Fractional Differential Equations

In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order. 

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Artificial neural networks: a practical review of applications involving fractional calculus

By imposing some conditions on the nonlinear term f , we construct a lower solution and an upper solution to prove the existence of a solution for a type of nonlinear third-order nonlocal boundary value problem. Our main tools are the upper and the lower solution method and the Schauder fixed point theorem. The results are illustrated by an example. MSC 2010. 34B10, 34B15.

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Existence of solution for third-order three-point boundary value problem

The aim of this paper is to investigate the existence, uniqueness and continuous dependence of solutions for a class of third order iterative differential equations with integral boundary conditions. The method applied here is based on Schauder’s fixed point theorem. The main idea consists to convert the considered equation into an integral one before using the fixed point theorem. Moreover, an example is given to illustrate our main results.

Bounded positive solutions of an iterative three-point boundary-value problem with integral boundary condtions

In this paper we consider the Green function for a boundary value problem of generic order. For a specific case, the Leray–Schauder form of the fixed point theorem has been used to prove the existence of a solution for this particular equation. Our theoretical approach generalizes, extends, complements, and enriches several results in the existing literature.

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Tenth order boundary value problem solution existence by fixed point theorem

In this paper, the existence of positive solutions for a nonlinear fourth-order two-point boundary value problem with integral condition is investigated. By using Krasnoselskii's fixed point theorem on cones, sufficient conditions for the existence of at least one positive solutions are obtained.

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Positive solutions of a nonlinear fourth-order integral boundary value problem

In this paper, we study the boundary value problem of a class of fractional differential equations involving the Riemann-Liouville fractional derivative with nonlocal integral boundary conditions. To establish the existence results for the given problems, we use the properties of the Green’s function and the monotone iteration technique, one shows the existence of positive solutions and constructs two successively iterative sequences to approximate the solutions. The results are illustrated with an example.

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Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions

We are concerned with positive solutions of two types of nonlinear elliptic boundary value problems (BVPs). We present conditions for existence, uniqueness and multiple positive solutions of a first type of elliptic BVPs. For a second type of elliptic BVPs, we obtain conditions for existence and uniqueness of positive global solutions. We employ mathematical tools including strictly upper (SU) and strictly lower (SL) solutions, iterative sequence method and Amann theorem. We present our research findings in new original theorems. Finally, we summarize and indicate areas of future study and possible applications of the research work.

New Positive Solutions of Nonlinear Elliptic PDEs

This paper is concerned with the following fourth-order three-point boundary value problem BVP \[ u^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right)\right),\quad t\in\left[0,1\right], \] \[ u'\left(0\right)=u''\left(0\right)=u\left(1\right)=0,\;u'''\left(\eta\right)+\alpha u\left(0\right)=0, \] where $f\in C\left(\left[0,1\right]\times\left[0,+\infty\right),\left[0,+\infty\right)\right)$ , $\alpha\in\left[0,6\right)$ and $\eta\in\left[\frac{2}{3},1\right)$. Although corresponding Green\textquoteright s function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions on $f$ by applying iterative method. An example is also given to illustrate the main results.

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Existence and Iteration of Monotone Positive Solution for a Fourth-Order Nonlinear Boundary Value Problem

In this paper, we study the existence of at least one positive solution for a nonlinear third-order two-point boundary value problem with integral condition. By employing the Krasnoselskii's fixed point theorem on cones, the existence results of the problem are established.

Slimane Benaicha

Papers

Positive solutions of a nonlinear fourth-order integral boundary value problem

Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions

New Positive Solutions of Nonlinear Elliptic PDEs

Existence and Iteration of Monotone Positive Solution for a Fourth-Order Nonlinear Boundary Value Problem

Existence of Positive Solutions for a Nonlinear Third-order Integral Boundary Value Problem