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Axioms
Special Issues
Applied Mathematics and Numerical Analysis: Theory and Applications
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Applied Mathematics and Numerical Analysis: Theory and Applications

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

Deadline for manuscript submissions: 30 May 2025 | Viewed by 4198

Special Issue Editor

Dr. Avrilia Konguetsof

E-Mail Website
Guest Editor
Department of Civil Engineering, Polytechnic School, Democritus University of Thrace, Kimmeria Campus, 671 00 Xanthi, Greece
Interests: applied mathematics; numerical analysis; numerical solution of differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Numerical analysis is a major branch of mathematics which consists of mathematical approximation techniques and computational methods.

Numerical methods are applied in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business, etc. The main interests of numerical schemes include approximation, simulation, and estimation, and they are used in virtually every scientific field.

In this Special Issue, original research articles and reviews are welcome. Research areas may include (but are not limited to) the following:

Numerical approaches and solutions of ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), delay differential equations (DDEs), and differential algebraic equations (DAEs); numerical stability; interpolation; approximation; quadrature methods; numerical linear algebra; initial and boundary conditions; numerical fractional analyses; optimization; integral equations; iterative methods for solving nonlinear equations and systems; and applications for solving real problems in science and engineering.

I look forward to receiving your contributions.

Dr. Avrilia Konguetsof
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All 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. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • ordinary differential equations (ODEs)
  • partial differential equations (PDEs)
  • stochastic differential equations (SDEs)
  • delay differential equations (DDEs)
  • differential algebraic equations (DAEs)
  • integral equations
  • iterative methods
  • fluid dynamics
  • thermodynamics
  • quantum dynamics
  • control theory

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Published Papers (5 papers)

