Differential Equations and Related Topics

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 2 July 2024 | Viewed by 23313

Special Issue Editors


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Guest Editor
Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania
Interests: differential equations; stability; convexity

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Guest Editor
Department of Mathematics, Faculty of Basic Sciences, University of Maragheh, Maragheh 55181-83111, Iran
Interests: numerical analysis; scientific computing
Special Issues, Collections and Topics in MDPI journals

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Guest Editor
Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania
Interests: differential equations; stability; varational inequalities

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Guest Editor
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Interests: special functions; theory of distributions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue aims to collect new and original results in relation to differential equations. Papers related to theoretical aspects, numerical schemes, and various types of stability (Ulam stability, numerical stability, etc.) are welcome. Differential equations model a wide range of problems in engineering, economics, biology, chemistry, medicine, etc. Over the last few years, fractional differential equations have been successfully applied to the study of numerous physical problems in the areas of electronics, chemistry, biology, mechanics, chaos, fluid mechanics, epidemiology, and modelling. Real-world applications, therefore, facilitate interdisciplinary research. Hence, this Special Issue aims to bring mathematicians together with physicists, engineers, and other scientists, for whom differential equations are valuable research tools.

Dr. Daniela Marian
Prof. Dr. Ali Shokri
Dr. Daniela Inoan
Dr. Kamsing Nonlaopon
Guest Editors

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Keywords

  • differential equations
  • partial differential equations
  • fractional differential equations
  • initial-value problems
  • boundary value problems
  • ulam’s type stability
  • fixed-point theory
  • approximation theory
  • numerical schemes
  • integral transforms
  • real-world applications
  • other related topics

Published Papers (26 papers)

