B. Yu. Irgashev
The mixed problem for a degenerate high order equation with a fractional derivative in a rectangular domain is considered in the article. The existence of a solution and its uniqueness are shown by the spectral method.
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The article constructs a general solution of a degenerate equation with a fractional derivative of Dzhrbashyan-Nersesyan. Particular solutions are presented through the Kilbas-Saigo function.
On nth Level Fractional Derivatives: An Equivalent Representation and Applications to Inverse Problem
本文研究n阶分数阶导数，给出其基于Riemann-Liouville分数阶导数的等价表示，并将其用于求解含二阶分数阶导数扩散方程的反问题。
We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Δ)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Ω$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Ω)$ and in the weighted Hölder space $C^0_s(\overlineΩ)$, respectively, do coincide.
In this paper, for a degenerate higher order equation with a fractional derivative in the sense of Caputo, a nonlocal problem with conjugation conditions in a rectangular domain is studied. The solution is constructed in the form of a Fourier series in the eigenfunctions of a one-dimensional problem. A criterion for the uniqueness of a solution is given.
Fractional derivative in spaces of continuous functions
本文是关于含分数阶时间导数的抽象发展方程的第一篇文章。给出了连续函数空间和Hoelder连续函数空间中的极大正则性结果.
We prove the uniqueness in determining both orders of fractional time derivatives and spatial derivatives in diffusion equations by pointwise data. The proof relies on the eigenfunction expansion and the asymptotics of the Mittag-Leffler function.
We are concerned in this paper with the degenerate fractional diffusion advection equations posed in bounded domains. Due to a suitable formulation, we show the existence of weak entropy solutions for measurable and bounded initial and Dirichlet boundary data. Moreover, we prove a $L^1-$type contraction property for weak entropy solutions obtained via parabolic perturbation. This is a weak selection principle which means that the weak entropy solutions are stable in this class.
Numerical computation of a fractional derivative with non-local and non-singular kernel
提出了一种求解包含Atangana-Baleanu分数阶导数的分数阶初值问题的数值格式。所提出的方法的一些例子，包括方程和系统的分数初值问题。
A Nondivergence Parabolic Problem with a Fractional Time derivative
研究了一类带分数阶时间导数的非局部非线性抛物问题。我们证明了一个Krylov-Safonov型结果，主要证明了H解决方案的老规律。当分数阶时间导数的阶数接近1时，我们的估计保持一致。
Operational Calculus for the nth-Level Prabhakar Type Fractional Derivative with Applications
本文提出了一种用于第n级Prabhakar型分数导数的运算微积分方法，并通过两个实例展示了其在求解分数阶微分方程中的应用。
We consider the Cauchy problem for a degenerate fractional conservation laws driven by a noise. In particular, making use of an adapted kinetic formulation, a result of existence and uniqueness of solution is established. Moreover, a unified framework is also established to develop the continuous dependence theory. More precisely, we demonstrate L^1-continuous dependence estimates on the initial data, the order of fractional Laplacian, the diffusion matrix, the flux function, and the multiplicative noise function present in the equation.
This paper is a generalization of the author's previous work [14]. We extend the argument [14] for any uniformly elliptic operator in divergence form $\mathcal{L}u=-div(A(x)\nabla u)$, more precisely, we study a fractional type degenerate elliptic equation posed in bounded domains $$ \partial_t u=div(u\,A(x)\nabla \mathcal{L}^{-s}u) $$ where $\mathcal{L}^{-s}$ is the inverse $s$-fractional elliptic operator for any $s \in (0,1)$. We show the existence of solutions for measurable and bounded non-negative initial data, and homogeneous Dirichlet boundary condition.
On the stability of degenerate Schrödinger equation under boundary fractional damping
本文研究了退化薛定谔方程在分数边界阻尼下的适定性与稳定性，通过乘子方法证明了解的指数和多项式衰减率。
To study the existence and uniqueness of solutions to Cauchy-type problems for fractional q-difference equations with the bi-ordinal Hilfer fractional q-derivative which is an extension of the Hilfer fractional q-derivative. An approach is based on the equivalence of the nonlinear Cauchy-type problem with a nonlinear Volterra q-integral equation of the second kind. Applying an analog of Banach fixed point theorem we prove the uniqueness and the existence of the solution. Moreover, we present an explicit solution to the q-analog of the Cauchy problem for the linear case.
