Special Issue "Recent Advances in 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: 24 February 2023 | Viewed by 13344

Special Issue Editors

Prof. Dr. Juan Ramón Torregrosa Sánchez
E-Mail Website
Guest Editor
Prof. Dr. Alicia Cordero Barbero
E-Mail Website
Guest Editor
Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain
Interests: iterative processes; numerical analysis; dynamic analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Juan Carlos Cortés López
E-Mail Website
Guest Editor
Department of Applied Mathematics and Institute for Multidisciplinary Mathematics (im2), Universitat Politècnica de València, 46022 Valencia, Spain
Interests: differential equations with randomness; mathematical modelling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential equations play a key role in modelling the dynamics of many phenomena belonging to different realms including physics, chemistry, finance, and social sciences. Since their classical formulation, via ordinary derivatives, a number of other classes of differential equations have been proposed, such as delay, fractional, functional, or integro-differential equations. The mathematical and numerical analyses of all these types of differential equations are still a hot topic in mathematics. This interest increased when the aforementioned types of differential equations consider the randomness often present in mathematical modelling, which lead to random and stochastic differential equations.

In this Special Issue, we encourage submissions providing new results in the setting of differential equations and their applications. Potential topics include, but are not limited to the next keywords (see below).

Prof. Dr. Juan Ramón Torregrosa Sánchez
Prof. Dr. Alicia Cordero Barbero
Prof. Dr. Juan Carlos Cortés López
Guest Editors

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 1800 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 and partial differential equations
  • Differential-difference equations
  • Delay differential equations
  • Fractional differential equations
  • Algebraic-differential equations
  • Integro-differential equations
  • Complex differential equations
  • Functional differential equations
  • Numerical methods for differential equations
  • Stability theory
  • Random and stochastic differential equations
  • Mathematical modelling using differential equations

