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Mathematics
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Numerical Analysis and Mathematical Modeling in Computational Mechanics and Engineering
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Numerical Analysis and Mathematical Modeling in Computational Mechanics and Engineering

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 31 July 2025 | Viewed by 1394

Special Issue Editor

Dr. Chung-Lun Kuo

E-Mail Website
Guest Editor
Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Interests: meshless method; inverse problem; dynamical system; algebraic equation; Lie-group method

Special Issue Information

Dear Colleagues,

Numerical analysis and mathematical modeling are important and powerful tools in scientific research and are widely applied in many fields. Additionally, the development of big data analytics has become an extremely significant trend in the technology sector in recent years. With the rapid increase in data volume, extracting value and insights from data has become crucial. This Special Issue aims to cover the latest developments in numerical analysis and mathematical modeling, as well as their applications in computational mechanics and engineering. Topics listed below, as well as other related topics, are welcome:

  • Development and application of numerical methods;
  • Development and application of mathematical modeling;
  • High-performance computing;
  • Multiphysics problems;
  • Multiscale modeling;
  • Big data analytics and machine learning.

Dr. Chung-Lun Kuo
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. 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 2600 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

  • differential equation
  • integral equation
  • algebraic equation
  • optimization problem
  • inverse problem
  • interpolation
  • big data analytics
  • machine learning

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (1 paper)

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Research

29 pages, 1395 KiB  
Article
Decomposition–Linearization–Sequential Homotopy Methods for Nonlinear Differential/Integral Equations
by Chein-Shan Liu, Chung-Lun Kuo and Chih-Wen Chang
Mathematics 2024, 12(22), 3557; https://doi.org/10.3390/math12223557 - 14 Nov 2024
Cited by 2 | Viewed by 857
Abstract
In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of [...] Read more.
In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of certain function obtained at the previous step for obtaining a sequential analytic solution of nonlinear differential equation, which provides quite accurate analytic solution. Some first- and second-order nonlinear differential equations display the fast convergence and accuracy of the DLSM. An analytic approximation for the Volterra differential–integral equation model of the population growth of a species is obtained by using the LHPM. In addition, the LHPM is also applied to the first-, second-, and third-order nonlinear ordinary differential equations. To reduce the cost of computation of He’s homotopy perturbation method and enhance the accuracy for solving cubically nonlinear jerk equations, the LHPM is implemented by invoking a linearization technique in advance is developed. A generalization of the LHPM to the nth-order nonlinear differential equation is involved, which can greatly simplify the work to find an analytic solution by solving a set of second-order linear differential equations. A remarkable feature of those new analytic methods is that just a few steps and lower-order approximations are sufficient for producing reasonably accurate analytic solutions. For all examples, the second-order analytic solution x2(t) is found to be a good approximation of the real solution. The accuracy of the obtained approximate solutions are identified by the fourth-order Runge–Kutta method. The major objection is to unify the analytic solution methods of different nonlinear differential equations by simply solving a set of first-order or second-order linear differential equations. It is clear that the new technique considerably saves computational costs and converges faster than other analytical solution techniques existing in the literature, including the Picard iteration method. Moreover, the accuracy of the obtained analytic solution is raised. Full article
(This article belongs to the Special Issue Numerical Analysis and Mathematical Modeling in Computational Mechanics and Engineering)
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