Special Issue "Ordinary and Partial Differential Equations: Theory and Applications"

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics and Symmetry".

Deadline for manuscript submissions: 31 December 2020.

Special Issue Editor

Prof. Dr. Clemente Cesarano
Website
Guest Editor
Section of Mathematics - Uninettuno University, C.so Vittorio Emanuele II, 39, 00186 Roma, Italy
Interests: special functions; orthogonal polynomials; differential equations; numerical analysis; fractional calculus; operator theory; Lie algebra
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Special Issue Information

Dear Colleagues,

The study of differential equations is useful for understanding natural phenomena. In this Special Issue, we aim to present the latest research on the properties of ODE (Ordinary Differential Equations) and PDE (Partial Differential Equations) related to different techniques for finding solutions and methods describing the nature of these solutions or their related approximations.

In addition, we welcome papers on numerical aspects using classical or non-standard approaches, for example, the concepts and related formalism of special functions. Furthermore, articles on fractional differential equations are of interest, as are contributions related to the symmetry approach to problems of integrability in the field of differential equations.

Prof. Clemente Cesarano
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 papers will be 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. Symmetry is an international peer-reviewed open access monthly 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 1400 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

  • symmetries
  • ODE
  • PDE
  • numerical methods
  • fractional calculus

Published Papers (18 papers)

