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Mathematics
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Recent Investigations of Differential and Fractional Equations and Inclusions
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Recent Investigations of Differential and Fractional Equations and Inclusions

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

Deadline for manuscript submissions: closed (31 October 2020) | Viewed by 20600

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Snezhana Hristova

E-Mail Website
Guest Editor
Department of Applied Mathematics and Modeling, University of Plovdiv, Paisii Hilendarski, 4002 Plovdiv, Bulgaria
Interests: differential equations; delays; impulses; difference equations; fractional differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

During the past decades the subject of the calculus of integrals and derivatives of any arbitrary real or complex order has gained considerable popularity and importance. This is mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering.

This Special Issue invites papers that focus on recent and novel developments in the theory of any types of differential and fractional differential equations and inclusions, especially on analytical and numerical results for fractional ordinary and partial differential equations.

This Special Issue will accept high-quality papers containing original research results and survey articles of exceptional merit in the following fields:

  • Differential equations and inclusions;
  • Differential equations and inclusions with impulses;
  • Delay differential equations;
  • Fuzzy differential and integral equations;
  • Fractional differential equations and inclusions;
  • Difference equations;
  • Discrete fractional equations;
  • Dynamical models with differential, fractional, difference, or fuzzy equations.

Prof. Dr. Snezhana Hristova
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

  • differential equations
  • differential inclusions
  • fuzzy differential and integral equations
  • fractional differential equations
  • difference equations
  • dynamical models

Published Papers (12 papers)