Published Papers (19 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

Article
Blow-Up Time of Solutions for a Parabolic Equation with Exponential Nonlinearity
Mathematics 2022, 10(16), 2887; https://doi.org/10.3390/math10162887 - 12 Aug 2022
Viewed by 127
Abstract
This paper studies a parabolic equation with exponential nonlinearity, which has several applications, for example the self-trapped beams in plasma. Based on a modified concavity method we prove the blow-up of the solution for initial data with high initial energy. We also proposed [...] Read more.
This paper studies a parabolic equation with exponential nonlinearity, which has several applications, for example the self-trapped beams in plasma. Based on a modified concavity method we prove the blow-up of the solution for initial data with high initial energy. We also proposed the solution’s lower and upper bound of the blow-up time for the equation. Our results complement the existing results. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Article
A Novel Projection Method for Cauchy-Type Systems of Singular Integro-Differential Equations
Mathematics 2022, 10(15), 2694; https://doi.org/10.3390/math10152694 - 29 Jul 2022
Viewed by 214
Abstract
This article introduces a new projection method via shifted Legendre polynomials and an efficient procedure for solving a system of integro-differential equations of the Cauchy type. The proposed computational process solves two systems of linear equations. We demonstrate the existence of the solution [...] Read more.
This article introduces a new projection method via shifted Legendre polynomials and an efficient procedure for solving a system of integro-differential equations of the Cauchy type. The proposed computational process solves two systems of linear equations. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests provide theoretical results. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
Semi-Hyers–Ulam–Rassias Stability via Laplace Transform, for an Integro-Differential Equation of the Second Order
Mathematics 2022, 10(11), 1893; https://doi.org/10.3390/math10111893 - 01 Jun 2022
Viewed by 322
Abstract
The Laplace transform method is applied to study the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of the second order. A general equation is formulated first; then, some particular cases for the function from the kernel are considered. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Article
New Monotonic Properties of Positive Solutions of Higher-Order Delay Differential Equations and Their Applications
Mathematics 2022, 10(10), 1786; https://doi.org/10.3390/math10101786 - 23 May 2022
Viewed by 480
Abstract
In this work, new criteria were established for testing the oscillatory behavior of solutions of a class of even-order delay differential equations. We follow an approach that depends on obtaining new monotonic properties for the decreasing positive solutions of the studied equation. Moreover, [...] Read more.
In this work, new criteria were established for testing the oscillatory behavior of solutions of a class of even-order delay differential equations. We follow an approach that depends on obtaining new monotonic properties for the decreasing positive solutions of the studied equation. Moreover, we use these properties to provide new oscillation criteria of an iterative nature. We provide an example to support the significance of the results and compare them with the related previous work. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Article
Hyers–Ulam–Rassias Stability of Hermite’s Differential Equation
Mathematics 2022, 10(6), 964; https://doi.org/10.3390/math10060964 - 17 Mar 2022
Viewed by 327
Abstract
In this paper, we studied the Hyers–Ulam–Rassias stability of Hermite’s differential equation, using Pachpatte’s inequality. We compared our results with those obtained by Blaga et al. Our estimation for zxyx, where z is an approximate solution and y [...] Read more.
In this paper, we studied the Hyers–Ulam–Rassias stability of Hermite’s differential equation, using Pachpatte’s inequality. We compared our results with those obtained by Blaga et al. Our estimation for zxyx, where z is an approximate solution and y is an exact solution of Hermite’s equation, was better than that obtained by the authors previously mentioned, in some parts of the domain, especially in a neighborhood of the origin. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
The Impact of the Wiener Process on the Analytical Solutions of the Stochastic (2+1)-Dimensional Breaking Soliton Equation by Using Tanh–Coth Method
Mathematics 2022, 10(5), 817; https://doi.org/10.3390/math10050817 - 04 Mar 2022
Cited by 7 | Viewed by 550
Abstract
The stochastic (2+1)-dimensional breaking soliton equation (SBSE) is considered in this article, which is forced by the Wiener process. To attain the analytical stochastic solutions such as the polynomials, hyperbolic and trigonometric functions of the SBSE, we use the tanh–coth method. The results [...] Read more.
The stochastic (2+1)-dimensional breaking soliton equation (SBSE) is considered in this article, which is forced by the Wiener process. To attain the analytical stochastic solutions such as the polynomials, hyperbolic and trigonometric functions of the SBSE, we use the tanh–coth method. The results provided here extended earlier results. In addition, we utilize Matlab tools to plot 2D and 3D graphs of analytical stochastic solutions derived here to show the effect of the Wiener process on the solutions of the breaking soliton equation. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
Semi-Hyers–Ulam–Rassias Stability of the Convection Partial Differential Equation via Laplace Transform
Mathematics 2021, 9(22), 2980; https://doi.org/10.3390/math9222980 - 22 Nov 2021
Cited by 7 | Viewed by 426
Abstract