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Research

Open AccessArticle
Analytical Solution of Nonlinear Fractional Volterra Population Growth Model Using the Modified Residual Power Series Method
Symmetry 2020, 12(11), 1779; https://doi.org/10.3390/sym12111779 - 27 Oct 2020
Abstract
In this paper, we introduce an analytical approximate solution of nonlinear fractional Volterra population growth model based on the Caputo fractional derivative and the Riemann fractional integral of the symmetry order. The residual power series method and Adomain decomposition method are implemented to [...] Read more.
In this paper, we introduce an analytical approximate solution of nonlinear fractional Volterra population growth model based on the Caputo fractional derivative and the Riemann fractional integral of the symmetry order. The residual power series method and Adomain decomposition method are implemented to find an approximate solution of this problem. The convergence analysis of the proposed technique has been proved. A numerical example is given to illustrate the method. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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Open AccessArticle
On Optimization Techniques for the Construction of an Exponential Estimate for Delayed Recurrent Neural Networks
Symmetry 2020, 12(10), 1731; https://doi.org/10.3390/sym12101731 - 20 Oct 2020
Abstract
This work is devoted to the modeling and investigation of the architecture design for the delayed recurrent neural network, based on the delayed differential equations. The usage of discrete and distributed delays makes it possible to model the calculation of the next states [...] Read more.
This work is devoted to the modeling and investigation of the architecture design for the delayed recurrent neural network, based on the delayed differential equations. The usage of discrete and distributed delays makes it possible to model the calculation of the next states using internal memory, which corresponds to the artificial recurrent neural network architecture used in the field of deep learning. The problem of exponential stability of the models of recurrent neural networks with multiple discrete and distributed delays is considered. For this purpose, the direct method of stability research and the gradient descent method is used. The methods are used consequentially. Firstly we use the direct method in order to construct stability conditions (resulting in an exponential estimate), which include the tuple of positive definite matrices. Then we apply the optimization technique for these stability conditions (or of exponential estimate) with the help of a generalized gradient method with respect to this tuple of matrices. The exponential estimates are constructed on the basis of the Lyapunov–Krasovskii functional. An optimization method of improving estimates is offered, which is based on the notion of the generalized gradient of the convex function of the tuple of positive definite matrices. The search for the optimal exponential estimate is reduced to finding the saddle point of the Lagrange function. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
Open AccessArticle
Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type
Symmetry 2020, 12(10), 1613; https://doi.org/10.3390/sym12101613 - 29 Sep 2020
Abstract
The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose [...] Read more.
The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened Ho¨lder condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We first show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
Open AccessArticle
Exact Solutions and Continuous Numerical Approximations of Coupled Systems of Diffusion Equations with Delay
Symmetry 2020, 12(9), 1560; https://doi.org/10.3390/sym12091560 - 21 Sep 2020
Abstract
In this work, we obtain exact solutions and continuous numerical approximations for mixed problems of coupled systems of diffusion equations with delay. Using the method of separation of variables, and based on an explicit expression for the solution of the separated vector initial-value [...] Read more.
In this work, we obtain exact solutions and continuous numerical approximations for mixed problems of coupled systems of diffusion equations with delay. Using the method of separation of variables, and based on an explicit expression for the solution of the separated vector initial-value delay problem, we obtain exact infinite series solutions that can be truncated to provide analytical–numerical solutions with prescribed accuracy in bounded domains. Although usually implicit in particular applications, the method of separation of variables is deeply correlated with symmetry ideas. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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Open AccessArticle
Indirect Contributions to Tumor Dynamics in the First Stage of the Avascular Phase
Symmetry 2020, 12(9), 1546; https://doi.org/10.3390/sym12091546 - 18 Sep 2020
Abstract
A continuum model for tumor invasion in a two-dimensional spatial domain based on the interaction of the urokinase plasminogen activation system with a model for cancer cell dynamics is proposed. The arising system of partial differential equations is numerically solved using the finite [...] Read more.
A continuum model for tumor invasion in a two-dimensional spatial domain based on the interaction of the urokinase plasminogen activation system with a model for cancer cell dynamics is proposed. The arising system of partial differential equations is numerically solved using the finite element method. We simulated a portion of biological tissue imposing no flux boundary conditions. We monitored the cancer cell dynamics, as well the degradation of an extra cellular matrix representative, vitronectin, and the evolution of a specific degrading enzyme, plasmin, inside the biological tissue. The computations were parameterized as a function of the indirect cell proliferation induced by a plasminogen activator inhibitor binding to vitronectin and of the indirect plasmin deactivation due to the plasminogen activator inhibitor binding to the urokinase plasminogen activator. Their role during the cancer dynamical evolution was identified, together with a possible marker helping the mapping of the cancer invasive front. Our results indicate that indirect cancer cell proliferation biases the speed of the tumor invasive front as well as the heterogeneity of the cancer cell clustering and networking, as it ultimately acts on the proteolytic activity supporting cancer formation. Because of the initial conditions imposed, the numerical solutions of the model show a symmetrical dynamical evolution of heterogeneities inside the simulated domain. Moreover, an increase of up to about 12% in the invasion speed was observed, increasing the rate of indirect cancer cell proliferation, while increasing the plasmin deactivation rate inhibits heterogeneities and networking. As cancer cell proliferation causes vitronectin consumption and plasmin formation, the intensities of the concentration maps of both vitronectin and plasmin are superimposable to the cancer cell concentration maps. The qualitative imprinting that cancer cells leave on the extra cellular matrix during the time evolution as well their activity area is identified, framing the numerical results in the context of a methodology aimed at diagnostic and therapeutic improvement. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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Open AccessArticle
