Differential and Integro–Differential Equations: Theory 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: 31 August 2024 | Viewed by 8317

Special Issue Editors

Department of Mechanical and Industrial Engineering, Tallinn University of Technology, 19086 Tallinn, Estonia
Interests: design optimization; numerical methods; Haar wavelet methods; composite structures; fractional differential equations; nanocomposites; graphene structures; nonlocal elasticity theories; laminated glass panels; solar panels
Special Issues, Collections and Topics in MDPI journals
Department of Cybernetics, School of Science, Tallinn University of Technology, 12616 Tallinn, Estonia
Interests: internal variable; solitary wave solution; higher-order asymptotics

Special Issue Information

Dear Colleagues, 

It is our pleasure to announce the launch of a new Special Issue of Mathematics on the topic of “Differential and Integro –Differential Equations: Theory and Applications”. 

The purpose of this Special Issue is to present recent advances in the development and application of differential and integro-differential equations covering different research areas. Papers focused on the development and validation of new methods (algorithms, convergence and complexity analysis, etc.), as well as the adaption of existing methods for solving particular problems are welcome.  

The topics considered include but are not restricted to: 

  • Ordinary differential equations; 
  • Partial differential equations;
  • Fractional differential equations;
  • Delay differential equations; 
  • Integro-differential equations; 
  • Applications of differential equations in physical, engineering, live sciences, economics, etc. 

We look forward to your contributions. 

Prof. Dr. Jüri Majak
Prof. Dr. Andrus Salupere
Guest Editors

Manuscript Submission Information

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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

  • ordinary differential equations
  • partial differential equation
  • fractional differential equations
  • delay differential equations
  • integro-differential equations
  • applications of differential equations

Published Papers (7 papers)

