B. Yu. Irgashev
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
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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.
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.
This article contains a new discussion for the generalized fractional Cauchy-type problem involving Hilfer-Katugampola-type fractional derivative. We study an existence and continuation of its solution. Firstly, we establish a new theorems of local existence. Then, we deduce a continuation theorems for a general fractional differential equations. By applying continuation theorems deduced in this article, we present several global existence results. Moreover, the examples are given to illustrate our main results.
In this paper, we consider the Cauchy-type problem (1.1) involving Hilfer-Hadamard-type fractional derivative for a nonlinear fractional differential equation. We prove an equivalence between the Cauchy-type problem (1.1) and Volterra integral equation(VIE), existence, and uniqueness. We present a slight generalization for the Gronwall inequality which was used in studying the continuous dependence of a solution for the Cauchy-type problem (1.2).
For the following semilinear equation with Hilfer- Hadamard fractional derivative \begin{equation*} \mathcal{D}^{α_1,β}_{a^+} u-Δ\mathcal{D}^{α_2,β}_{a^+} u-Δu =\vert u\vert^p, \qquad t>a>0, \qquad x\inΩ, \end{equation*} where $Ω\subset \mathbb{R}^N$ $(N\geqslant 1)$, $p>1$, $0<α_{2}<α_{1}<1$ and $0<β<1$. $\mathcal{D}^{α_i,β}_{a^+}$ $(i=1,2)$ is the Hilfer- Hadamard fractional derivative of order $α_i$ and of type $β$, we establish the necessary conditions for the existence of global solutions.
In this paper, we used some theorems of fixed point for studying the results of Existence and Uniqueness For Hilfer-Hadamard-Type Fractional Differential Equations, \[_{H}D^{α,β}x(t)+f(t,x(t))=0, ~~~~~~ on~~the~~ interval~~ J:=(1,e]\] with Boundary Value Problems \[x(1+ε)=0,~~~~~~~~~~_{H}D^{1,1}x(e)=ν~_{H}D^{1,1}x(ζ)\]
This paper is committed to establishing the assumptions essential for the existence and uniqueness results of a fractional functional integrodifferential equation (FFIDE) having a derivative of generalized Hilfer type. Using the Picard operator method, and Banach fixed point theorem, we obtain the existence and uniqueness solution to the proposed problem. Along with this, the Ulam-Hyers Mittag-Leffler (UHML) stability is discussed via Pachpatte's inequality. For supporting our results, an illustrative example will be introduced.
This paper deals with the existence and uniqueness of solutions for a nonlinear boundary value problem involving a sequential $ψ$-Hilfer fractional integro-differential equations with nonlocal boundary conditions. The existence and uniqueness of solutions are established for the considered problem by using the Banach contraction principle, Sadovski's fixed point theorem, and Krasnoselskii-Schaefer fixed point theorem due to Burton and Kirk. In addition, the Ulam-Hyers stability of solutions is discussed. Finally, the obtained results are illustrated by examples.
In this paper, we used some theorems of fixed point for studying the results of existence and uniqueness for Hilfer-Hadamard-Type fractional differential equations, \[_{H}D^{α,β}x(t)+f(t,x(t))=0, \hbox{ on the interval } J:=(1,e]\] with nonlinear boundary value problems \[x(1+ε)=\sum_{i=1}^{n-2}ν_{i}x(ζ_{i}), \quad\quad\quad~_{H}D^{1,1}x(e) = \sum_{i=1}^{n-2} σ_i~_{H} D^{1,1}x(ζ_{i})\]
Non-local boundary value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional differential operators
本文通过特征函数方法研究矩形域上含双阶Hilfer分数阶导数的混合型方程的非局部边值问题，证明了解的唯一性和存在性。
This paper deals with the existence and uniqueness of solution for a coupled system of Hilfer fractional Langevin equation with non local integral boundary value conditions. The novelty of this work is that it is more general than the works based on the derivative of Caputo and Riemann-Liouville, because when $β=0$ we find the Riemann-Liouville fractional derivative and when $β=1$ we find the Caputo fractional derivative. Initially, we give some definitions and notions that will be used throughout the work, after that we will establish the existence and uniqueness results by employing the fixe…
Direct and inverse source problems for two-term time-fractional diffusion equation with Hilfer derivative
本文研究矩形域中具有两项广义分数阶导数（Hilfer导数）的扩散方程的正问题和反源问题，通过谱展开方法将问题转化为两项分数阶微分方程，并利用正弦函数和二元多项Mittag-Leffler函数表示解，在给定数据条件下证明无穷级数的一致收敛性。
In this paper, we consider the nonlinear $Ψ$-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of results. The acquired results are extended to the nonlocal $Ψ$-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we procured. Further, examples are provided in support of the results we got.
In this paper we study the approximate controllability of fractional partial differential equations associated with the so-called Hilfer type time fractional derivative and a non-negative selfadjoint operator $A$ with a compact resolvent on $L^2(Ω)$, where $Ω\subset\RR^N$ ($N\geq 1$) is an open set. More precisely, we show that if $0\leν\le 1$, $1<μ\le 2$ and $Ω\subset\RR^N$ is an open set, then the system \begin{equation*} \begin{cases} \D^{μ,ν}_tu+Au=fχ_ω\;\;&\mbox{ in }\;Ω\times(0,T),\\ (I_t^{(1-ν)(2-μ)}u)(\cdot,0)=u_0 &\mbox{ in }\;Ω,\\ (\partial_tI_t^{(1-ν)(2-μ)}u)(\cdot,0)=u_1 &\mbox{ in…
This paper is concerned with the existence and uniqueness, and Ulam--Hyers stabilities of solutions of nonlinear impulsive $φ$--Hilfer fractional differential equations. Further, we investigate the dependence of the solution on the initial conditions, order of derivative and the functions involved in the equations. The outcomes are acquired in the space of weighted piecewise continuous functions by means of fixed point theorems and the generalized version of Gronwall inequality.
