Solvability of the boundary-value problem for a mixed equation involving hyper-Bessel fractional differential operator and bi-ordinal Hilfer fractional derivative

In a rectangular domain, a boundary-value problem is considered for a mixed equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. By using the method of separation of variables a unique solvability of the considered problem has been established. Moreover, we have found the explicit solution of initial-boundary problems for the heat equation with the regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting point.     

Numerical solution of two-dimensional time fractional cable equation with Mittag-Leffler kernel

The main motive of this article is to study the recently developed Atangana-Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag-Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two-dimensional cable equation in which time-fractional derivative is of Mittag-Leffler kernel type. First, we derive an approximation formula of the fractional-order ABC derivative of a function t^k using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two-dimensional model of the time-fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two-dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag-Leffler fractional derivative.     

On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.     

On a final value problem for fractional reaction-diffusion equation with Riemann-Liouville fractional derivative

In this paper, we study a backward problem for a fractional diffusion equation with nonlinear source in a bounded domain. By applying the properties of Mittag-Leffler functions and Banach fixed point theorem, we establish some results above the existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space. Moreover, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given, and the convergence rate between the regularized solution and the exact solution is also obtained.     

New bilingualism model based on fractional operators with Mittag-Leffler kernel

A model of one unilingual component and one bilingual component of a population is studied using fractional derivatives with Mittag-Leffler kernel in Liouville-Caputo sense. Numerical simulations were obtained using iterative schemes. We studied in detail the existence and uniqueness of the solutions. Numerical simulations for different values of the fractional order were obtained.     

Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative

In this paper, we investigate the existence of solutions of the periodic boundary value problem for nonlinear impulsive fractional differential equation involving Riemann-Liouville sequential fractional derivative by using monotone iterative method. An example is presented to illustrate our main result.     

On a jump-type stochastic fractional partial differential equation with fractional noises

We study a class of stochastic fractional partial differential equations of order α>1$\alpha >1$ driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions.     

On a backward problem for fractional diffusion equation with Riemann-Liouville derivative

In the present paper, we study the initial inverse problem (backward problem) for a two-dimensional fractional differential equation with Riemann-Liouville derivative. Our model is considered in the random noise of the given data. We show that our problem is not well-posed in the sense of Hadamard. A truncated method is used to construct an approximate function for the solution (called the regularized solution). Furthermore, the error estimate of the regularized solution in L² and H^τ norms is considered and illustrated by numerical example.     

Space-time fractional derivative operators

Evolution equations for anomalous diffusion employ fractional derivatives in space and time. Linkage between the space-time variables leads to a new type of fractional derivative operator. This paper develops the mathematical foundations of those operators.

     

On the properties of a Cauchy-type problem for an abstract differential equation with fractional derivatives

We study the relationship between the solutions of abstract differential equations with fractional derivatives and their stability with respect to the perturbation by a bounded operator. Besides, we obtain representations for the solution of an inhomogeneous equation and for an equation containing a fractional power of the generator of a cosine operator function.     

具有逐项分数阶导数的微分方程边值问题解的存在性

研究了一类具有逐项分数阶导数的微分方程边值问题.对参数的各种取值情况进行了全面的分析,运用Banach压缩映射原理和Schauder不动点定理,得到并证明了边值问题解的存在性定理.最后,给出了两个例子来证明结论有效.     

Oscillatory behavior of a fractional partial differential equation

In this paper, a fractional partial differential equation subject to the Robin boundary condition is considered. Based on the properties of Riemann-Liouville fractional derivative and a generalized Riccati technique, we obtained sufficient conditions for oscillation of the solutions of such equation. Examples are given to illustrate the main results.     

Regularization of a terminal value problem for time fractional diffusion equation

In this article, we study an inverse problem with inhomogeneous source to determine an initial data from the time fractional diffusion equation. In general, this problem is ill-posed in the sense of Hadamard, so the quasi-boundary value method is proposed to solve the problem. In the theoretical results, we propose a priori and a posteriori parameter choice rules and analyze them. Finally, two numerical results in the one-dimensional and two-dimensional case show the evidence of the used regularization method.     

分数阶变分迭代法处理分数阶类Boussinesq方程

分数阶变分迭代法（FVIM）是一种处理分数阶微分方程的有效工具.用分数阶变分迭代法求解了时间分数阶类Boussinesq方程,并且作为一种特殊情况,得到了类Boussinesq方程B（2.2）的单孤子解.     

Well-posedness of equations with fractional derivative

We study the well-posedness of the equations with fractional derivative D^αu（t） = Au（t） ＋ f（t）,0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.     

Nonexistence for the Laplace equation with a dynamical boundary condition of fractional type

We consider the Laplace equation in ? ^d?1 × ?⁺ × (0,+∞) with a dynamical nonlinear boundary condition of order between 1 and 2. Namely, the boundary condition is a fractional differential inequality involving derivatives of noninteger order as well as a nonlinear source. Nonexistence results and necessary conditions are established for local and global existence. In particular, we show that the critical exponent depends only on the fractional derivative of the least order.     

Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

A class of inverse problems for restoring the right‐hand side of a fractional heat equation with involution is considered. The results on existence and uniqueness of solutions of these problems are presented.     

On the local fractional derivative

We present the necessary conditions for the existence of the Kolwankar-Gangal local fractional derivatives (KG-LFD) and introduce more general but weaker notions of LFDs by using limits of certain integral averages of the difference-quotient. By applying classical results due to Stein and Zygmund (1965) [16] we show that the KG-LFD is almost everywhere zero in any given intervals. We generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing.