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2021 | Book

Existence Results for Impulsive Partial Functional Fractional Differential Equation with State Dependent Delay Read chapter Read first chapter

Nonlinear Analysis: Problems, Applications and Computational Methods

Editors: Dr. Zakia Hammouch, Dr. Hemen Dutta, Dr. Said Melliani, Prof. Dr. Michael Ruzhansky

Publisher: Springer International Publishing

Book Series : Lecture Notes in Networks and Systems

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book is a collection of original research papers as proceedings of the 6th International Congress of the Moroccan Society of Applied Mathematics organized by Sultan Moulay Slimane University, Morocco, during 7th–9th November 2019. It focuses on new problems, applications and computational methods in the field of nonlinear analysis. It includes various topics including fractional differential systems of various types, time-fractional systems, nonlinear Jerk equations, reproducing kernel Hilbert space method, thrombin receptor activation mechanism model, labour force evolution model, nonsmooth vector optimization problems, anisotropic elliptic nonlinear problem, viscous primitive equations of geophysics, quadratic optimal control problem, multi-orthogonal projections and generalized continued fractions.

The conference aimed at fostering cooperation among students, researchers and experts from diverse areas of applied mathematics and related sciences through fruitful deliberations on new research findings. This book is expected to be resourceful for researchers, educators and graduate students interested in applied mathematics and interactions of mathematics with other branches of science and engineering.