Indirect boundary stabilization for weakly coupled degenerate wave equations under fractional damping
本文针对一维弱耦合退化波动方程系统，在仅一端施加分数阶边界阻尼下，证明了最优多项式能量衰减速率 $1/t^{(3-τ)}$，方法采用频域与乘子技术。
The present article deals with the similarity method to tackle the fractional Schrodinger equation where the derivative is defined in the Riesz sense. Moreover the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform to the proposed fractional equation. Furthermore the ODEs are treated effectively via the Fourier transform. The graphical solutions are also depicted for different fractional derivatives .
We consider fractional diffusion equation with the distributed order Caputo derivative. We prove existence of a weak and regular solution for general uniformly elliptic operator under the assumption that the weight function is only integrable.
In this paper we investigate fractional differential equations with Hilfer fractional derivative of order $1<γ<2$ and type $δ\in [0,1]$ in a Banach space. We introduce a family of general fractional cosine operator functions of order $1<γ<2$ and type $δ\in [0,1]$ and discuss their properties to give a suitable definition of mild solution of a semilinear evolution equation. In last section an example is presented.
Complex order fractional derivatives in viscoelasticity
我们引入复数阶分数导数模型描述粘弹性材料。这不能无限制地进行，因此，我们得出，第一次，真正的价值相容性约束，以及物理约束，导致可接受的模型。因此，我们引入了一种新的形式的复阶分数阶导数。同时，我们考虑了一个具有复数导数的分数阶微分方程，并研究了它的可解性。应力松弛和蠕变得到的结果说明了几个数值例子。
We consider an initial mixed-boundary value problem for anisotropic fractional type degenerate parabolic equations posed in bounded domains. Namely, we consider that the boundary of the domain splits into two parts. In one of them, we impose a Dirichlet boundary condition and in the another one a Neumann condition. Under this mixed-boundary condition, we show the existence of solutions for measurable and bounded non-negative initial data. The nonlocal anisotropic diffusion effect relies on an inverse of a s-fractional type elliptic operator, and the solvability is proved for any s in (0, 1).
In the article, a general solution of an equation with a generalized Hilfer derivative, which has a degeneration, is constructed. Particular solutions are presented through the Kilbas-Saigo function. A representation of the solution of the initial problem of the type of the Cauchy problem is obtained.
A Parabolic Problem with a Fractional-Time Derivative
研究了一类具有分数阶时间导数和空间分数阶扩散的抛物型方程的正则性。我们的主要结果是散度型方程解的De Giorgi-Nash-Moser Holder正则性定理。我们还证明了结果的存在性，唯一性和更高的规律性的时间。
Holder continuity of weak solutions to evolution equations with distributed order fractional time derivative
本文研究分布阶分数时间导数演化方程弱解的正则性，证明了非负弱上解的弱Harnack不等式和弱解的Hölder连续性，推广了单阶分数时间导数情形的已知结果。
We prove uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.
We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Ω$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Hölder regularity up to the boundary, that is, $u/d^s\in C^α(\overlineΩ)$ for some $α\in(0,1)$, $d$ being the distance from the boundary.
In this paper, with the help of previously constructed self-similar solutions, we construct a solution to a Cauchy-type problem for an even-order high-order equation with a fractional derivative in the sense of Hilfer
Operator theoretic approach to the Caputo derivative and the fractional diffusion equations
Caputo时间导数通常被定义为行为良好的函数，比如连续可微函数。因此，在含Caputo导数的部分分数阶微分方程理论中，寻找解的函数空间往往是太窄的光滑函数空间。本文在分数索伯列夫空间中引入了Caputo导数的一个合适的定义，并从算子论的角度研究了它.特别地，给出了分数阶索伯列夫空间中与分数阶积分算子和微分算子有关的范数的一些重要等价性.利用这些结果证明了分数阶索伯列夫空间中具有Caputo导数的时间分数阶扩散方程的某些初边值问题解的极大正则性.