Hyperbolic Equations with Unknown Coefficients
Symmetry 2020, 12(9), 1539; https://doi.org/10.3390/sym12091539 - 17 Sep 2020
Abstract
We study the solvability of nonlinear inverse problems of determining the low order coefficients in the second order hyperbolic equation. The overdetermination condition is specified as an integral condition with final data. Existence and uniqueness theorems for regular solutions are proved (i.e., for [...] Read more.
We study the solvability of nonlinear inverse problems of determining the low order coefficients in the second order hyperbolic equation. The overdetermination condition is specified as an integral condition with final data. Existence and uniqueness theorems for regular solutions are proved (i.e., for solutions having all weak derivatives in the sense of Sobolev, occuring in the equation). Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
Open AccessArticle
Exact and Numerical Solitary Wave Structures to the Variant Boussinesq System
Symmetry 2020, 12(9), 1473; https://doi.org/10.3390/sym12091473 - 08 Sep 2020
Cited by 3
Abstract
Solutions such as symmetric, periodic, and solitary wave solutions play a significant role in the field of partial differential equations (PDEs), and they can be utilized to explain several phenomena in physics and engineering. Therefore, constructing such solutions is significantly essential. This article [...] Read more.
Solutions such as symmetric, periodic, and solitary wave solutions play a significant role in the field of partial differential equations (PDEs), and they can be utilized to explain several phenomena in physics and engineering. Therefore, constructing such solutions is significantly essential. This article concentrates on employing the improved exp(ϕ(η))-expansion approach and the method of lines on the variant Boussinesq system to establish its exact and numerical solutions. Novel solutions based on the solitary wave structures are obtained. We present a comprehensible comparison between the accomplished exact and numerical results to testify the accuracy of the used numerical technique. Some 3D and 2D diagrams are sketched for some solutions. We also investigate the L2 error and the CPU time of the used numerical method. The used mathematical tools can be comfortably invoked to handle more nonlinear evolution equations. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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Open AccessArticle
On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives
Symmetry 2020, 12(8), 1355; https://doi.org/10.3390/sym12081355 - 13 Aug 2020
Abstract
In this paper, two forms of an exact solution and an analytical–numerical solution of the three-term fractional differential equation with the Caputo derivatives are presented. The Prabhakar function and an asymptotic expansion are utilized to present the double series solution. Using properties of [...] Read more.
In this paper, two forms of an exact solution and an analytical–numerical solution of the three-term fractional differential equation with the Caputo derivatives are presented. The Prabhakar function and an asymptotic expansion are utilized to present the double series solution. Using properties of the Pochhammer symbol, a solution is obtained in the form of an infinite series of generalized hypergeometric functions. As an alternative for the series solutions of the fractional commensurate equation, a solution received by an analytical–numerical method based on the Laplace transform technique is proposed. This solution is obtained in the form of a finite sum of the Mittag-Leffler type functions. Numerical examples and a discussion are presented. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
Open AccessArticle
Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method
Symmetry 2020, 12(7), 1195; https://doi.org/10.3390/sym12071195 - 19 Jul 2020
Cited by 14
Abstract
Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative [...] Read more.
Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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Open AccessArticle
Perturbation of One-Dimensional Time Independent Schrödinger Equation With a Symmetric Parabolic Potential Wall
Symmetry 2020, 12(7), 1089; https://doi.org/10.3390/sym12071089 - 01 Jul 2020
Cited by 1
Abstract
The first author has recently investigated a type of Hyers-Ulam stability of the one-dimensional time independent Schrödinger equation when the relevant system has a rectangular potential barrier of finite height. In the present paper, we will investigate a type of Hyers-Ulam stability of [...] Read more.
The first author has recently investigated a type of Hyers-Ulam stability of the one-dimensional time independent Schrödinger equation when the relevant system has a rectangular potential barrier of finite height. In the present paper, we will investigate a type of Hyers-Ulam stability of the Schrödinger equation with the symmetric parabolic wall potential. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
Open AccessArticle
Boundary Layer Flow and Heat Transfer of Al2O3-TiO2/Water Hybrid Nanofluid over a Permeable Moving Plate
Symmetry 2020, 12(7), 1064; https://doi.org/10.3390/sym12071064 - 28 Jun 2020
Abstract
Hybrid nanofluid is considered a new type of nanofluid and is further used to increase the heat transfer efficiency. This paper explores the two-dimensional steady axisymmetric boundary layer which contains water (base fluid) and two different nanoparticles to form a hybrid nanofluid over [...] Read more.
Hybrid nanofluid is considered a new type of nanofluid and is further used to increase the heat transfer efficiency. This paper explores the two-dimensional steady axisymmetric boundary layer which contains water (base fluid) and two different nanoparticles to form a hybrid nanofluid over a permeable moving plate. The plate is suspected to move to the free stream in the similar or opposite direction. Similarity transformation is introduced in order to convert the nonlinear partial differential equation of the governing equation into a system of ordinary differential equations (ODEs). Then, the ODEs are solved using bvp4c in MATLAB 2019a software. The mathematical hybrid nanofluid and boundary conditions under the effect of suction, S, and the concentration of nanoparticles, ϕ 1 (Al2O3) and ϕ 2 (TiO2) are taken into account. Numerical results are graphically described for the skin friction coefficient, C f , and local Nusselt number, N u x , as well as velocity and temperature profiles. The results showed that duality occurs when the plate and the free stream travel in the opposite direction. The range of dual solutions expand widely for S and closely reduce for ϕ . Thus, a stability analysis is performed. The first solution is stable and realizable compared to the second solution. The C f and N u x increase with the increment of S. It is also noted that the increase of ϕ 2 leads to an increase in C f and decrease in N u x . Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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Open AccessArticle