Table of Contents

Frontmatter
Existence Results for Impulsive Partial Functional Fractional Differential Equation with State Dependent Delay
Abstract
In this paper, we study the existence of mild solutions of impulsive fractional semilinear differential equation with state dependent delay of order \(0<\alpha <1\). We shall rely on fixed point theorem for the sum of completely continuous and contraction operators due to Burton and Kirk. An example is given to illustrate the theory.
Nadjet Abada, Helima Chahdane, Hadda Hammouche
A Novel Method for Solving Nonlinear Jerk Equations
Abstract
In this article, reproducing kernel method for solving Jerk equations is given. Convergence of the solution is shown. This method is applied to the equation for chosen values of the parameters that seem in the model and some numerical experiments prove that the reproducing kernel method is very effective method.
Ali Akgül, Esra Karatas Akgül
Solving a New Type of Fractional Differential Equation by Reproducing Kernel Method
Abstract
The aim of this work is to get the solutions of the fractional counterpart of a boundary value problem by implementing the reproducing kernel Hilbert space method. Convergence of the solution problem discussed has been shown. The efficiency of the proposed technique is demonstrated by some tables.
Ali Akgül, Esra Karatas Akgül
An Efficient Approach for the Model of Thrombin Receptor Activation Mechanism with Mittag-Leffler Function
Abstract
In the present work, we haired an efficient technique called, \( q \)-homotopy analysis transform method (\( q \)-HATM) in order to find the solution for the model of thrombin receptor activation mechanism (TRAM) and examine the nature of \( q \)-HATM solution with distinct fractional order. The considered model elucidates the TRA mechanism in calcium signalling, and this mechanism plays a vital role in the human body. We defined fractional derivative defined with Atangana-Baleanu (AB) operator and the projected scheme is an amalgamation of Laplace transform with \( q \)-homotopy analysis scheme. For the achieved results, to present the existence and uniqueness we hired the fixed point hypothesis. To validate and illustrate the effectiveness of the considered scheme, we examined the projected model with arbitrary order. The behaviour of the achieved results is captured in terms of plots and also showed the importance of the parameters offered by the considered solution procedure. The attained results illuminate, the projected scheme is easy to employ and more effective in order to analyse the behaviour of fractional order differential systems exemplifying real word problems associated with science and technology.
P. Veeresha, D. G. Prakasha, Zakia Hammouch
Stability Analysis of Bifurcated Limit Cycles in a Labor Force Evolution Model
Abstract
This article focuses on the fluctuations observed in the labor markets. We divide the total population into three categories: employed, unemployed and inactive, then we describe the entry-exit flows between these different categories by two delay differential equations. Our contribution is to compute an indicator for determining the behavior of the model variables, in a neighborhood of the critical delay. Our findings show that the model can undergo a Hopf bifurcation and the bifurcated limit cycles is stable (or unstable), according to the crossing direction of critical delay.
Sanaa ElFadily, Najib Khalid, Abdelilah Kaddar
Existence and Uniqueness Results of Fractional Differential Equations with Fuzzy Data
Abstract
In this paper, we are going to study the existence and uniqueness solutions of fractional differential equations with fuzzy data, involving the fuzzy fractional differential operators of the order \(\gamma \in \mathrm {R}_{+}\). The aid method of successive approximation is provided with adequate conditions for the existence and uniqueness solution. Examples are given to explain the theory obtained.
Atimad Harir, Said Melliani, Lalla Saadia Chadli
Approximate Efficient Solutions of Nonsmooth Vector Optimization Problems via Approximate Vector Variational Inequalities
Abstract
In this work, we demonstrate the connection between the solutions of approximate vector variational inequalities and approximate efficient solutions of corresponding nonsmooth vector optimization problems via generalized approximate invex functions. The underlying variational inequalities are stated under the Clarke’s generalized Jacobian.
Mohsine Jennane, El Mostafa Kalmoun
Existence of Entropy Solutions for Anisotropic Elliptic Nonlinear Problem in Weighted Sobolev Space
Abstract
In this paper, we will study the existence of an entropy solution to the unilateral problem for a class of nonlinear anisotropic elliptic equation, with second term being an element of \(L^1(\varOmega )\). Our technical approach is based on a monotony method and the truncation techniques in the framework of the weighted anisotropic Sobolev space.
Adil Abbassi, Chakir Allalou, Abderrazak Kassidi
Well-Posedness and Stability for the Viscous Primitive Equations of Geophysics in Critical Fourier-Besov-Morrey Spaces
Abstract
In this paper we study the Cauchy problem of the viscous primitive equations of geophysics in critical Fourier-Besov-Morrey spaces. By using the Fourier localization argument and the Littlewood-Paley theory, we prove that the Cauchy problem with Prankster number \(P=1\) is local well-posedness and global well-posedness when the initial data \((u_0,\theta _0)\) are small and we give a stability result for global solutions.
A. Abbassi, C. Allalou, Y. Oulha
Regional Controllability of a Class of Time-Fractional Systems
Abstract
The main purpose of this paper is to develop the concept of regional controllability for an important class of Caputo time-fractional semi-linear systems using the analytical approach, where the dynamic of the considered system is generates by an analytical semigroup. This approach use the fixed point techniques and semigroup theory. Finally, we present some numerical simulations to approve our theoretical results.
Asmae Tajani, Fatima-Zahrae El Alaoui, Ali Boutoulout
Quadratic Optimal Control for Bilinear Systems
Abstract
In this work, we will investigate the quadratic optimal control for bilinear systems. We will first study the existence of a solution for the considered optimal control. Then, we will focus on a special class of bilinear systems for which the quadratic optimal control can be expressed in a feedback law form. The approach relies on the conditions of optimality and linear semi-group theory.
Soufiane Yahyaoui, Mohamed Ouzahra
Regional Observability of Linear Fractional Systems Involving Riemann-Liouville Fractional Derivative
Abstract
In this paper, we study the concept of regional observability, more precisely the regional reconstruction of the initial state of a linear fractional system on a subregion \(\omega \) of the evolution domain \(\varOmega \). We use the Hilbert uniqueness method in order to reconstruct the initial state of the given system, which consists of transforming the reconstruction problem into a solvability one. After presenting an algorithm that allows us to reconstruct the regional initial state, we give, at the end, two successful numerical results, in order to backup our theoretical work, each with a different type of sensor and with a reasonable value of error.
Khalid Zguaid, Fatima Zahrae El Alaoui, Ali Boutoulout
Stability Analysis of Fractional Differential Systems Involving Riemann–Liouville Derivative
Abstract
We introduce the stability notion of the fractional differential systems under Riemann–Liouville time derivative of order \(\alpha \in (0,1)\), evolving on a spatial domain \(\varOmega \). Then, we characterize the asymptotic behavior of the state. Also, we present sufficient and necessary conditions to achieve the exponential stability of this important class of systems. Hence, we study the state stabilization of fractional differential systems by means of decomposition method. Several examples and simulations are given to show the applicability of our presented results.
Hanaa Zitane, Fatima Zahrae El Alaoui, Ali Boutoulout
Deformed Joint Free Distributions of Semicircular Elements Induced by Multi Orthogonal Projections
Abstract
In this paper, we consider (i) how to establish semicircular elements \(\{U_{k}\}_{k=1}^{N}\) induced by N-many mutually orthogonal projections \(\{q_{k}\}_{k=1}^{N}\), for N \(\in \) \((\mathbb {N}\setminus \{1\})\cup \{\infty \},\) and the corresponding free product Banach \(*\)-probability space \(\mathbb {L}_{Q}^{(N)}\) generated by \(\{U_{k}\}_{k=1}^{N}\), (ii) the free-distributional data on \(\mathbb {L}_{Q}^{(N)}\), (iii) certain \(*\)-homomorphisms on \(\mathbb {L}_{Q}^{(N)},\) and (iv) how the \(*\)-homomophisms of (iii) deform the original free-distributional data of (ii).
Ilwoo Cho
Several Explicit and Recurrent Formulas for Determinants of Tridiagonal Matrices via Generalized Continued Fractions
Abstract
In the paper, by the aid of mathematical induction and some properties of determinants, the authors present several explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of finite generalized continued fractions and apply these newly-established formulas to evaluations for determinants of the Sylvester matrix and two Sylvester type matrices.
Feng Qi, Wen Wang, Bai-Ni Guo, Dongkyu Lim
Backmatter
Metadata
Title
Nonlinear Analysis: Problems, Applications and Computational Methods
Editors
Dr. Zakia Hammouch
Dr. Hemen Dutta
Dr. Said Melliani
Prof. Dr. Michael Ruzhansky
Copyright Year
2021
Electronic ISBN
978-3-030-62299-2
Print ISBN
978-3-030-62298-5
DOI
https://doi.org/10.1007/978-3-030-62299-2

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