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14 pages, 279 KiB  
Article
Second-Order Damped Differential Equations with Superlinear Neutral Term: New Criteria for Oscillation
by Asma Al-Jaser, Clemente Cesarano, Belgees Qaraad and Loredana Florentina Iambor
Axioms 2024, 13(4), 234; https://doi.org/10.3390/axioms13040234 - 01 Apr 2024
Viewed by 462
Abstract
This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. [...] Read more.
This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. To prove these results, we employed various analysis methods, establishing new relationships to address certain problems that have hindered previous research. Consequently, by applying the principles of comparison and the Riccati transformation, we obtained findings that develop and complement those reported in earlier literature. The significance of our results is illustrated with several examples. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
16 pages, 1461 KiB  
Article
Accurate Approximations for a Nonlinear SIR System via an Efficient Analytical Approach: Comparative Analysis
by Mona Aljoufi
Axioms 2024, 13(3), 167; https://doi.org/10.3390/axioms13030167 - 04 Mar 2024
Viewed by 720
Abstract
The homotopy perturbation method (HPM) is one of the recent fundamental methods for solving differential equations. However, checking the accuracy of this method has been ignored by some authors in the literature. This paper reanalyzes the nonlinear system of ordinary differential equations (ODEs) [...] Read more.
The homotopy perturbation method (HPM) is one of the recent fundamental methods for solving differential equations. However, checking the accuracy of this method has been ignored by some authors in the literature. This paper reanalyzes the nonlinear system of ordinary differential equations (ODEs) describing the SIR epidemic model, which has been solved in the literature utilizing the HPM. The main objective of this work is to obtain a highly accurate analytical solution for this model via a direct technique. The proposed technique is mainly based on reducing the given system to a single nonlinear ODE that can be easily solved. Numerical results are conducted to compare our approach with the previous HPM, where the Runge–Kutta numerical method is chosen as a reference solution. The obtained results reveal that the current technique exhibits better accuracy over HPM in the literature. Moreover, some physical properties are introduced and discussed in detail regarding the influence of the transmission rate on the behavior of the SIR model. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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23 pages, 604 KiB  
Article
Comparing the Performance of Two Butcher-Based Block Hybrid Algorithms for the Solution of Initial Value Problems
by Richard Olatokunbo Akinola, Ali Shokri, Joshua Sunday, Daniela Marian and Oyindamola D. Akinlabi
Axioms 2024, 13(3), 165; https://doi.org/10.3390/axioms13030165 - 01 Mar 2024
Viewed by 760
Abstract
In this paper, we compare the performances of two Butcher-based block hybrid methods for the numerical integration of initial value problems. We compare the condition numbers of the linear system of equations arising from both methods and the absolute errors of the solution [...] Read more.
In this paper, we compare the performances of two Butcher-based block hybrid methods for the numerical integration of initial value problems. We compare the condition numbers of the linear system of equations arising from both methods and the absolute errors of the solution obtained. The results of the numerical experiments illustrate that the better conditioned method outperformed its less conditioned counterpart based on the absolute errors. In addition, after applying our method on some examples, it was discovered that the absolute errors in this work were better than those of a recent study in the literature. Hence, we recommend this method for the numerical solution of stiff and non-stiff initial value problems. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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12 pages, 903 KiB  
Article
Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation
by A. A. Al Qarni
Axioms 2024, 13(1), 1; https://doi.org/10.3390/axioms13010001 - 19 Dec 2023
Viewed by 877
Abstract
The standard pantograph delay equation (SPDDE) is one of the famous delay models. This standard model is basically homogeneous in nature and it has been extensively studied in the literature. However, the studies on the general inhomogeneous form of such a model seem [...] Read more.
The standard pantograph delay equation (SPDDE) is one of the famous delay models. This standard model is basically homogeneous in nature and it has been extensively studied in the literature. However, the studies on the general inhomogeneous form of such a model seem rare. This paper considers the inhomogeneous pantograph delay equation (IPDDE) with a kind of arbitrary inhomogeneous term. This arbitrary inhomogeneous term is used in different forms to generate various classes of IPDDEs. The solutions of the present classes are obtained in closed series forms which satisfy the criteria of convergence. Also, the exact solutions are determined for these classes under a certain relation between the given initial condition of the model and the initial value of the inhomogeneous term. Several classes are generated and solved when the inhomogeneous term takes the form of trigonometric, exponential, and hyperbolic functions. Some existing results in the literature are recovered as special cases of the present ones. Moreover, the behaviors of the obtained solutions are demonstrated through graphs for various kinds of IPDDEs. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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13 pages, 327 KiB  
