Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.     

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.     

Exact solution of a class of fractional integro‐differential equations with the weakly singular kernel based on a new fractional reproducing kernel space

In this paper, we construct a new fractional weighted reproducing kernel space, which is the minimum space containing the exact solution. The closed form of the reproducing kernel is obtained. Using this fractional reproducing kernel space, a class of fractional integro‐differential equations with a weakly singular kernel is solved. The error estimation is given. The final numerical experiments demonstrate the correctness of the theory and the effectiveness of the method.     

抽象空间缓增分数阶微分方程的吸引性

研究了抽象空间中缓增分数阶微分方程解的吸引性.建立了Cauchy问题存在全局吸引解的充分条件.揭示了缓增分数阶导数求解分数微分方程解的特征.     

分数k-因子临界图的条件

设G是-个连通简单无向图,如果删去G的任意k个项点后的图有分数完美匹配,则称G是分数k-因子临界图．给出了G是分数k-因子临界图的韧度充分条件与度和充分条件,这些条件中的界是可达的,并给出G是分数k-因子临界图的一个关于分数匹配数的充分必要条件．     

关于分数k-因子临界图与分数k-可扩图的若干结果

一个简单图G, 如果对于V(G)的任意k元子集S, 子图G-S都包含分数完美匹配, 那么称G为分数k-因子临界图. 如果图G的每个k-匹配M都包含在一个分数完美匹配中, 那么称图G为分数k-可扩图. 给出一个图是分数k-因子临界图和分数k-可扩图的充分条件, 并给出一个图是分数k-因子临界图的充分必要条件.     

New approach to a generalized fractional integral

The paper presents a new fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form. Conditions are given for such a fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. We give a general definition of the fractional derivatives and give some examples.     

Pontryagin maximum principle for fractional ordinary optimal control problems

In the paper, fractional systems with Riemann–Liouville derivatives are studied. A theorem on the existence and uniqueness of a solution of a fractional ordinary Cauchy problem is given. Next, the Pontryagin maximum principle for nonlinear fractional control systems with a nonlinear integral performance index is proved. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximate analytical solution of two-dimensional space-time fractional diffusion equation

This work presents an iterative scheme for the numerical solution of the space-time fractional two-dimensional advection–reaction–diffusion equation applying homotopy perturbation with Laplace transform using Caputo fractional-order derivatives. The solution obtained is beneficial and significant to analyze the modeling of superdiffusive systems and subdiffusive system, anomalous diffusion, transport process in porous media. This iterative technique presents the combination of homotopy perturbation technique, and Laplace transforms with He's polynomials, which can further be applied to numerous linear/nonlinear two-dimensional fractional models to computes the approximate analytical solution. In the present method, the nonlinearity can be tackle by He's polynomials. The salient features of the present scientific work are the pictorial presentations of the approximate numerical solution of the two-dimensional fractional advection–reaction–diffusion equation for different particular cases of fractional order and showcasing of the damping effect of reaction terms on the nature of probability density function of the considered two-dimensional nonlinear mathematical models for various situations.     

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.     

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.     

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.     

Dual characterization of fractional capacity via solution of fractional p-Laplace equation

In terms of weak solutions of the fractional p-Laplace equation with measure data, this paper offers a dual characterization for the fractional Sobolev capacity on bounded domain. In addition, two further results are given: one is an equivalent estimate for the fractional Sobolev capacity; the other is the removability of sets of zero capacity and its relation to solutions of the fractional p-Laplace equation.     

Solution of fractional order heat equation via triple Laplace transform in 2 dimensions

In this article, we introduce the triple Laplace transform for the solution of a class of fractional order partial differential equations. As a consequence, fractional order homogeneous heat equation in 2 dimensions is investigated in detail. The corresponding solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. Numerical plots to the concerned solutions are provided to demonstrate our results.     

On a semilinear double fractional heat equation driven by fractional Brownian sheet

In this paper, we consider the stochastic heat equation of the form $$\frac{\partial u}{\partial t}=(\Delta_\alpha+\Delta_\beta)u+\frac{\partial f}{\partial x}(t,x,u)+\frac{\partial^2W}{\partial t\partial x},$$ where $1<\beta<\alpha< 2$, $W(t,x)$ is a fractional Brownian sheet, $\Delta_\theta:=-(-\Delta)^{\theta/2}$ denotes the fractional Lapalacian operator and $f:[0,T]\times \mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear measurable function. We introduce the existence, uniqueness and H\"older regularity of the solution. As a related question, we consider also a large deviation principle associated with the above equation with a small perturbation via an equivalence relationship between Laplace principle and large deviation principle.     

Blow-up of solutions for a time-space fractional evolution system

In this paper, firstly, we study the local existence and uniqueness of mild solutions for fractional evolution systems with nonlocal in time nonlinearity. Then, we claim that such a mild solution is weak solution of this system. Finally, we prove a blow-up result under some conditions.     

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 the fractional derivatives of radial basis functions: Theories and applications

The paper provides the fractional integrals and derivatives of the Riemann‐Liouville and Caputo type for the five kinds of radial basis functions, including the Powers, Gaussian, Multiquadric, Matérn, and Thin‐plate splines, in one dimension. It allows to use high‐order numerical methods for solving fractional differential equations. The results are tested by solving two test problems. The first test case focuses on the discretization of the fractional differential operator while the second considers the solution of a fractional order differential equation.     

Integral representation of solutions of fractional system with distributed delays

The aim of the present paper is to obtain an integral representation of the solution of the Cauchy problem with discontinuous and continuous initial conditions for linear fractional differential system with Caputo-type derivatives and distributed delay. The obtained results are new even in the particular case of fractional system with constant delays.