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Computational Mathematics Methods and Applications in Engineering Science
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Computational Mathematics Methods and Applications in Engineering Science

A Special Issue of Mathematics (ISSN 2227-7390) belonging to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 25 February 2027 | Viewed by 3364

Editor

Prof. Dr. Youness El Hamzaoui

E-Mail Website
Guest Editor
Engineering College, Carmen Autonomous University, Calle 56, 4, Esq. Avenida Concordia, Col. Benito Juárez, Campeche, Mexico
Interests: artificial neural network architectures and optimization; advanced backpropagation algorithm development; statistical modeling for environmental systems; process parameter optimization; machine learning in environmental engineering; neural network development; statistical methods; computational techniques; advanced process technologies; methodological expertise
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Engineering and applied sciences are undergoing a profound transformation, driven by the increasing power and sophistication of computational mathematics. The ability to model, simulate, and optimize complex systems has become indispensable for innovation and problem-solving across all engineering disciplines. From designing next-generation materials to optimizing sustainable energy systems and developing intelligent infrastructure, advanced computational methods are at vital for modern scientific discovery and technological advancement.

This Special Issue, "Computational Mathematics Methods and Applications in Engineering Science," will bring together leading researchers, scientists, and engineers to share their latest theoretical advancements and practical applications in this dynamic field. We seek to create a comprehensive collection of high-impact articles that not only showcase novel mathematical techniques but also demonstrate their successful application to solve pressing real-world engineering challenges. This Special Issue’s scope is intentionally broad to foster cross-disciplinary collaboration, highlighting the universal power of computational mathematics as a foundational tool for modern engineering.

We invite submissions of original research articles, comprehensive reviews, and insightful communications that bridge the gap between mathematical theory and engineering practice.

Prof. Dr. Youness El Hamzaoui
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 250 words) can be sent to the Editorial Office for assessment.

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-anonymized 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

  • numerical methods for differential equations: the finite element method (FEM) and finite difference method (FDM)
  • the boundary element method (BEM)
  • mesh-free methods and particle methods
  • numerical solutions for partial differential equations (PDEs) and ordinary differential equations (ODEs) in engineering models
  • computational optimization and operations research: linear and nonlinear programming
  • heuristic and metaheuristic algorithms (e.g., genetic algorithms, particle swarm optimization)
  • applications in logistics, structural design, and resource management
  • machine learning and artificial intelligence in engineering: neural networks, deep learning, and reinforcement learning for system modeling and control
  • data-driven modeling and surrogate models for complex simulations
  • applications in predictive maintenance, process control, and materials discovery
  • modeling and simulation: multiphysics and multiscale modeling
  • computational fluid dynamics (CFD)
  • solid mechanics and structural analysis
  • simulation of transport phenomena (heat, mass, and momentum)
  • computational statistics and data analysis: bayesian methods and uncertainty quantification
  • high-dimensional data analysis and signal processing
  • statistical modeling for engineering reliability and risk assessment
  • applications in engineering disciplines: civil and environmental engineering (e.g., water resource management, structural health monitoring)
  • mechanical and aerospace engineering (e.g., aerodynamics, robotics, thermodynamics)
  • chemical and process engineering (e.g., reactor design, separation processes)
  • electrical engineering (e.g., electromagnetics, circuit simulation, control systems). materials science (e.g., computational materials design)

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Published Papers (7 papers)