Article
Fixed Point Theorems in Rectangular b-Metric Space Endowed with a Partial Order
by Andrei Horvat-Marc, Mariana Cufoian, Adriana Mitre and Ioana Taşcu
Axioms 2023, 12(11), 1050; https://doi.org/10.3390/axioms12111050 - 15 Nov 2023
Viewed by 826
Abstract
The purpose of this paper is to present some fixed-point results for self-generalized contractions in ordered rectangular b-metric spaces. We also provide some examples that illustrate the non-triviality and richness of this area of research. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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17 pages, 349 KiB  
Article
Multiple Positive Solutions for a System of Fractional Order BVP with p-Laplacian Operators and Parameters
by Abdullah Ali H. Ahmadini, Mahammad Khuddush and Sabbavarapu Nageswara Rao
Axioms 2023, 12(10), 974; https://doi.org/10.3390/axioms12100974 - 17 Oct 2023
Cited by 1 | Viewed by 925
Abstract
In this paper, we investigate the existence of positive solutions to a system of fractional differential equations that include the (r1,r2,r3)-Laplacian operator, three-point boundary conditions, and various fractional derivatives. We use a combination [...] Read more.
In this paper, we investigate the existence of positive solutions to a system of fractional differential equations that include the (r1,r2,r3)-Laplacian operator, three-point boundary conditions, and various fractional derivatives. We use a combination of techniques, including cone expansion and compression of the functional type, and the Leggett–Williams fixed point theorem, to prove the existence of positive solutions. Finally, we provide two examples to illustrate our main results. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
9 pages, 286 KiB  
Article
Existence Results for a Class of Fractional Differential Beam Type Equations
by Imed Bachar, Hassan Eltayeb and Said Mesloub
Axioms 2023, 12(10), 939; https://doi.org/10.3390/axioms12100939 - 29 Sep 2023
Viewed by 503
Abstract
Fractional differential beam type equations are considered. By using an efficient approach, we prove the existence and uniqueness of continuous solutions. An iterative scheme for approximating the solution is given. Some examples are presented. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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28 pages, 404 KiB  
Article
Existence and Uniqueness of Variable-Order φ-Caputo Fractional Two-Point Nonlinear Boundary Value Problem in Banach Algebra
by Yahia Awad, Hussein Fakih and Yousuf Alkhezi
Axioms 2023, 12(10), 935; https://doi.org/10.3390/axioms12100935 - 29 Sep 2023
Cited by 1 | Viewed by 606
Abstract
Using variable-order fractional derivatives in differential equations is essential. It enables more precise modeling of complex phenomena with varying memory and long-range dependencies, improving our ability to describe real-world processes reliably. This study investigates the properties of solutions for a two-point boundary value [...] Read more.
Using variable-order fractional derivatives in differential equations is essential. It enables more precise modeling of complex phenomena with varying memory and long-range dependencies, improving our ability to describe real-world processes reliably. This study investigates the properties of solutions for a two-point boundary value problem associated with φ-Caputo fractional derivatives of variable order. The primary objectives are to establish the existence and uniqueness of solutions, as well as explore their stability through the Ulam-Hyers concept. To achieve these goals, Banach’s and Krasnoselskii’s fixed point theorems are employed as powerful mathematical tools. Additionally, we provide numerical examples to illustrate results and enhance comprehension of theoretical findings. This comprehensive analysis significantly advances our understanding of variable-order fractional differential equations, providing a strong foundation for future research. Future directions include exploring more complex boundary value problems, studying the effects of varying fractional differentiation orders, extending the analysis to systems of equations, and applying these findings to real-world scenarios, all of which promise to deepen our understanding of Caputo fractional differential equations with variable order, driving progress in both theoretical and applied mathematics. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
16 pages, 285 KiB  
Article
Best Constant in Ulam Stability for the Third Order Linear Differential Operator with Constant Coefficients
by Alina Ramona Baias and Dorian Popa
Axioms 2023, 12(10), 922; https://doi.org/10.3390/axioms12100922 - 27 Sep 2023
Cited by 1 | Viewed by 542
Abstract
The authors of the present paper previously proved the Ulam stability for the n-th-order linear differential operator with constant coefficients. They obtained its best Ulam constant for the case of distinct roots of the characteristic equation. However, a complete answer to the problem [...] Read more.
