of a certain two-dimensional fractional differential system with Hadamard derivative

By means of the de-singular method and a result on two-dimensional Gronwall–Bellman type integral inequalities, the component-wise (instead of being on some norms) a prior bounds are obtained for solutions of a class of the nonlinear two-dimensional system of fractional differential equations with the Hadamard derivative. The uniqueness and continuous dependence of the solutions for the systems are also discussed here.     

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．     

Existence of solutions for a fractional advection-dispersion equation with impulsive effects via variational approach

In this paper, based on the variational approach and iterative technique, the existence of nontrivial weak solutions is derived for a fractional advection-dispersion equation with impulsive effects, and the nonlinear term of fractional advection-dispersion equation contain the fractional order derivative. In addition, an example is presented as an application of the main result.     

Solvability for a higher-order Hadamard fractional differential model with a sign-changing nonlinearity dependent on the parameter $\varrho$

Until now most of the results are obtained in the sense of fractional derivatives such as Caputo and Riemann-Liouville, and there are few models using the Hadamard fractional derivatives. In this paper, based on the properties of the Green"s function, the existence of positive solutions are obtained for a Hadamard fractional differential equation with a higher-order sign-changing nonlinearity under some conditions by the fixed point theorem, and the existence of positive solutions is dependent on the parameter $\varrho$ for the Semipositive problem.