In this paper we consider space-time fractional telegraph equations, where the time derivatives are intended in the sense of Hilfer and Hadamard while the space fractional derivatives are meant in the sense of Riesz-Feller. We provide the Fourier transforms of the solutions of some Cauchy problems for these fractional equations. Probabilistic interpretations of some specific cases are also provided.
In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam--Hyers--Mittag--Leffler stability results for impulsive implicit $Ψ$--Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam--Hyers and generalized Ulam--Hyers stability are the specific cases of Ulam--Hyers--Mittag--Leffler stability. Extended version of Gronwall inequality, abstract Gronwall lemma and Picard operator theory are the primary devices in our investigation. We give an example to illustrate the obtained results.
In the current paper, we derive the comparison results for the homogeneous and non-homogeneous linear initial value problem (IVP) for $Ψ$-Hilfer fractional differential equations. In the presence of upper and lower solutions, the obtained comparison results and the location of roots theorem utilized to prove the existence and uniqueness of the solution for the linear $Ψ$-Hilfer boundary value problem (BVP) through the linear non-homogeneous $Ψ$-Hilfer IVP. Assuming the existence of lower solution $w_0 $ and upper solution $z_0 $, we establish the existence of minimal and maximal solutions for …
We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator $A_B$ with a compact resolvent on $L^2(Ω)$, where $Ω\subset\mathbb{R}^N$ ($N\ge 1$) is a bounded open set. More precisely, we show that if $0\leν\le 1$, $0<μ\le 1$ and $Ω\subset\mathbb R^N$ is a bounded open set, then the system $$\mathbb D_t^{μ,ν} u+A_Bu=f|_ω\;\; \mbox{ in }\; Ω\times (0,T),\,\, (\mathbb I_t^{(1-ν)(1-μ)}u)(\cdot,0)=u_0 \mbox{ in }\;Ω,$$ is approximately…
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…
Optimal Control and Approximate controllability of fractional semilinear differential inclusion involving $ψ$- Hilfer fractional derivatives
本文研究含 $ψ$-Hilfer 分数阶导数的半线性微分包含，利用紧半群、线性系统近似能控性及对偶映射构造控制，克服自反 Banach 空间中的凸性难题，并给出最优控制与示例验证。
In this paper, we investigate the existence and uniqueness of mild and strong solutions of fractional semilinear evolution equations in the Hilfer sense, by means of Banach fixed point theorem and the Gronwall inequality.
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.
Fractional Dzherbashian-Nersesian operator is considered and three famous fractional order derivatives namely Riemann-Liouville, Caputo and Hilfer derivatives are shown to be special cases of the earlier one. The expression for Laplace transform of fractional Dzherbashian-Nersesian operator is constructed. Inverse problems of recovering space dependent and time dependent source terms of a time fractional diffusion equation with involution and involving fractional Dzherbashian-Nersesian operator are considered. The results on existence and uniqueness for the solutions of inverse problems are es…
In this paper we derive a sufficient condition for the existence of a unique solution of a Cauchy type q-fractional problem (involving the fractional q-derivative of Riemann-Liouville type) for some nonlinear differential equations. The key technique is to first prove that this Cauchy type q-fractional problem is equivalent to a corresponding Volterra q-integral equation. Moreover, we define the $q$-analogue of the Hilfer fractional derivative or composite fractional derivative operator and prove some similar new equivalence, existence and uniqueness results as above. Finally, some examples ar…
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。
This is a first version of a paper concerning abstract evolution equation with fractional time derivatives. Maximal regularity results in spaces of continuous and Hoelder continuous functions are described.
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.
A numerical scheme for solving fractional initial value problems involving the Atangana-Baleanu fractional derivative is presented. Some examples for the proposed method are included, both for equations and systems of fractional initial value problems.
We study a nonlocal nonlinear parabolic problem with a fractional time derivative. We prove a Krylov-Safonov type result; mainly, we prove Holder regularity of solutions. Our estimates remain uniform as the order of the fractional time derivative approaches 1.
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。
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.
We introduce complex order fractional derivatives in models that describe viscoelastic materials. This can not be carried out unrestrictedly, and therefore we derive, for the first time, real valued compatibility constraints, as well as physical constraints that lead to acceptable models. As a result, we introduce a new form of complex order fractional derivative. Also, we consider a fractional differential equation with complex derivatives, and study its solvability. Results obtained for stress relaxation and creep are illustrated by several numerical examples.
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.
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.
We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi-Nash-Moser Holder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.
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.
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function phi(x,t) which is a nonlinear function overning reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. T…
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 …
In this paper we introduce Lipschitz spaces with respect to the Gaussian measure, and study the boundedness of the fractional integral and fractional derivative operators on them.The methods are general enough to provide alternative proofs of the ones given in the classical case and moreover can be extended to the case of Laguerre and Jacobi expansions too
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…
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.
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