In this paper, we study the semi-Hyers–Ulam–Rassias stability and the generalized semi-Hyers–Ulam–Rassias stability of some partial differential equations using Laplace transform. One of them is the convection partial differential equation. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Article
Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil
Mathematics 2021, 9(20), 2617; https://doi.org/10.3390/math9202617 - 17 Oct 2021
Viewed by 586
Abstract
The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general [...] Read more.
The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
The Polynomial Least Squares Method for Nonlinear Fractional Volterra and Fredholm Integro-Differential Equations
Mathematics 2021, 9(18), 2324; https://doi.org/10.3390/math9182324 - 19 Sep 2021
Viewed by 574
Abstract
We present a relatively new and very efficient method to find approximate analytical solutions for a very general class of nonlinear fractional Volterra and Fredholm integro-differential equations. The test problems included and the comparison with previous results by other methods clearly illustrate the [...] Read more.
We present a relatively new and very efficient method to find approximate analytical solutions for a very general class of nonlinear fractional Volterra and Fredholm integro-differential equations. The test problems included and the comparison with previous results by other methods clearly illustrate the simplicity and accuracy of the method. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
Arbitrary Coefficient Assignment by Static Output Feedback for Linear Differential Equations with Non-Commensurate Lumped and Distributed Delays
Mathematics 2021, 9(17), 2158; https://doi.org/10.3390/math9172158 - 04 Sep 2021
Cited by 1 | Viewed by 483
Abstract
We consider a linear control system defined by a scalar stationary linear differential equation in the real or complex space with multiple non-commensurate lumped and distributed delays in the state. In the system, the input is a linear combination of multiple variables and [...] Read more.
We consider a linear control system defined by a scalar stationary linear differential equation in the real or complex space with multiple non-commensurate lumped and distributed delays in the state. In the system, the input is a linear combination of multiple variables and its derivatives, and the output is a multidimensional vector of linear combinations of the state and its derivatives. For this system, we study the problem of arbitrary coefficient assignment for the characteristic function by linear static output feedback with lumped and distributed delays. We obtain necessary and sufficient conditions for the solvability of the arbitrary coefficient assignment problem by the static output feedback controller. Corollaries on arbitrary finite spectrum assignment and on stabilization of the system are obtained. We provide an example illustrating our results. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
Application of Said Ball Curve for Solving Fractional Differential-Algebraic Equations
Mathematics 2021, 9(16), 1926; https://doi.org/10.3390/math9161926 - 12 Aug 2021
Cited by 2 | Viewed by 751
Abstract
The aim of this paper is to apply the Said Ball curve (SBC) to find the approximate solution of fractional differential-algebraic equations (FDAEs). This method can be applied to solve various types of fractional order differential equations. Convergence theorem of the method is [...] Read more.
The aim of this paper is to apply the Said Ball curve (SBC) to find the approximate solution of fractional differential-algebraic equations (FDAEs). This method can be applied to solve various types of fractional order differential equations. Convergence theorem of the method is proved. Some examples are presented to show the efficiency and accuracy of the method. Based on the obtained results, the SBC is more accurate than the Bezier curve method. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
Alternating Polynomial Reconstruction Method for Hyperbolic Conservation Laws
Mathematics 2021, 9(16), 1885; https://doi.org/10.3390/math9161885 - 08 Aug 2021
Cited by 3 | Viewed by 920
Abstract
We propose a new multi-moment numerical solver for hyperbolic conservation laws by using the alternating polynomial reconstruction approach. Unlike existing multi-moment schemes, our approach updates model variables by implementing two polynomial reconstructions alternately. First, Hermite interpolation reconstructs the solution within the cell by [...] Read more.
We propose a new multi-moment numerical solver for hyperbolic conservation laws by using the alternating polynomial reconstruction approach. Unlike existing multi-moment schemes, our approach updates model variables by implementing two polynomial reconstructions alternately. First, Hermite interpolation reconstructs the solution within the cell by matching the point-based variables containing both physical values and their spatial derivatives. Then the reconstructed solution is updated by the Euler method. Second, we solve a constrained least-squares problem to correct the updated solution to preserve the conservation laws. Our method enjoys the advantages of a compact numerical stencil and high-order accuracy. Fourier analysis also indicates that our method allows a larger CFL number compared with many other high-order schemes. By adding a proper amount of artificial viscosity, shock waves and other discontinuities can also be computed accurately and sharply without solving an approximated Riemann problem. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
Local Dynamics of Logistic Equation with Delay and Diffusion
Mathematics 2021, 9(13), 1566; https://doi.org/10.3390/math9131566 - 03 Jul 2021
Cited by 2 | Viewed by 991
Abstract
The behavior of all the solutions of the logistic equation with delay and diffusion in a sufficiently small positive neighborhood of the equilibrium state is studied. It is assumed that the Andronov–Hopf bifurcation conditions are met for the coefficients of the problem. Small [...] Read more.