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Research

15 pages, 302 KiB  
Article
On a Non Local Initial Boundary Value Problem for a Semi-Linear Pseudo-Hyperbolic Equation in the Theory of Vibration
by Eman Alhazzani and Said Mesloub
Mathematics 2023, 11(23), 4803; https://doi.org/10.3390/math11234803 - 28 Nov 2023
Viewed by 564
Abstract
This research article addresses a nonclassical initial boundary value problem characterized by a non-local constraint within the framework of a pseudo-hyperbolic equation. Employing rigorous analytical techniques, the paper establishes the existence, uniqueness, and continuous dependence of a strong solution to the problem at [...] Read more.
This research article addresses a nonclassical initial boundary value problem characterized by a non-local constraint within the framework of a pseudo-hyperbolic equation. Employing rigorous analytical techniques, the paper establishes the existence, uniqueness, and continuous dependence of a strong solution to the problem at hand. With respect to the associated linear problem, the uniqueness of its solution is ascertained through an energy inequality, which provides an a priori bound for the solution. Moreover, the solvability of this linear problem is verified by proving that the operator range engendered by the problem is indeed dense. Extending the analysis to the nonlinear problem, an iterative methodology is utilized. This approach is predicated on the insights gained from the linear problem and facilitates the demonstration of both the existence and uniqueness of a solution for the nonlinear problem under study. Consequently, the paper contributes a robust mathematical framework for solving both linear and nonlinear variants of complex initial boundary value problems with non-local constraints. Full article
21 pages, 5223 KiB  
Article
Analytical Solution of Stability Problem of Nanocomposite Cylindrical Shells under Combined Loadings in Thermal Environments
by Mahmure Avey, Nicholas Fantuzzi and Abdullah H. Sofiyev
Mathematics 2023, 11(17), 3781; https://doi.org/10.3390/math11173781 - 03 Sep 2023
Cited by 2 | Viewed by 604
Abstract
The mathematical modeling of the stability problem of nanocomposite cylindrical shells is one of the applications of partial differential equations (PDEs). In this study, the stability behavior of inhomogeneous nanocomposite cylindrical shells (INH-NCCSs), under combined axial compression and hydrostatic pressure in the thermal [...] Read more.
The mathematical modeling of the stability problem of nanocomposite cylindrical shells is one of the applications of partial differential equations (PDEs). In this study, the stability behavior of inhomogeneous nanocomposite cylindrical shells (INH-NCCSs), under combined axial compression and hydrostatic pressure in the thermal environment, is investigated by means of the first-order shear deformation theory (FSDT). The nanocomposite material is modeled as homogeneous and heterogeneous and is based on a carbon nanotube (CNT)-reinforced polymer with the linear variation of the mechanical properties throughout the thickness. In the heterogeneous case, the mechanical properties are modeled as the linear function of the thickness coordinate. The basic equations are derived as partial differential equations and solved in a closed form, using the Galerkin procedure, to determine the critical combined loads for the selected structure in thermal environments. To test the reliability of the proposed formulation, comparisons with the results obtained by finite element and numerical methods in the literature are accompanied by a systematic study aimed at testing the sensitivity of the design response to the loading parameters, CNT models, and thermal environment. Full article
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16 pages, 822 KiB  
Article
Generalized Moment Method for Smoluchowski Coagulation Equation and Mass Conservation Property
by Md. Sahidul Islam, Masato Kimura and Hisanori Miyata
Mathematics 2023, 11(12), 2770; https://doi.org/10.3390/math11122770 - 19 Jun 2023
Cited by 1 | Viewed by 1028 | Correction
Abstract
In this paper, we develop a generalized moment method with a continuous weight function for the Smoluchowski coagulation equation in its continuous form to study the mass conservation property of this equation. We first establish some basic inequalities for the generalized moment and [...] Read more.
In this paper, we develop a generalized moment method with a continuous weight function for the Smoluchowski coagulation equation in its continuous form to study the mass conservation property of this equation. We first establish some basic inequalities for the generalized moment and prove the mass conservation property under a sufficient condition on the kernel and an initial condition, utilizing these inequalities. Additionally, we provide some concrete examples of coagulation kernels that exhibit mass conservation properties and show that these kernels exhibit either polynomial or exponential growth along specific particular curves. Full article
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34 pages, 446 KiB  
Article
Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations
by Sergey Kashchenko
Mathematics 2023, 11(9), 2047; https://doi.org/10.3390/math11092047 - 26 Apr 2023
Cited by 2 | Viewed by 1590
Abstract
Chains of coupled van der Pol equations are considered. The main assumption that motivates the use of special asymptotic methods is that the number of elements in the chain is sufficiently large. This allows moving from a discrete system of equations to the [...] Read more.
Chains of coupled van der Pol equations are considered. The main assumption that motivates the use of special asymptotic methods is that the number of elements in the chain is sufficiently large. This allows moving from a discrete system of equations to the use of a continuity argument and obtaining an integro-differential boundary value problem as the initial model. In the study of the behaviour of all its solutions in a neighbourhood of the equilibrium state, infinite-dimensional critical cases arise in the problem of the stability of solutions. The main results include the construction of special families of quasi-normal forms, namely non-linear boundary value problems of either Schrödinger or Ginzburg–Landau type. Their solutions make it possible to determine the main terms of the asymptotic expansion of both regular and irregular solutions to the original system. The main goal is the study of chains with diffusion- and advective-type couplings, as well as fully connected chains. Full article
7 pages, 248 KiB  
Article
Nonlinear Volterra Integrodifferential Equations from above on Unbounded Time Scales
by Andrejs Reinfelds and Shraddha Christian
Mathematics 2023, 11(7), 1760; https://doi.org/10.3390/math11071760 - 06 Apr 2023
Viewed by 1049
Abstract
The paper is devoted to studying the existence, uniqueness and certain growth rates of solutions with certain implicit Volterra-type integrodifferential equations on unbounded from above time scales. We consider the case where the integrand is estimated by the Lipschitz type function with respect [...] Read more.
The paper is devoted to studying the existence, uniqueness and certain growth rates of solutions with certain implicit Volterra-type integrodifferential equations on unbounded from above time scales. We consider the case where the integrand is estimated by the Lipschitz type function with respect to the unknown variable. Lipschitz coefficient is an unbounded rd-function and the Banach fixed-point theorem at a functional space endowed with a suitable Bielecki-type norm. Full article
17 pages, 343 KiB  
Article
Inverse Problem to Determine Two Time-Dependent Source Factors of Fractional Diffusion-Wave Equations from Final Data and Simultaneous Reconstruction of Location and Time History of a Point Source
by Jaan Janno
Mathematics 2023, 11(2), 456; https://doi.org/10.3390/math11020456 - 14 Jan 2023
Viewed by 1513
Abstract
In this paper, two inverse problems for the fractional diffusion-wave equation that use final data are considered. The first problem consists in the determination of two time-dependent source terms. Uniqueness for this inverse problem is established under an assumption that given space-dependent factors [...] Read more.
In this paper, two inverse problems for the fractional diffusion-wave equation that use final data are considered. The first problem consists in the determination of two time-dependent source terms. Uniqueness for this inverse problem is established under an assumption that given space-dependent factors of these terms are “sufficiently different”. The proof uses asymptotical properties of Mittag–Leffler functions. In the second problem, the aim is to reconstruct a location and time history of a point source. The uniqueness for this problem is deduced from the uniqueness theorem for the previous problem in the one-dimensional case. Full article
11 pages, 283 KiB  
Article
On a Lyapunov-Type Inequality for Control of a ψ-Model Thermostat and the Existence of Its Solutions
by Shahram Rezapour, Sina Etemad, Ravi P. Agarwal and Kamsing Nonlaopon
Mathematics 2022, 10(21), 4023; https://doi.org/10.3390/math10214023 - 30 Oct 2022
Cited by 6 | Viewed by 866
Abstract
In this paper, a new structure of an applied model of thermostat is defined using the generalized ψ-operators with three-point boundary conditions. Some useful properties of the relevant Green’s function are established, and based on these properties, the Lyapunov-type inequality is constructed [...] Read more.
In this paper, a new structure of an applied model of thermostat is defined using the generalized ψ-operators with three-point boundary conditions. Some useful properties of the relevant Green’s function are established, and based on these properties, the Lyapunov-type inequality is constructed for the given extended ψ-model thermostat with the help of Jensen’s inequality. By defining mild solutions for such an extended system, the existence and non-existence conditions are discussed. Full article
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