The authors of the present paper previously proved the Ulam stability for the n-th-order linear differential operator with constant coefficients. They obtained its best Ulam constant for the case of distinct roots of the characteristic equation. However, a complete answer to the problem of the best Ulam constant was later obtained only for the second-order linear differential operator. This paper deals with the Ulam stability of the third-order linear differential operator with constant coefficients acting in a Banach space. The paper’s main purpose is to obtain the best Ulam constant of this operator, thus completing the previous research in the field. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
27 pages, 551 KiB  
Article
Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies
by Ezekiel Olaoluwa Omole, Emmanuel Oluseye Adeyefa, Victoria Iyadunni Ayodele, Ali Shokri and Yuanheng Wang
Axioms 2023, 12(9), 891; https://doi.org/10.3390/axioms12090891 - 18 Sep 2023
Viewed by 788
Abstract
A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major [...] Read more.
A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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15 pages, 424 KiB  
Article
New Results for the Investigation of the Asymptotic Behavior of Solutions of Nonlinear Perturbed Differential Equations
by Osama Moaaz and Wedad Albalawi
Axioms 2023, 12(9), 841; https://doi.org/10.3390/axioms12090841 - 30 Aug 2023
Viewed by 623
Abstract
This study focuses on investigating the oscillatory properties of a particular class of perturbed differential equations in the noncanonical case. Our research aims to establish more effective criteria for evaluating the absence of positive solutions to the equation under study and subsequently investigate [...] Read more.
This study focuses on investigating the oscillatory properties of a particular class of perturbed differential equations in the noncanonical case. Our research aims to establish more effective criteria for evaluating the absence of positive solutions to the equation under study and subsequently investigate its oscillatory behavior. We also perform a comparative analysis, contrasting the oscillation of the studied equation with another equation in the canonical case. To achieve this, we employ the Riccati technique along with other methods to obtain several sufficient criteria. Furthermore, we apply these new conditions to specific instances of the considered equation, assessing their performance. The significance of our work lies in its extension and broadening of the existing body of literature, contributing novel insights into this field of study. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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15 pages, 1998 KiB  
Article
Application of the Triple Sumudu Decomposition Method for Solving 1+1 and 2+1-Dimensional Boussinesq Equations
by Huda Alsaud, Hassan Eltayeb and Imed Bachar
Axioms 2023, 12(9), 829; https://doi.org/10.3390/axioms12090829 - 28 Aug 2023
Viewed by 543
Abstract
The triple Sumudu transform decomposition method (TSTDM) is a combination of the Adomian decomposition method (ADM) and the triple Sumudu transform. It is a computational method that can be appropriate for solving linear and nonlinear partial differential equations. The existence analysis of the [...] Read more.
The triple Sumudu transform decomposition method (TSTDM) is a combination of the Adomian decomposition method (ADM) and the triple Sumudu transform. It is a computational method that can be appropriate for solving linear and nonlinear partial differential equations. The existence analysis of the method and partial derivatives theorems are proven. Finally, we solve the 1+1 and 2+1-dimensional Boussinesq equations by applying the (TSTDM)technique, which gives the approximate solution with quick convergence. It is more precise than using ADM alone. In addition, three examples are offered to examine the performance and precision of our method. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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33 pages, 759 KiB  
Article
Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options
by Chinonso Nwankwo and Weizhong Dai
Axioms 2023, 12(8), 762; https://doi.org/10.3390/axioms12080762 - 02 Aug 2023
Viewed by 683
Abstract
The trade-off between numerical accuracy and computational cost is always an important factor to consider when pricing options numerically, due to the inherent irregularity and existence of non-linearity in many models. In this work, we first present fast and accurate (1,2) and (2,2) [...] Read more.
The trade-off between numerical accuracy and computational cost is always an important factor to consider when pricing options numerically, due to the inherent irregularity and existence of non-linearity in many models. In this work, we first present fast and accurate (1,2) and (2,2) predictor–corrector methods with a fourth-order compact finite difference scheme for pricing coupled system of the non-linear free boundary option pricing problem consisting of the option value and delta sensitivity. To predict the optimal exercise boundary, we set up a high-order boundary scheme, which is strategically derived using a combination of the fourth-order Robin boundary scheme and the fourth-order compact finite difference scheme near boundary. Furthermore, we implement a three-step high-order correction scheme for computing interior values of the option value and delta sensitivity. The discrete matrix system of this correction scheme has a tri-diagonal structure and strictly diagonal dominance. This nice feature allows for the implementation of the Thomas algorithm, thereby enabling fast computation. The optimal exercise boundary value is also corrected in each of the three correction steps with the derived Robin boundary scheme. Our implementations are fast on both coarse and very refined grids and provide highly accurate numerical approximations. Moreover, we recover a reasonable convergence rate. Further extensions to high-order predictor two-step corrector schemes are elaborated. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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22 pages, 3547 KiB  