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Research

14 pages, 2944 KiB  
Article
The Effect of Fractional Time Derivative on Two-Dimension Porous Materials Due to Pulse Heat Flux
by Tareq Saeed and Ibrahim A. Abbas
Mathematics 2021, 9(3), 207; https://doi.org/10.3390/math9030207 - 20 Jan 2021
Cited by 9 | Viewed by 1736
Abstract
In the present article, the generalized thermoelastic wave model with and without energy dissipation under fractional time derivative is used to study the physical field in porous two-dimensional media. By applying the Fourier-Laplace transforms and eigenvalues scheme, the physical quantities are presented analytically. [...] Read more.
In the present article, the generalized thermoelastic wave model with and without energy dissipation under fractional time derivative is used to study the physical field in porous two-dimensional media. By applying the Fourier-Laplace transforms and eigenvalues scheme, the physical quantities are presented analytically. The surface is shocked by heating (pulsed heat flow problem) and application of free traction on its outer surface (mechanical conditions) by the process of temperature transport (diffusion) to observe the full analytical solutions of the main physical fields. The magnesium (Mg) material is used to make the simulations and obtain numerical outcomes. The basic physical field quantities are graphed and discussed. Comparisons are made in the results obtained under the strong (SC), the weak (WC) and the normal (NC) conductivities. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
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20 pages, 351 KiB  
Article
Dissipativity of Fractional Navier–Stokes Equations with Variable Delay
by Lin F. Liu and Juan J. Nieto
Mathematics 2020, 8(11), 2037; https://doi.org/10.3390/math8112037 - 16 Nov 2020
Cited by 2 | Viewed by 1542
Abstract
We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity [...] Read more.
We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problem. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
14 pages, 302 KiB  
Article
On Caputo–Riemann–Liouville Type Fractional Integro-Differential Equations with Multi-Point Sub-Strip Boundary Conditions
by Ahmed Alsaedi, Amjad F. Albideewi, Sotiris K. Ntouyas and Bashir Ahmad
Mathematics 2020, 8(11), 1899; https://doi.org/10.3390/math8111899 - 31 Oct 2020
Cited by 4 | Viewed by 1344
Abstract
In this paper, we derive existence and uniqueness results for a nonlinear Caputo–Riemann–Liouville type fractional integro-differential boundary value problem with multi-point sub-strip boundary conditions, via Banach and Krasnosel’skii⏝’s fixed point theorems. Examples are included for the illustration of the obtained [...] Read more.
In this paper, we derive existence and uniqueness results for a nonlinear Caputo–Riemann–Liouville type fractional integro-differential boundary value problem with multi-point sub-strip boundary conditions, via Banach and Krasnosel’skii⏝’s fixed point theorems. Examples are included for the illustration of the obtained results. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
18 pages, 352 KiB  
Article
Existence of Positive Solutions for a System of Singular Fractional Boundary Value Problems with p-Laplacian Operators
by Ahmed Alsaedi, Rodica Luca and Bashir Ahmad
Mathematics 2020, 8(11), 1890; https://doi.org/10.3390/math8111890 - 31 Oct 2020
Cited by 13 | Viewed by 1740
Abstract
We investigate the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with singular nonnegative nonlinearities and p-Laplacian operators, subject to nonlocal boundary conditions which contain fractional derivatives and Riemann–Stieltjes integrals. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
27 pages, 418 KiB  
Article
Optimal Feedback Control Problem for the Fractional Voigt-α Model
by Victor Zvyagin, Andrey Zvyagin and Anastasiia Ustiuzhaninova
Mathematics 2020, 8(7), 1197; https://doi.org/10.3390/math8071197 - 21 Jul 2020
Cited by 7 | Viewed by 1715
Abstract
The study of the existence of an optimal feedback control problem for the initial-boundary value problem that describes the motion of the fractional Voigt- α model of a viscoelastic medium is investigated in this paper. In this model, the Voigt rheological relation is [...] Read more.
The study of the existence of an optimal feedback control problem for the initial-boundary value problem that describes the motion of the fractional Voigt- α model of a viscoelastic medium is investigated in this paper. In this model, the Voigt rheological relation is considered with the left-side fractional Riemann-Liouville derivative, which allows to take into account the memory of the medium. Also in this model, the memory is considered along the trajectory of the motion of fluid particles, determined by the velocity field. Due to the insufficient smoothness of the velocity field and, as a consequence, the impossibility of uniquely determining the trajectory for the velocity field for any initial value, a weak solution to the problem under study is introduced using regular Lagrangian flows. Based on the approximation-topological approach to the study of fluid dynamic problems, the existence of an optimal solution that gives a minimum to a given cost functional is proved. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
19 pages, 388 KiB  
Article
Generalized Fixed Point Results with Application to Nonlinear Fractional Differential Equations
by Hanadi Zahed, Hoda A. Fouad, Snezhana Hristova and Jamshaid Ahmad
Mathematics 2020, 8(7), 1168; https://doi.org/10.3390/math8071168 - 16 Jul 2020
Cited by 5 | Viewed by 1625
Abstract
The main objective of this paper is to introduce the ( α , β )-type ϑ -contraction, ( α , β )-type rational ϑ -contraction, and cyclic ( α - ϑ ) contraction. Based on these definitions we prove fixed point theorems in [...] Read more.