The behavior of all the solutions of the logistic equation with delay and diffusion in a sufficiently small positive neighborhood of the equilibrium state is studied. It is assumed that the Andronov–Hopf bifurcation conditions are met for the coefficients of the problem. Small perturbations of all coefficients are considered, including the delay coefficient and the coefficients of the boundary conditions. The conditions are studied when these perturbations depend on the spatial variable and when they are time-periodic functions. Equations on the central manifold are constructed as the main results. Their nonlocal dynamics determines the behavior of all the solutions of the original boundary value problem in a sufficiently small neighborhood of the equilibrium state. The ability to control the dynamics of the original problem using the phase change in the perturbing force is set. The numerical and analytical results regarding the dynamics of the system with parametric perturbation are obtained. The asymptotic formulas for the solutions of the original boundary value problem are given. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Article
Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs
Mathematics 2021, 9(13), 1535; https://doi.org/10.3390/math9131535 - 30 Jun 2021
Cited by 3 | Viewed by 559
Abstract
In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving [...] Read more.
In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator
Mathematics 2021, 9(12), 1323; https://doi.org/10.3390/math9121323 - 08 Jun 2021
Cited by 2 | Viewed by 714
Abstract
In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. [...] Read more.
In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Article
Generalization of Quantum Ostrowski-Type Integral Inequalities
Mathematics 2021, 9(10), 1155; https://doi.org/10.3390/math9101155 - 20 May 2021
Cited by 3 | Viewed by 652
Abstract
In this paper, we prove some new Ostrowski-type integral inequalities for q-differentiable bounded functions. It is also shown that the results presented in this paper are a generalization of know results in the literarure. Applications to special means are also discussed. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Article
On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method
Mathematics 2021, 9(1), 78; https://doi.org/10.3390/math9010078 - 31 Dec 2020
Viewed by 752
Abstract
Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a [...] Read more.
Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
Nonlinear Observability Algorithms with Known and Unknown Inputs: Analysis and Implementation
Mathematics 2020, 8(11), 1876; https://doi.org/10.3390/math8111876 - 29 Oct 2020
Cited by 3 | Viewed by 997
Abstract
The observability of a dynamical system is affected by the presence of external inputs, either known (such as control actions) or unknown (disturbances). Inputs of unknown magnitude are especially detrimental for observability, and they also complicate its analysis. Hence, the availability of computational [...] Read more.
The observability of a dynamical system is affected by the presence of external inputs, either known (such as control actions) or unknown (disturbances). Inputs of unknown magnitude are especially detrimental for observability, and they also complicate its analysis. Hence, the availability of computational tools capable of analysing the observability of nonlinear systems with unknown inputs has been limited until lately. Two symbolic algorithms based on differential geometry, ORC-DF and FISPO, have been recently proposed for this task, but their critical analysis and comparison is still lacking. Here we perform an analytical comparison of both algorithms and evaluate their performance on a set of problems, while discussing their strengths and limitations. Additionally, we use these analyses to provide insights about certain aspects of the relationship between inputs and observability. We found that, while ORC-DF and FISPO follow a similar approach, they differ in key aspects that can have a substantial influence on their applicability and computational cost. The FISPO algorithm is more generally applicable, since it can analyse any nonlinear ODE model. The ORC-DF algorithm analyses models that are affine in the inputs, and if those models have known inputs it is sometimes more efficient. Thus, the optimal choice of a method depends on the characteristics of the problem under consideration. To facilitate the use of both algorithms, we implemented the ORC-DF condition in a new version of STRIKE-GOLDD, a MATLAB toolbox for structural identifiability and observability analysis. Since this software tool already had an implementation of the FISPO algorithm, the new release allows modellers and model users the convenience of choosing between different algorithms in a single tool, without changing the coding of their model. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Article
Group Invariant Solutions and Conserved Quantities of a (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation
Mathematics 2020, 8(6), 1012; https://doi.org/10.3390/math8061012 - 20 Jun 2020
Cited by 3 | Viewed by 716
Abstract
In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general [...] Read more.
In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general since they contain elliptic functions. Finally, we derive the conserved quantities of this equation by employing two approaches—the general multiplier approach and Ibragimov’s theorem. The importance of conservation laws is explained in the introduction. It should be pointed out that the investigation of higher dimensional nonlinear partial differential equations is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
Show Figures

Figure 1

Back to TopTop