On a higher order multi-term time-fractional partial differential equation involving Caputo-Fabrizio derivative
本文讨论了高阶多项偏微分方程在时间上具有Caputo-Fabrizio分数阶导数的情形。研究一类含高阶Caputo-Fabrizio导数的分数阶热传导方程的边值问题.利用分离变量法和分部积分法，将分数阶偏微分方程降阶为整数阶。我们用傅里叶级数表示特殊情况下公式化问题的显式解。
In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior Hölder estimates are obtained when the order of the fractional diffusion is less or equal than one, and Lipschitz estimates when it is bigger than one. In the latter case, the estimates are robust enough to conclude interior $C^{1, α}$ regularity by an improvement of the flatness procedure, which is possible when the nonlocal term is close enough to a second-order diffusion.
This paper is devoted to study the nonlinear sequential fractional boundary value problems involving generalized $ψ$-Caputo fractional derivatives with nonlocal boundary conditions. We investigate the Green function and some of its properties, from which we obtain a new Lyapunov-type inequality for our problem. A lower bound for the possible eigenvalues of our problem is derived. Furthermore, we apply some properties of the Green function to obtain some existence results for our problem. It is worth mentioning that our results still work with some source functions including singularities.
This work aims to use the homotopy analysis method to obtain analytical solutions of linear time-fractional Navier-Stokes equations with cylindrical coordinates and of a system of nonlinear time-fractional Navier-Stokes equations with Cartesian coordinates. These equations are described in the $ψ$-Caputo time-fractional derivative. The solutions obtained for time-fractional Navier-Stokes equations are graphically presented.
We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton-Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem.
Non-local fractional derivatives. Discrete and continuous
我们证明了整数中分数阶离散导数的最大值原理和比较原理。正则性的结果时，空间是一个网格的长度$h$，和逼近定理的连续分数阶导数。当函数足够好时，这些近似过程给出了近似阶的度量。这些结果也使我们能够证明的巧合，足够好的功能，Marchaud和Grünwald-Letnikov衍生物在每一个点和收敛速度的Grünwald-Letnikov衍生物。分数阶离散导数也将被描述为由半离散扩展问题定义的Neumann-Dirichlet算子。一些与调和分析有关的算子也将被考虑，特别是它们在Lebesgue空间$\ell^p(\mathbb{Z}).$中的行为
An extension problem for the fractional derivative defined by Marchaud
我们证明了$\mathbb{R}$中的（非局部）Marchaud分数阶导数可以从带有额外（正）变量的抛物型延拓问题中得到，作为将热传导方程映射到Neumann条件的算子.从局部算子的性质出发，导出了分数阶导数的一些性质。特别地，我们证明了Marchaud-平稳函数的Harnack原理.
In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The first problem involves a space-degenerate partial differential equation and the second one involves a time-degenerate partial differential equation. Solutions to both problem are expressed in series expansions. For the first problem, we obtained solutions in the form of Fourier-Legendre series. Convergence and uniqueness of solutions have been discussed. Solutions to the second problem are expressed in the form of Fourier-Sine series and they involv…
In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved.
We define fractional derivatives $\pppa$ in Sobolev spaces based on $L^p(0,T)$ by an operator theory, and characterize the domain of $\pppa$ in subspaces of the Sobolev-Slobodecki spaces $W^{α,p}(0,T)$. Moreover we define $\pppa u$ for $u\in L^p(0,T)$ in a sense of distribution. Then we discuss initial value problems for linear fractional ordinary differential equations by means of such $\pppa$ and establish several results on the unique existence of solutions within specified classes according to the regularity of the coefficients and the non-homogeneous terms in the equations.
We study a general class of discrete $p$-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional $p$-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Laplace operator to the continuous fractional Laplace operator.