Article
Exact and Approximate Solutions for Linear and Nonlinear Partial Differential Equations via Laplace Residual Power Series Method
by Haneen Khresat, Ahmad El-Ajou, Shrideh Al-Omari, Sharifah E. Alhazmi and Moa’ath N. Oqielat
Axioms 2023, 12(7), 694; https://doi.org/10.3390/axioms12070694 - 17 Jul 2023
Cited by 4 | Viewed by 1408
Abstract
The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds [...] Read more.
The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein–Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach’s reliability, capability, and efficiency. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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19 pages, 340 KiB  
Article
A Non-Local Non-Homogeneous Fractional Timoshenko System with Frictional and Viscoelastic Damping Terms
by Said Mesloub, Eman Alhazzani and Gadain Hassan Eltayeb
Axioms 2023, 12(7), 689; https://doi.org/10.3390/axioms12070689 - 16 Jul 2023
Viewed by 682
Abstract
We are devoted to the study of a non-local non-homogeneous time fractional Timoshenko system with frictional and viscoelastic damping terms. We are concerned with the well-posedness of the given problem. The approach relies on some functional analysis tools, operator theory, a priori estimates [...] Read more.
We are devoted to the study of a non-local non-homogeneous time fractional Timoshenko system with frictional and viscoelastic damping terms. We are concerned with the well-posedness of the given problem. The approach relies on some functional analysis tools, operator theory, a priori estimates and density arguments. This work can be considered as a contribution to the development of energy inequality methods, the so-called a priori estimate method inspired from functional analyses and used to prove the well-posedness of mixed problems with integral boundary conditions. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
22 pages, 378 KiB  
Article
On Modified Interpolative Almost EType Contraction in Partial Modular bMetric Spaces
by Dilek Kesik, Abdurrahman Büyükkaya and Mahpeyker Öztürk
Axioms 2023, 12(7), 669; https://doi.org/10.3390/axioms12070669 - 07 Jul 2023
Viewed by 690
Abstract
The current study attempts to identify a new generalized metric space structure, referred to as partial modular bmetric, that extends both partial modular metric space via bmetric space and explains the topological aspects of the new space implementing examples. [...] Read more.
The current study attempts to identify a new generalized metric space structure, referred to as partial modular bmetric, that extends both partial modular metric space via bmetric space and explains the topological aspects of the new space implementing examples. In addition, a new contraction mapping referred to as modified interpolative almost Etype contraction is determined, which is an interpretation of interpolative contraction bestowed with almost contraction and Econtraction as well as a simulation function and a fixed point theorem that encompass such mappings in the context of partial modular bmetric space is demonstrated. In conclusion, an example and an application that endorse the main theorem’s outcomes are offered. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
16 pages, 1894 KiB  
Communication
General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions
by Zoltan Vizvari, Mihaly Klincsik, Peter Odry, Vladimir Tadic and Zoltan Sari
Axioms 2023, 12(7), 633; https://doi.org/10.3390/axioms12070633 - 27 Jun 2023
Cited by 1 | Viewed by 770
Abstract
In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an [...] Read more.
In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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18 pages, 338 KiB  
Article
A Fixed Point Theorem for Generalized Ćirić-Type Contraction in Kaleva–Seikkala’s Type Fuzzy b-Metric Spaces
by Jiaojiao Wu, Fei He and Shufang Li
Axioms 2023, 12(7), 616; https://doi.org/10.3390/axioms12070616 - 21 Jun 2023
Viewed by 776
Abstract
In this paper, we state and establish a new fixed point theorem for generalized Ćirić-type contraction in Kaleva-Seikkala’s type fuzzy b-metric space. Our results improve and extend some well-known results in the literature. Some examples are given to support our result. Finally, [...] Read more.
In this paper, we state and establish a new fixed point theorem for generalized Ćirić-type contraction in Kaleva-Seikkala’s type fuzzy b-metric space. Our results improve and extend some well-known results in the literature. Some examples are given to support our result. Finally, as an application, we show the existence and uniqueness of solution to Volterra integral equation formulated in Kaleva–Seikkala’s type fuzzy b-metric space. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
26 pages, 1126 KiB  
Article
Local Refinement and Adaptive Strategy for a System of Free Boundary Power Options with High Order Compact Differencing
by Chinonso Nwankwo and Weizhong Dai