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Research

26 pages, 2567 KB  
Article
A Recentered-Domain Yau–Yau Filter with Reduced-Order FKE Propagation for Nonlinear State Estimation
by Lei Ma, Yuzhong Hu and Xiaoming John Zhang
Mathematics 2026, 14(18), 3401; https://doi.org/10.3390/math14183401 (registering DOI) - 19 Sep 2026
Abstract
Nonlinear filtering can be formulated as the propagation and update of conditional probability densities, but direct numerical propagation of the associated Forward Kolmogorov equation (FKE) over a large fixed domain is computationally expensive. This paper proposes a Recentered-Domain Yau–Yau Filter (RD-YYF) with reduced-order [...] Read more.
Nonlinear filtering can be formulated as the propagation and update of conditional probability densities, but direct numerical propagation of the associated Forward Kolmogorov equation (FKE) over a large fixed domain is computationally expensive. This paper proposes a Recentered-Domain Yau–Yau Filter (RD-YYF) with reduced-order FKE propagation for nonlinear state estimation. The method solves the FKE on a fixed-size local computational window centered at the latest state estimate, thereby concentrating numerical resolution near the dominant posterior density. In the offline stage, physics-informed neural networks (PINNs) generate FKE solution snapshots, principal component analysis constructs a low-dimensional representation of density evolution, and a lightweight residual surrogate maps initial-condition coefficients and the domain center to terminal-solution coefficients. In the online stage, the pretrained surrogate performs per-timestep density prediction within the recentered window, followed by observation update and state estimation. Numerical experiments on two geometrically constrained target-tracking models show that RD-YYF achieves lower tracking errors than the extended Kalman filter and particle filter under matched online evaluation conditions. A fixed-domain ablation further shows that recentering improves density approximation in high-probability regions and reduces offline PINN training epochs. These results indicate that recentered-domain reduced-order FKE propagation is a practical computational strategy for nonlinear density-based filtering. Full article
(This article belongs to the Special Issue Computational Mathematics Methods and Applications in Engineering Science)
25 pages, 3912 KB  
Article
Optimal Thermal Design of a Micro Pin-Fin Heat Sink Using Hybrid Fin Heights and Various Perforated Fin Shapes
by Cheng-Hung Huang and Ching-Ping Hsu
Mathematics 2026, 14(17), 3227; https://doi.org/10.3390/math14173227 - 7 Sep 2026
Viewed by 189
Abstract
This study presents a numerical optimization framework for a micro pin-fin heat sink (MPFHS). Three-dimensional conjugate heat transfer and fluid flow are simulated using the commercial CFD code CFD-ACE+. Coupled with the Levenberg–Marquardt method (LMM), the geometric parameters are optimized to minimize the [...] Read more.
This study presents a numerical optimization framework for a micro pin-fin heat sink (MPFHS). Three-dimensional conjugate heat transfer and fluid flow are simulated using the commercial CFD code CFD-ACE+. Coupled with the Levenberg–Marquardt method (LMM), the geometric parameters are optimized to minimize the base wall temperature Tbw under a constant total fin-volume constraint. The investigation assesses four pin-fin architectures—solid square (MPFHS-S), solid cylindrical (MPFHS-C), perforated square (MPFHS-SP), and perforated cylindrical (MPFHS-CP)—across five distinct fin-height distributions: uniform, constant-step (Design #1), increasing-step (Design #2), decreasing-step (Design #3), and hybrid-step (Design #4). The results demonstrate that perforated fins significantly enhance heat transfer by disrupting the thermal boundary layer and mitigating heat accumulation in the wake region. Within the allowable geometric constraints, larger perforation radii provide superior cooling performance. Among all examined configurations, the LMM-optimized Design #4 consistently achieves the lowest base temperature across the entire Reynolds number range. Specifically, for the MPFHS-CP configuration, the optimized Design #4 yields Tbw values of 326.279 K, 311.617 K, and 309.142 K at Reynolds numbers of 200, 800, and 1200, respectively. As the Reynolds number increases, the temperature differences among the configurations gradually diminish, indicating that forced convection increasingly dominates the flow and reduces the sensitivity to fin geometry. Pressure-drop analysis reveals that perforated configurations incur an approximate 10–15% higher pressure loss compared to their solid counterparts. To evaluate the overall thermo-fluid behavior, the thermal performance factor η is employed to evaluate the trade-off between heat transfer enhancement and the associated pressure-drop penalty. The results confirm that cylindrical fins consistently outperform square fins, with perforations providing an additional reduction in Tbw. Overall, combining a hybrid fin-height distribution with perforated pin fins presents an effective strategy for optimizing MPFHS performance, offering practical guidelines for the thermal management of high-power electronic devices. Full article
(This article belongs to the Special Issue Computational Mathematics Methods and Applications in Engineering Science)
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31 pages, 521 KB  
Article
Skew-Symmetric MFS and Harmonic-Polynomial-Type Methods to Solve the Three-Dimensional Stokes Equations
by Chein-Shan Liu and Chia-Cheng Tsai
Mathematics 2026, 14(17), 3109; https://doi.org/10.3390/math14173109 - 29 Aug 2026
Viewed by 203
Abstract
Based on a new general solution, we derive three skew-symmetric variants of the method of fundamental solutions (MFS) for solving the three-dimensional Stokes equations: the symmetric/skew-symmetric MFS (SSMFS), the separated MFS (SeMFS) as an extension of the conventional MFS that incorporates an additional [...] Read more.