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as such paves a way for application to many other areas of analysis involving distributions. This includes (but is not limited to): the fractional Fourier series (i.e. an orthonormal basis for fractional derivatives), the fractional derivative of Fourier transforms, and fundamental solutions to differential equations such as the wave equation. This paper observes …
Fractional telegraph equation, Mittag-Leffler function, Hilfer derivative, Hadamard fractional derivative, Riesz-Feller space-fractional derivative
本文考虑时空分数阶电报方程，其中时间导数是Hilfer和Hadamard意义下的，空间分数阶导数是Riesz-Feller意义下的.我们给出了这类分数阶方程柯西问题解的傅里叶变换。并对一些具体案例作了可能的解释。
On Gaussian Lipschitz spaces and the boundedness of Fractional Integrals and Fractional Derivatives on them
本文引入了关于Gauss测度的Lipschitz空间，并研究了其上的分数次积分和分数次导数算子的有界性.所用的方法是一般的，足以提供经典情形的另一种证明，而且还可以推广到Laguerre和Jacobi展开的情形
Boundedness of fractional integrals and fractional derivatives on Laguerre Lipschitz spaces
该研究受高斯Lipschitz空间相关结果启发，通过发展拉盖尔框架下的泊松积分理论，证明了与拉盖尔多项式展开相关的一类分数次积分与导数在拉盖尔Lipschitz空间上的有界性。
In the present paper, we determine the estimations on Atangana-Baleanu-Caputo fractional derivative at extreme points. With the assistance of the estimations obtained, we derive the comparison results. Peano's type existence results established for nonlinear fractional differential equations involving Atangana-Baleanu-Caputo fractional derivative. The acquired comparison results are then utilized to deal with the existence of local, extremal and global solution.
Necessary and sufficient stability and instability conditions are obtained for multi-term homogeneous linear fractional differential equations with three Caputo derivatives and constant coefficients. In both cases, fractional-order-dependent as well as fractional-order-independent characterisations of stability and instability properties are obtained, in terms of the coefficients of the multi-term fractional differential equation. The theoretical results are exemplified for the particular cases of the Basset and Bagley-Torvik equations, as well as for a multi-term fractional differential equat…
On numerical methods and error estimates for degenerate fractional convection-diffusion equations
针对一大类非线性非局部可能退化的对流扩散方程，提出了一种收敛的数值方法，并开发了新的Kuznetsov型理论，得到了主导数阶数在1到2之间的最优误差估计。
In this paper, we develop the theory of nonlinear hybrid fractional differential equations involving Atangana--Baleanu--Caputo (ABC) fractional derivative. We construct the equivalent fractional integral equation and establish the existence results through it. Further, we build up the theory of inequalities for ABC--hybrid fractional differential equations and use it to examine the uniqueness, existence of a maximal and minimal solution and the comparison results.
The mapping properties of fractional derivatives in weighted fractional Sobolev space
本文利用Calderón–Lions插值定理，证明了Riemann–Liouville和Marchaud分数阶导数在加权分数阶Sobolev空间中的具体映射关系。
In this paper we develop a method to solve evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and $p$-Laplace equations as special cases.
Computational solutions of unified fractional reaction-diffusion equations with composite fractional time derivative
研究了一类统一的分数阶反应扩散方程的计算解，该方程是由标准扩散方程通过将一阶时间导数替换为Hilfer（2000）定义的广义分数阶时间导数，将二阶空间导数替换为Riesz-Feller分数阶导数，并加上函数phi（x，t），其是控制反应的非线性函数。该解是通过应用拉普拉斯变换和傅里叶变换以H函数的紧凑和封闭形式导出的。本文得到的主要结果是Mainardi等人（2001，2005）和Tomovski等人（2011）最近给出的时空分数阶扩散方程的基本解的一个优美推广.计算表示的基本解也得到明确的。推导了分布的分数阶矩。最后，对有限个Riesz-Feller空间分数阶导数的结果进行了适当的推广.
The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about the solution at a fixed time instant at a monitoring location, as "the observation data", identifies uniquely the order of the fractional derivative.
In a rectangular domain, a boundary-value problem is considered for a mixed-type equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. Using the method of separation of variables, Laplace transform, a unique solvability of the considered problem has been established. Moreover, we have found the explicit solution of initial problems for a differential equation with the bi-ordinal Hilfer's derivative and regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting po…
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