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Mathematics
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Functional Differential Equations and Applications
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Functional Differential Equations and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (15 November 2020) | Viewed by 10532

Special Issue Editor

Prof. Dr. Alexander Domoshnitsky

E-Mail Website
Guest Editor
Department of Mathematics, Ariel University, Ariel 40700, Israel
Interests: functional differential equations; general theory; boundary value problems; positivity of solutions; nonoscillation; distances between adjacent zeros of solutions; distribution of zeros; Sturm’s theorem; stability; feedback control; delay differential equations; integro-differential equations; impulsive equations; applications of equations with memory in technology and medicine
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue, “Functional Differential Equations and Applications”, will be destined for the publication of high-quality mathematical papers in the area of functional differential equations. Emphasis are placed on developments in the theory of delay differential, integro-differential, impulsive differential and difference equations and their applications. Among possible topics of the papers we can note, for example, the following: Various boundary value problems, positivity/negativity of their solutions, Green’s functions and their properties, existence and uniqueness solutions of nonlinear boundary value problems, optimization and control theory, stability theory, oscillation and non-oscillation, variational problems, the use of functional differential equations in technology, economics, biology and medicine.

Prof. Dr. Alexander Domoshnitsky
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • functional differential equations
  • oscillation/nonoscillation
  • feedback control
  • stability
  • boundary value problems

Published Papers (5 papers)

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Research

16 pages, 3499 KiB  
Article
Using Diffusion Map for Visual Navigation of a Ground Robot
by Oleg Kupervasser, Hennadii Kutomanov, Michael Mushaelov and Roman Yavich
Mathematics 2020, 8(12), 2175; https://0-doi-org.brum.beds.ac.uk/10.3390/math8122175 - 06 Dec 2020
Cited by 1 | Viewed by 1639
Abstract
This paper presents the visual navigation method for determining the position and orientation of a ground robot using a diffusion map of robot images (obtained from a camera in an upper position—e.g., tower, drone) and for investigating robot stability with respect to desirable [...] Read more.
This paper presents the visual navigation method for determining the position and orientation of a ground robot using a diffusion map of robot images (obtained from a camera in an upper position—e.g., tower, drone) and for investigating robot stability with respect to desirable paths and control with time delay. The time delay appears because of image processing for visual navigation. We consider a diffusion map as a possible alternative to the currently popular deep learning, comparing the possibilities of these two methods for visual navigation of ground robots. The diffusion map projects an image (described by a point in multidimensional space) to a low-dimensional manifold preserving the mutual relationships between the data. We find the ground robot’s position and orientation as a function of coordinates of the robot image on the low-dimensional manifold obtained from the diffusion map. We compare these coordinates with coordinates obtained from deep learning. The algorithm has higher accuracy and is not sensitive to changes in lighting, the appearance of external moving objects, and other phenomena. However, the diffusion map needs a larger calculation time than deep learning. We consider possible future steps for reducing this calculation time. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications)
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13 pages, 6144 KiB  
Article
Using Deep Learning for Visual Navigation of Drone with Respect to 3D Ground Objects
by Oleg Kupervasser, Hennadii Kutomanov, Ori Levi, Vladislav Pukshansky and Roman Yavich
Mathematics 2020, 8(12), 2140; https://0-doi-org.brum.beds.ac.uk/10.3390/math8122140 - 01 Dec 2020
Cited by 4 | Viewed by 1794
Abstract
In the paper, visual navigation of a drone is considered. The drone navigation problem consists of two parts. The first part is finding the real position and orientation of the drone. The second part is finding the difference between desirable and real position [...] Read more.
In the paper, visual navigation of a drone is considered. The drone navigation problem consists of two parts. The first part is finding the real position and orientation of the drone. The second part is finding the difference between desirable and real position and orientation of the drone and creation of the correspondent control signal for decreasing the difference. For the first part of the drone navigation problem, the paper presents a method for determining the coordinates of the drone camera with respect to known three-dimensional (3D) ground objects using deep learning. The algorithm has two stages. It causes the algorithm to be easy for interpretation by artificial neural network (ANN) and consequently increases its accuracy. At the first stage, we use the first ANN to find coordinates of the object origin projection. At the second stage, we use the second ANN to find the drone camera position and orientation. The algorithm has high accuracy (these errors were found for the validation set of images as differences between positions and orientations, obtained from a pretrained artificial neural network, and known positions and orientations), it is not sensitive to interference associated with changes in lighting, the appearance of external moving objects and the other phenomena where other methods of visual navigation are not effective. For the second part of the drone navigation problem, the paper presents a method for stabilization of drone flight controlled by autopilot with time delay. Indeed, image processing for navigation demands a lot of time and results in a time delay. However, the proposed method allows to get stable control in the presence of this time delay. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications)
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12 pages, 232 KiB  
Article
The Local Representation Formula of Solution for the Perturbed Controlled Differential Equation with Delay and Discontinuous Initial Condition
by A. Nachaoui, T. Shavadze and T. Tadumadze
Mathematics 2020, 8(10), 1845; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101845 - 20 Oct 2020
Cited by 2 | Viewed by 1480
Abstract
For the perturbed controlled nonlinear delay differential equation with the discontinuous initial condition, a formula of the analytic representation of solution is proved in the left neighborhood of the endpoint of the main interval. In the formula, the effects of perturbations of the [...] Read more.
For the perturbed controlled nonlinear delay differential equation with the discontinuous initial condition, a formula of the analytic representation of solution is proved in the left neighborhood of the endpoint of the main interval. In the formula, the effects of perturbations of the delay parameter, the initial vector, the initial and control functions are detected. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications)
29 pages, 837 KiB  
Article
On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations
by Clement McCalla
Mathematics 2020, 8(9), 1418; https://0-doi-org.brum.beds.ac.uk/10.3390/math8091418 - 24 Aug 2020
Viewed by 2317
Abstract
This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of [...] Read more.
This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications)
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13 pages, 298 KiB  
Article
Qualitative Analysis of Multi-Terms Fractional Order Delay Differential Equations via the Topological Degree Theory
by Muhammad Sher, Kamal Shah, Michal Fečkan and Rahmat Ali Khan
Mathematics 2020, 8(2), 218; https://0-doi-org.brum.beds.ac.uk/10.3390/math8020218 - 08 Feb 2020
Cited by 26 | Viewed by 2365
Abstract
With the help of the topological degree theory in this manuscript, we develop qualitative theory for a class of multi-terms fractional order differential equations (FODEs) with proportional delay using the Caputo derivative. In the same line, we will also study various forms of [...] Read more.
With the help of the topological degree theory in this manuscript, we develop qualitative theory for a class of multi-terms fractional order differential equations (FODEs) with proportional delay using the Caputo derivative. In the same line, we will also study various forms of Ulam stability results. To clarify our theocratical analysis, we provide three different pertinent examples. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications)
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