Based on a new general solution, we derive three skew-symmetric variants of the method of fundamental solutions (MFS) for solving the three-dimensional Stokes equations: the symmetric/skew-symmetric MFS (SSMFS), the separated MFS (SeMFS) as an extension of the conventional MFS that incorporates an additional skew-symmetric part, and the symmetric/skew-symmetric separated MFS (SSeMFS) as an extension of the SSMFS by further adding a skew-symmetric component. In these methods, a skew-symmetric tensor interlaced with the Oseen tensor is employed to regularize the Stokeslet. Numerical examinations show that the SeMFS and SSeMFS outperform the conventional MFS, achieving faster convergence and an improvement in accuracy of approximately one order of magnitude. We construct six families of kth-order homogeneous harmonic polynomials that possess rich properties. These polynomials are arranged into two harmonic vectors, which are then inserted into the Liu–Hsu–Tsai formula to yield the LHT method (LHTM), into the Liu–Tsai formula to yield the LT method (LTM), and into the Slabodyanskii formula to yield the Slabodyanskii method (SM). The resulting methods solve the Stokes equations via a simple collocation technique that enforces the prescribed boundary conditions. In addition, we derive a particular solution of the Stokes equations by means of an integration method (IM). Six numerical examples, including a benchmark three-dimensional lid-driven cavity problem, are tested to assess the efficiency and accuracy of these newly developed harmonic-polynomial-type methods. They are superior to the MFS-type methods, delivering improvements in accuracy of several orders of magnitude together with much faster convergence rates. Full article
(This article belongs to the Special Issue Computational Mathematics Methods and Applications in Engineering Science)
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21 pages, 4739 KB  
Article
Image Reconstruction by Frequency Extrapolation and Deep Learning in Three-Layer Medium
by Chien-Ching Chiu, Po-Hsiang Chen, Guan-Jang Li and Eng Hock Lim
Mathematics 2026, 14(14), 2605; https://doi.org/10.3390/math14142605 - 17 Jul 2026
Viewed by 538
Abstract
This paper proposes a novel multi-frequency extended Deep Learning (DL) model for electromagnetic image reconstruction under Transverse Magnetic (TM) wave incidence in layered media, inspired by conventional microwave imaging techniques that combine nonlinear inversion algorithms with neural networks to improve reconstruction performance. The [...] Read more.
This paper proposes a novel multi-frequency extended Deep Learning (DL) model for electromagnetic image reconstruction under Transverse Magnetic (TM) wave incidence in layered media, inspired by conventional microwave imaging techniques that combine nonlinear inversion algorithms with neural networks to improve reconstruction performance. The proposed framework adopts a two-stage neural network architecture. In the first stage, a Deep Residual Convolutional Neural Network (DRCNN) is employed to extrapolate multi-frequency scattered fields from single-frequency input data, thereby enriching the frequency-dependent scattering information available for reconstruction. Subsequently, the extrapolated multi-frequency scattered fields are fed into a Deep Convolutional Encoder–Decoder (DCED) network to reconstruct an accurate dielectric constant distribution within the imaging domain. To validate the effectiveness of the proposed approach, two representative comparison methods are considered: (1) a hybrid framework combining the Back-Propagation Scheme (BPS) with a Convolutional Neural Network (CNN), and (2) a framework integrating the Dominant Current Scheme (DCS) with a CNN. In both approaches, conventional inversion algorithms are first utilized to generate coarse initial reconstructions, which are subsequently refined by the neural network. Numerical simulations and experimental results show that the proposed multi-frequency extension model achieves lower reconstruction error and higher structural similarity than the reference methods. These results confirm the effectiveness and potential of the proposed framework for advanced electromagnetic imaging applications. Full article
(This article belongs to the Special Issue Computational Mathematics Methods and Applications in Engineering Science)
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22 pages, 372 KB  
Article
An α-Cut Optimization Framework for Modular EV Charging Station Design Under Fuzzy Uncertainty
by Nikolay Hinov, Reni Kabakchieva and Plamen Stanchev
Mathematics 2026, 14(10), 1638; https://doi.org/10.3390/math14101638 - 12 May 2026
Viewed by 446
Abstract
This paper develops a unified α-cut optimization framework for modular electric vehicle (EV) fast-charging station design under fuzzy uncertainty. Uncertain peak demand, annual delivered energy, electricity price, ambient temperature, arrival rate, and energy per session are represented by triangular or trapezoidal fuzzy numbers [...] Read more.
This paper develops a unified α-cut optimization framework for modular electric vehicle (EV) fast-charging station design under fuzzy uncertainty. Uncertain peak demand, annual delivered energy, electricity price, ambient temperature, arrival rate, and energy per session are represented by triangular or trapezoidal fuzzy numbers and reformulated through α-cut bounds. The resulting design problem is expressed as a hybrid discrete–continuous model in which the number of modules, the selected catalog module rating, installed power, cooling provision, and a station-volume proxy are jointly optimized. An aggregated representation of interchangeable modules is adopted to remove permutation-equivalent descriptions and preserve a compact search space. Three planning views are examined: minimum CAPEX at a prescribed α-cut level, minimum loss-driven OPEX under a CAPEX budget, and a service-oriented admissibility/coverage analysis that avoids interpreting larger α values as greater robustness. The strengthened numerical study includes a deterministic nominal benchmark, peak demand sensitivity regimes, feasibility threshold and budget sweep results, explicit service stress scenarios, and a queueing sensitivity check against Erlang-C and discrete-event simulation indicators. The results show that baseline CAPEX designs may be dominated by catalog thresholds, whereas OPEX and service-oriented conclusions become informative once budget and traffic regimes are varied. The proposed framework is therefore positioned as a tractable α-cut-based design screening and comparative optimization tool for representative modular EV charging station scenarios, rather than as a universally validated operational design rule. Full article