The main objective of this paper is to introduce the ( α , β )-type ϑ -contraction, ( α , β )-type rational ϑ -contraction, and cyclic ( α - ϑ ) contraction. Based on these definitions we prove fixed point theorems in the complete metric spaces. These results extend and improve some known results in the literature. As an application of the proved fixed point Theorems, we study the existence of solutions of an integral boundary value problem for scalar nonlinear Caputo fractional differential equations with a fractional order in (1,2). Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
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17 pages, 292 KiB  
Article
Evolution Inclusions in Banach Spaces under Dissipative Conditions
by Tzanko Donchev, Shamas Bilal, Ovidiu Cârjă, Nasir Javaid and Alina I. Lazu
Mathematics 2020, 8(5), 750; https://doi.org/10.3390/math8050750 - 09 May 2020
Cited by 1 | Viewed by 1352
Abstract
We develop a new concept of a solution, called the limit solution, to fully nonlinear differential inclusions in Banach spaces. That enables us to study such kind of inclusions under relatively weak conditions. Namely we prove the existence of this type of solutions [...] Read more.
We develop a new concept of a solution, called the limit solution, to fully nonlinear differential inclusions in Banach spaces. That enables us to study such kind of inclusions under relatively weak conditions. Namely we prove the existence of this type of solutions and some qualitative properties, replacing the commonly used compact or Lipschitz conditions by a dissipative one, i.e., one-sided Perron condition. Under some natural assumptions we prove that the set of limit solutions is the closure of the set of integral solutions. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
13 pages, 280 KiB  
Article
Double Fuzzy Sumudu Transform to Solve Partial Volterra Fuzzy Integro-Differential Equations
by Atanaska Georgieva
Mathematics 2020, 8(5), 692; https://doi.org/10.3390/math8050692 - 02 May 2020
Cited by 21 | Viewed by 2139
Abstract
In this paper, the double fuzzy Sumudu transform (DFST) method was used to find the solution to partial Volterra fuzzy integro-differential equations (PVFIDE) with convolution kernel under Hukuhara differentiability. Fundamental results of the double fuzzy Sumudu transform for double fuzzy convolution and fuzzy [...] Read more.
In this paper, the double fuzzy Sumudu transform (DFST) method was used to find the solution to partial Volterra fuzzy integro-differential equations (PVFIDE) with convolution kernel under Hukuhara differentiability. Fundamental results of the double fuzzy Sumudu transform for double fuzzy convolution and fuzzy partial derivatives of the n-th order are provided. By using these results the solution of PVFIDE is constructed. It is shown that DFST method is a simple and reliable approach for solving such equations analytically. Finally, the method is demonstrated with examples to show the capability of the proposed method. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
13 pages, 291 KiB  
Article
On the Exponents of Exponential Dichotomies
by Flaviano Battelli and Michal Fečkan
Mathematics 2020, 8(4), 651; https://doi.org/10.3390/math8040651 - 23 Apr 2020
Cited by 1 | Viewed by 2315
Abstract
An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. [...] Read more.
An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. This roughness result is crucial in developing a Melnikov bifurcation method for either discontinuous or implicit perturbed nonlinear differential equations. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
10 pages, 252 KiB  
Article
Infinitely Many Homoclinic Solutions for Fourth Order p-Laplacian Differential Equations
by Stepan Tersian
Mathematics 2020, 8(4), 505; https://doi.org/10.3390/math8040505 - 02 Apr 2020
Cited by 1 | Viewed by 1415
Abstract
The existence of infinitely many homoclinic solutions for the fourth-order differential equation [...] Read more.
The existence of infinitely many homoclinic solutions for the fourth-order differential equation φ p u ″ t ″ + w φ p u ′ t ′ + V ( t ) φ p u t = a ( t ) f ( t , u ( t ) ) , t ∈ R is studied in the paper. Here φ p ( t ) = t p − 2 t , p ≥ 2 , w is a constant, V and a are positive functions, f satisfies some extended growth conditions. Homoclinic solutions u are such that u ( t ) → 0 , | t | → ∞ , u ≠ 0 , known in physical models as ground states or pulses. The variational approach is applied based on multiple critical point theorem due to Liu and Wang. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
16 pages, 481 KiB  
Article
Computer Simulation and Iterative Algorithm for Approximate Solving of Initial Value Problem for Riemann-Liouville Fractional Delay Differential Equations
by Snezhana Hristova, Kremena Stefanova and Angel Golev
Mathematics 2020, 8(4), 477; https://doi.org/10.3390/math8040477 - 01 Apr 2020
Cited by 2 | Viewed by 1507
Abstract
The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme [...] Read more.
The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is computerized and applied to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval. The suggested and well-grounded algorithm is applied to a particular problem and the practical usefulness is illustrated. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
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6 pages, 224 KiB  
Article
A Note on the Topological Transversality Theorem for Weakly Upper Semicontinuous, Weakly Compact Maps on Locally Convex Topological Vector Spaces
by Donal O’Regan
Mathematics 2020, 8(3), 304; https://doi.org/10.3390/math8030304 - 25 Feb 2020
Viewed by 1404
Abstract
A simple theorem is presented that automatically generates the topological transversality theorem and Leray–Schauder alternatives for weakly upper semicontinuous, weakly compact maps. An application is given to illustrate our results. Full article
(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)
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