An Optimal Fourth Order Derivative-Free Numerical Algorithm for Multiple Roots
Symmetry 2020, 12(6), 1038; https://doi.org/10.3390/sym12061038 - 21 Jun 2020
Cited by 1
Abstract
A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to construct, and hence, such methods are almost [...] Read more.
A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to construct, and hence, such methods are almost non-existent. This motivated us to explore a derivative-free iterative scheme with optimal fourth order convergence. The applicability of the new scheme is shown by testing on different functions, which illustrates the excellent convergence. Moreover, the comparison of the performance shows that the new technique is a good competitor to existing optimal fourth order Newton-like techniques. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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Open AccessArticle
New Uniform Motion and Fermi–Walker Derivative of Normal Magnetic Biharmonic Particles in Heisenberg Space
Symmetry 2020, 12(6), 1017; https://doi.org/10.3390/sym12061017 - 16 Jun 2020
Abstract
In the present paper, we firstly discuss the normal biharmonic magnetic particles in the Heisenberg space. We express new uniform motions and its properties in the Heisenberg space. Moreover, we obtain a new uniform motion of Fermi–Walker derivative of normal magnetic biharmonic particles [...] Read more.
In the present paper, we firstly discuss the normal biharmonic magnetic particles in the Heisenberg space. We express new uniform motions and its properties in the Heisenberg space. Moreover, we obtain a new uniform motion of Fermi–Walker derivative of normal magnetic biharmonic particles in the Heisenberg space. Finally, we investigate uniformly accelerated motion (UAM), the unchanged direction motion (UDM), and the uniformly circular motion (UCM) of the moving normal magnetic biharmonic particles in Heisenberg space. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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Open AccessArticle
On the Oscillatory Behavior of a Class of Fourth-Order Nonlinear Differential Equation
Symmetry 2020, 12(4), 524; https://doi.org/10.3390/sym12040524 - 02 Apr 2020
Cited by 11
Abstract
In this work, we study the oscillatory behavior of a class of fourth-order differential equations. New oscillation criteria were obtained by employing a refinement of the Riccati transformations. The new theorems complement and improve a number of results reported in the literature. An [...] Read more.
In this work, we study the oscillatory behavior of a class of fourth-order differential equations. New oscillation criteria were obtained by employing a refinement of the Riccati transformations. The new theorems complement and improve a number of results reported in the literature. An example is provided to illustrate the main results. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
Open AccessArticle
A Philos-Type Oscillation Criteria for Fourth-Order Neutral Differential Equations
Symmetry 2020, 12(3), 379; https://doi.org/10.3390/sym12030379 - 03 Mar 2020
Cited by 20
Abstract
Some sufficient conditions are established for the oscillation of fourth order neutral differential equations of the form r t z t α + q t x β σ t = 0 , where z t : = x t + p [...] Read more.
Some sufficient conditions are established for the oscillation of fourth order neutral differential equations of the form r t z t α + q t x β σ t = 0 , where z t : = x t + p t x τ t . By using the technique of Riccati transformation and integral averaging method, we get conditions to ensure oscillation of solutions of this equation. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. Moreover, the importance of the obtained conditions is illustrated via some examples. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
Open AccessArticle
Asymptotic and Oscillatory Behavior of Solutions of a Class of Higher Order Differential Equation
Symmetry 2019, 11(12), 1434; https://doi.org/10.3390/sym11121434 - 21 Nov 2019
Cited by 24
Abstract
The objective of this paper is to study asymptotic behavior of a class of higher-order delay differential equations with a p-Laplacian like operator. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, [...] Read more.
The objective of this paper is to study asymptotic behavior of a class of higher-order delay differential equations with a p-Laplacian like operator. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and show us the correct direction for future developments. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and comparison principles. This new theorem complements and improves a number of results reported in the literature. Some examples are provided to illustrate the main results. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
Open AccessArticle
On Some Initial and Initial Boundary Value Problems for Linear and Nonlinear Boussinesq Models
Symmetry 2019, 11(10), 1273; https://doi.org/10.3390/sym11101273 - 11 Oct 2019
Abstract
The main concern of this paper is to apply the modified double Laplace decomposition method to a singular generalized modified linear Boussinesq equation and to a singular nonlinear Boussinesq equation. An a priori estimate for the solution is also derived. Some examples are [...] Read more.
The main concern of this paper is to apply the modified double Laplace decomposition method to a singular generalized modified linear Boussinesq equation and to a singular nonlinear Boussinesq equation. An a priori estimate for the solution is also derived. Some examples are given to validate and illustrate the method. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
Open AccessArticle
N-Soliton Solutions for the NLS-Like Equation and Perturbation Theory Based on the Riemann–Hilbert Problem
Symmetry 2019, 11(6), 826; https://doi.org/10.3390/sym11060826 - 22 Jun 2019
Cited by 9
Abstract
In this paper, a kind of nonlinear Schrödinger (NLS) equation, called an NLS-like equation, is Riemann–Hilbert investigated. We construct a 2 × 2 Lax pair associated with the NLS equation and combine the spectral analysis to formulate the Riemann–Hilbert (R–H) problem. Then, we [...] Read more.
In this paper, a kind of nonlinear Schrödinger (NLS) equation, called an NLS-like equation, is Riemann–Hilbert investigated. We construct a 2 × 2 Lax pair associated with the NLS equation and combine the spectral analysis to formulate the Riemann–Hilbert (R–H) problem. Then, we mainly use the symmetry relationship of potential matrix Q to analyze the zeros of det P + and det P ; the N-soliton solutions of the NLS-like equation are expressed explicitly by a particular R–H problem with an unit jump matrix. In addition, the single-soliton solution and collisions of two solitons are analyzed, and the dynamic behaviors of the single-soliton solution and two-soliton solutions are shown graphically. Furthermore, on the basis of the R–H problem, the evolution equation of the R–H data with the perturbation is derived. Full article
(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications)
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