(This article belongs to the Special Issue Computational Mathematics Methods and Applications in Engineering Science)
22 pages, 3781 KB  
Article
Reliability and Availability Analysis of k-out-of-M+S Retrial Machine Repair System with Two-Way Communication
by Chen-Hsiang Hsieh, Tzu-Hsin Liu, Fu-Min Chang and Yu-Tang Lee
Mathematics 2026, 14(8), 1400; https://doi.org/10.3390/math14081400 - 21 Apr 2026
Viewed by 467
Abstract
This paper studies the reliability and availability of a k-out-of-(M+S) retrial machine repair system with two-way communication, consisting of M primary components and S warm standby components. The system incorporates the retrial behavior of failed components. When the repairman becomes [...] Read more.
This paper studies the reliability and availability of a k-out-of-(M+S) retrial machine repair system with two-way communication, consisting of M primary components and S warm standby components. The system incorporates the retrial behavior of failed components. When the repairman becomes idle, he initiates outgoing calls after a random period either to failed components in the orbit for repair or to components outside the orbit for preventive maintenance. The main contribution of this study is the incorporation of proactive repairman behavior, which more realistically captures operational practices in certain engineering systems. By employing the matrix analytic method together with a recursive approach, the steady-state probabilities of the system are obtained, and several important performance measures are derived. Furthermore, the Runge–Kutta method is used to evaluate the system reliability and the mean time to failure. A sensitivity analysis is conducted to investigate the effects of key system parameters, supported by numerical experiments and graphical illustrations. Finally, a cost–benefit model is formulated, and a genetic algorithm is implemented to determine the optimal values of the decision variables that minimize the cost–benefit ratio. Full article
(This article belongs to the Special Issue Computational Mathematics Methods and Applications in Engineering Science)
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27 pages, 10557 KB  
Article
Numerical and Experimental Estimation of Heat Source Strengths in Multi-Chip Modules on Printed Circuit Boards
by Cheng-Hung Huang and Hao-Wei Su
Mathematics 2026, 14(2), 327; https://doi.org/10.3390/math14020327 - 18 Jan 2026
Viewed by 738
Abstract
In this study, a three-dimensional Inverse Conjugate Heat Transfer Problem (ICHTP) is numerically and experimentally investigated to estimate the heat-source strength of multiple chips mounted on a printed circuit board (PCB) using the Conjugate Gradient Method (CGM) and infrared thermography. The interfaces between [...] Read more.
In this study, a three-dimensional Inverse Conjugate Heat Transfer Problem (ICHTP) is numerically and experimentally investigated to estimate the heat-source strength of multiple chips mounted on a printed circuit board (PCB) using the Conjugate Gradient Method (CGM) and infrared thermography. The interfaces between the PCB and the surrounding air domain are assumed to exhibit perfect thermal contact, establishing a fully coupled conjugate heat transfer framework for the inverse analysis. Unlike the conventional Inverse Heat Conduction Problem (IHCP), which typically only accounts for conduction within solid domains, the present ICHTP formulation requires the simultaneous solution of the governing continuity, momentum, and energy equations in the air domain, along with the heat conduction equation in the chips and PCB. This coupling introduces substantial computational complexity due to the nonlinear interaction between convective and conductive heat transfer mechanisms, as well as the sensitivity of the inverse solution to measurement uncertainties. The numerical simulations are conducted first with error-free measurement data and an inlet velocity of uin = 4 m/s; the recovered heat-sources exhibit excellent agreement with the true values. The computed average errors for the estimated temperatures ERR1 and estimated heat sources ERR2 are as low as 0.0031% and 1.87%, respectively. The accuracy of the estimated heat sources is then experimentally validated under various prescribed inlet air velocities. During experimental verification at an inlet velocity of 4 m/s, the corresponding ERR1 and ERR2 values are obtained as 0.91% and 3.34%, while at 6 m/s, the values are 0.86% and 2.81%, respectively. Compared with the numerical results, the accuracy of the experimental estimations decreases noticeably. This discrepancy arises because the numerical simulations are free from measurement noise, whereas experimental data inherently include uncertainties due to thermal picture resolutions, environmental fluctuations, and other uncontrollable factors. These results highlight the inherent challenges associated with inverse problems and underscore the critical importance of obtaining precise and reliable temperature measurements to ensure accurate heat source estimation. Full article
(This article belongs to the Special Issue Computational Mathematics Methods and Applications in Engineering Science)
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