Axioms 2023, 12(6), 602; https://doi.org/10.3390/axioms12060602 - 17 Jun 2023
Viewed by 753
Abstract
In this research, we propose fourth-order non-uniform Hermitian differencing with a fifth-order adaptive time integration method for pricing system of free boundary exotic power put options consisting of the option value, delta sensitivity, and gamma. The main objective for implementing the above scheme [...] Read more.
In this research, we propose fourth-order non-uniform Hermitian differencing with a fifth-order adaptive time integration method for pricing system of free boundary exotic power put options consisting of the option value, delta sensitivity, and gamma. The main objective for implementing the above scheme is to carefully account for the irregularity in the locality of the left corner point after fixing the free boundary. Specifically and mainly, we stretch the performance of our proposed method threefold. First, we exploit the non-uniform fourth-order Hermitian scheme to locally concentrate space grid points arbitrarily close to the left boundary. Secondly, we further leverage the adaptive nature of the embedded time integration method, which allows optimal selection of a time step based on the space grid point distribution and regional variation. Thirdly, we introduce a fourth-order combined Hermitian scheme, which requires fewer grid points for computing the near boundary point of the delta sensitivity and gamma. Another novelty is how we approximate the optimal exercise boundary and its derivative using a fifth-order Robin boundary scheme and fourth-order combined Hermitian scheme. Our proposed method consistently achieves reasonable accuracy with very coarse grids and little runtime across the numerical experiments. We further compare the results with existing methods and the ones we obtained from the uniform space grid. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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13 pages, 306 KiB  
Article
Relational Contractions Involving Shifting Distance Functions with Applications to Boundary Value Problems
by Ebrahem Ateatullah Algehyne, Mounirah Areshi and Faizan Ahmad Khan
Axioms 2023, 12(5), 478; https://doi.org/10.3390/axioms12050478 - 15 May 2023
Viewed by 843
Abstract
This manuscript includes certain results on fixed points under a generalized contraction involving a pair of shifting distance functions in the framework of metric space endowed with a class of transitive relation. The results presented herein are illustrated by an example. Finally, we [...] Read more.
This manuscript includes certain results on fixed points under a generalized contraction involving a pair of shifting distance functions in the framework of metric space endowed with a class of transitive relation. The results presented herein are illustrated by an example. Finally, we apply our result to compute a unique solution of certain first order boundary value problems. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
20 pages, 2036 KiB  
Article
On the Simulations of Second-Order Oscillatory Problems with Applications to Physical Systems
by Lydia J. Kwari, Joshua Sunday, Joel N. Ndam, Ali Shokri and Yuanheng Wang
Axioms 2023, 12(3), 282; https://doi.org/10.3390/axioms12030282 - 08 Mar 2023
Cited by 3 | Viewed by 881
Abstract
Second-order oscillatory problems have been found to be applicable in studying various phenomena in science and engineering; this is because these problems have the capabilities of replicating different aspects of the real world. In this research, a new hybrid method shall be formulated [...] Read more.
Second-order oscillatory problems have been found to be applicable in studying various phenomena in science and engineering; this is because these problems have the capabilities of replicating different aspects of the real world. In this research, a new hybrid method shall be formulated for the simulations of second-order oscillatory problems with applications to physical systems. The proposed method shall be formulated using the procedure of interpolation and collocation by adopting power series as basis function. In formulating the method, off-step points were introduced within the interval of integration in order to bypass the Dahlquist barrier, improve the accuracy of the method and also upgrade the order of consistence of the method. The paper further validated the some properties of the hybrid method derived and from the results obtained; the new method was found to be consistent, convergent and stable. The simulation results generated as a result of the application of the new method on some second-order oscillatory differential equations also showed that the new hybrid method is computationally reliable. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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14 pages, 369 KiB  
Article
Semi-Hyers–Ulam–Rassias Stability of Some Volterra Integro-Differential Equations via Laplace Transform
by Daniela Inoan and Daniela Marian
Axioms 2023, 12(3), 279; https://doi.org/10.3390/axioms12030279 - 07 Mar 2023
Viewed by 856
Abstract
In this paper the semi-Hyers–Ulam–Rassias stability of some Volterra integro-differential equations is investigated, using the Laplace transform. This is a continuation of some previous work on this topic. The equation in the general form contains more terms, where the unknown function appears together [...] Read more.