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Research

25 pages, 1086 KiB  
Article
On the Existence, Uniqueness and a Numerical Approach to the Solution of Fractional Cauchy–Euler Equation
by Nazim I. Mahmudov, Suzan Cival Buranay and Mtema James Chin
Axioms 2024, 13(9), 627; https://doi.org/10.3390/axioms13090627 - 12 Sep 2024
Viewed by 573
Abstract
In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order 0<α<2. The problem also constitutes a class of examples of the Cauchy problem of the [...] Read more.
In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order 0<α<2. The problem also constitutes a class of examples of the Cauchy problem of the Bagley–Torvik equation with variable coefficients. For proving the existence and uniqueness of the solution of the given problem, the contraction mapping principle is utilized. Furthermore, a numerical method and an algorithm are developed for obtaining the approximate solution. Also, convergence analyses are studied, and simulations on some test problems are given. It is shown that the proposed method and the algorithm are easy to implement on a computer and efficient in computational time and storage. Full article
(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)
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20 pages, 904 KiB  
Article
Analytical and Numerical Approaches via Quadratic Integral Equations
by Jihan Alahmadi, Mohamed A. Abdou and Mohamed A. Abdel-Aty
Axioms 2024, 13(9), 621; https://doi.org/10.3390/axioms13090621 - 12 Sep 2024
Viewed by 409
Abstract
A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space C([0,T]×[0,T]). The existence of at least one solution to the QIE is [...] Read more.
A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space C([0,T]×[0,T]). The existence of at least one solution to the QIE is discussed in this article. Our evidence depends on a suitable combination of the measures of the noncompactness approach and the fixed-point principle of Darbo. The quadratic integral equation can be used to derive a system of integral equations of the second kind using the quadrature method. With the aid of two different polynomials, Laguerre and Hermite, the system of integral equations is solved using the collocation method. In each numerical approach, the estimation of the error is discussed. Finally, using some examples, the accuracy and scalability of the proposed method are demonstrated along with comparisons. Mathematica 11 was used to obtain all of the results from the techniques that were shown. Full article
(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)
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21 pages, 7804 KiB  
Article
Stability and Convergence Analysis of the Discrete Dynamical System for Simulating a Moving Bed
by Chao-Fan Xie, Hong Zhang and Rey-Chue Hwang
Axioms 2024, 13(9), 586; https://doi.org/10.3390/axioms13090586 - 28 Aug 2024
Viewed by 465
Abstract
The efficiency of controlling the simulated moving bed (SMB) has long been a critical issue in the chemical engineering industry. Most existing research relies on finite element methods, which often result in lower control efficiency and are unable to achieve online control. To [...] Read more.
The efficiency of controlling the simulated moving bed (SMB) has long been a critical issue in the chemical engineering industry. Most existing research relies on finite element methods, which often result in lower control efficiency and are unable to achieve online control. To enhance control over the SMB process, this paper employs the Crank–Nicolson method to develop a discrete dynamical model. This approach allows for the investigation of system stability and convergence, fundamentally addressing the sources of error. During the discretization of partial differential equations (PDEs), two main types of errors arise: intrinsic errors from the method itself and truncation errors due to derivative approximations and the discretization process. Research indicates that for the former, the iterative process remains convergent as long as the time and spatial steps are sufficiently small. Regarding truncation errors, studies have demonstrated that they exhibit second-order behavior relative to time and spatial steps. The theoretical validation shows that the iteration works effectively, and simulations confirm that the finite difference method is stable and performs well with varying SMB system parameters and controller processes. This provides a solid theoretical foundation for practical, real-time online control. Full article
(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)
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13 pages, 428 KiB  
Article
Homotopy Analysis Transform Method for a Singular Nonlinear Second-Order Hyperbolic Pseudo-Differential Equation
by Said Mesloub and Hassan Eltayeb Gadain
Axioms 2024, 13(6), 398; https://doi.org/10.3390/axioms13060398 - 14 Jun 2024
Cited by 1 | Viewed by 706
Abstract
In this study, we employed the homotopy analysis transform method (HATM) to derive an iterative scheme to numerically solve a singular second-order hyperbolic pseudo-differential equation. We evaluated the effectiveness of the derived scheme in solving both linear and nonlinear equations of similar nature [...] Read more.
In this study, we employed the homotopy analysis transform method (HATM) to derive an iterative scheme to numerically solve a singular second-order hyperbolic pseudo-differential equation. We evaluated the effectiveness of the derived scheme in solving both linear and nonlinear equations of similar nature through a series of illustrative examples. The stability of this scheme in handling the approximate solutions of these examples was studied graphically and numerically. A comparative analysis with existing methodologies from the literature was conducted to assess the performance of the proposed approach. Our findings demonstrate that the HATM-based method offers notable efficiency, accuracy, and ease of implementation when compared to the alternative technique considered in this study. Full article
(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)
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12 pages, 271 KiB  
Article
Optimal Fourth-Order Methods for Multiple Zeros: Design, Convergence Analysis and Applications
by Sunil Kumar, Janak Raj Sharma and Lorentz Jäntschi
Axioms 2024, 13(3), 143; https://doi.org/10.3390/axioms13030143 - 23 Feb 2024
Cited by 1 | Viewed by 1266
Abstract
Nonlinear equations are frequently encountered in many areas of applied science and engineering, and they require efficient numerical methods to solve. To ensure quick and precise root approximation, this study presents derivative-free iterative methods for finding multiple zeros with an ideal fourth-order convergence [...] Read more.
Nonlinear equations are frequently encountered in many areas of applied science and engineering, and they require efficient numerical methods to solve. To ensure quick and precise root approximation, this study presents derivative-free iterative methods for finding multiple zeros with an ideal fourth-order convergence rate. Furthermore, the study explores applications of the methods in both real-life and academic contexts. In particular, we examine the convergence of the methods by applying them to the problems, namely Van der Waals equation of state, Planck’s law of radiation, the Manning equation for isentropic supersonic flow and some academic problems. Numerical results reveal that the proposed derivative-free methods are more efficient and consistent than existing methods. Full article
(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)
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