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Open AccessArticle

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 291

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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Open AccessArticle

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 292

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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Open AccessArticle

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 190

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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Open AccessArticle

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 361

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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Open AccessArticle

Exact and Approximate Solutions for Linear and Nonlinear Partial Differential Equations via Laplace Residual Power Series Method

by

,

,

,

and

Axioms 2023, 12(7), 694; https://doi.org/10.3390/axioms12070694 - 17 Jul 2023

Viewed by 563

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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Open AccessArticle

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 362

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)

Open AccessArticle

On Modified Interpolative Almost

E -

Type Contraction in Partial Modular

b -

Metric 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 418

Abstract

The current study attempts to identify a new generalized metric space structure, referred to as partial modular

b -

metric, that extends both partial modular metric space via

b -

metric 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

b -

metric, that extends both partial modular metric space via

b -

metric space and explains the topological aspects of the new space implementing examples. In addition, a new contraction mapping referred to as modified interpolative almost

E -

type contraction is determined, which is an interpretation of interpolative contraction bestowed with almost contraction and

E -

contraction as well as a simulation function and a fixed point theorem that encompass such mappings in the context of partial modular

b -

metric 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)

Open AccessCommunication

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

by

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,

,

and

Axioms 2023, 12(7), 633; https://doi.org/10.3390/axioms12070633 - 27 Jun 2023

Viewed by 424

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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Open AccessArticle

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 436

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)

Open AccessArticle

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 440

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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Open AccessArticle

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 514

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)

Open AccessArticle

On the Simulations of Second-Order Oscillatory Problems with Applications to Physical Systems

by

,

,

,

and

Axioms 2023, 12(3), 282; https://doi.org/10.3390/axioms12030282 - 08 Mar 2023

Cited by 2 | Viewed by 588

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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Open AccessArticle

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 631

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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Open AccessArticle

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 824

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)

Open AccessArticle

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 677

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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Open AccessArticle

Third-Order Neutral Differential Equation with a Middle Term and Several Delays: Asymptotic Behavior of Solutions

by

,

,

and

Axioms 2023, 12(2), 166; https://doi.org/10.3390/axioms12020166 - 07 Feb 2023

Cited by 1 | Viewed by 776

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

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Open AccessReview

Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction

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,

and

Axioms 2023, 12(5), 483; https://doi.org/10.3390/axioms12050483 - 16 May 2023

Viewed by 790

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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