In this paper the semi-Hyers–Ulam–Rassias stability of some Volterra integro-differential equations is investigated, using the Laplace transform. This is a continuation of some previous work on this topic. The equation in the general form contains more terms, where the unknown function appears together with the derivative of order one and with two integral terms. The particular cases that are considered illustrate the main results for some polynomial and exponential functions. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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20 pages, 361 KiB  
Article
Infinitely Many Small Energy Solutions to the Double Phase Anisotropic Variational Problems Involving Variable Exponent
by Jun-Hyuk Ahn and Yun-Ho Kim
Axioms 2023, 12(3), 259; https://doi.org/10.3390/axioms12030259 - 02 Mar 2023
Viewed by 1119
Abstract
This paper is devoted to double phase anisotropic variational problems for the case of a combined effect of concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, on a class of superlinear term which [...] Read more.
This paper is devoted to double phase anisotropic variational problems for the case of a combined effect of concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, on a class of superlinear term which is different from the previous related works, is to discuss the multiplicity result of non-trivial solutions by applying the dual fountain theorem as the main tool. In particular, our main result is obtained without assuming the conditions on the nonlinear term at infinity. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
17 pages, 557 KiB  
Article
A New Framework for Numerical Techniques for Fuzzy Nonlinear Equations
by Fazlollah Abbasi, Tofigh Allahviranloo and Muhammad Akram
Axioms 2023, 12(2), 222; https://doi.org/10.3390/axioms12020222 - 20 Feb 2023
Viewed by 1202
Abstract
This paper describes a computational method for solving the nonlinear equations with fuzzy input parameters that we encounter in engineering system analysis. In addition to discussing the existence of solutions, the definition and formalization of numerical solutions is based on a new fuzzy [...] Read more.
This paper describes a computational method for solving the nonlinear equations with fuzzy input parameters that we encounter in engineering system analysis. In addition to discussing the existence of solutions, the definition and formalization of numerical solutions is based on a new fuzzy computation operation as a transmission average. Error analysis in numerical solutions is described. Finally, some examples are presented to implement the proposed method and its effectiveness compared to other previous methods. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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13 pages, 396 KiB  
Article
Third-Order Neutral Differential Equation with a Middle Term and Several Delays: Asymptotic Behavior of Solutions
by Barakah Almarri, Osama Moaaz, Mona Anis and Belgees Qaraad
Axioms 2023, 12(2), 166; https://doi.org/10.3390/axioms12020166 - 07 Feb 2023
Cited by 1 | Viewed by 1052
Abstract
This study aims to investigate the asymptotic behavior of a class of third-order delay differential equations. Here, we consider an equation with a middle term and several delays. We obtain an iterative relationship between the positive solution of the studied equation and the [...] Read more.
This study aims to investigate the asymptotic behavior of a class of third-order delay differential equations. Here, we consider an equation with a middle term and several delays. We obtain an iterative relationship between the positive solution of the studied equation and the corresponding function. Using this new relationship, we derive new criteria that ensure that all non-oscillatory solutions converge to zero. The new findings are an extension and expansion of relevant findings in the literature. We apply our results to a special case of the equation under study to clarify the importance of the new criteria. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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Review

Jump to: Research

19 pages, 2205 KiB  
Review
Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction
by Alit Kartiwa, Asep K. Supriatna, Endang Rusyaman and Jumat Sulaiman
Axioms 2023, 12(5), 483; https://doi.org/10.3390/axioms12050483 - 16 May 2023
Viewed by 1535
Abstract
Quaternion is a four-dimensional and an extension of the complex number system. It is often viewed from various fields, such as analysis, algebra, and geometry. Several applications of quaternions are related to an object’s rotation and motion in three-dimensional space in the form [...] Read more.
Quaternion is a four-dimensional and an extension of the complex number system. It is often viewed from various fields, such as analysis, algebra, and geometry. Several applications of quaternions are related to an object’s rotation and motion in three-dimensional space in the form of a differential equation. In this paper, we do a systematic literature review on the development of quaternion differential equations. We utilize PRISMA (preferred reporting items for systematic review and meta-analyses) framework in the review process as well as content analysis. The expected result is a state-of-the-art and the gap of concepts or problems that still need to develop or answer. It was concluded that there are still some opportunities to develop a quaternion differential equation using a quaternion function domain. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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