Water level monitoring by using magnetic optical sensor by Jonatan Daniel Morales Arrieta, Jesus Vargas Ortiz, Jesús Patricio Ordaz Oliver, Juan Carlos González Islas, Omar Samperio Vázquez

“…Firstly, mathematical modeling of the water level measurement in a tank using differential equations is described. This system comprises a printed circuit board (PCB), a novel float mechanism, an optical sensor, a microcontroller, and a visualization stage. …”
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Non-local piezoelasticity to incorporate the influence of small-scale factors on the resonance behavior of the Mindlin piezoelectric polymeric nanoplates by Narinderjit Singh Sawaran Singh, Waqed H. Hassan, Zainab Mоhammed Ameen Ahmed, Younis Mohamed Atiah Al-zahy, Soheil Salahshour, Mostafa Pirmoradian

“…The Galerkin method is utilized to solve the partial differential equations governing the dynamics of the piezoelectric polymeric nanoplate, marking a significant methodological contribution to the field. …”
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Qualitative financial modelling in fractal dimensions by Rami Ahmad El-Nabulsi, Waranont Anukool

“…Abstract The Black–Scholes equation is one of the most important partial differential equations governing the value of financial derivatives in financial markets. …”
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A Mathematical Model for the Growth Dynamics of Demand in the Fashion Industry within the Era of the COVID-19 Pandemic by John Awuah Addor, Anthony Joe Turkson, Douglas Yenwon Kparib

“…A five-compartment susceptible-infection-recovery-susceptible-based model was formulated in an integrated environment with application of fashion-based personal protective equipment (FPPEs) and government policy regulation, using ordinary differential equations. Solution techniques included a mix of qualitative analysis and simulations with data from various publications on COVID-19. …”
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Models for an arenavirus infection in a rodent population: consequences of horizontal, vertical and sexual transmission by Chandrani Banerjee, Linda J. S. Allen, Jorge Salazar-Bravo

“…We formulate a general deterministic model for male and female rodents consisting of eight differential equations, four for females and four for males. …”
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Displacement of hydrocarbon liquids by water using the models of zonally heterogeneous deformable formations by Marat Ya. Khabibullin, Rustem I. Suleymanov, Razifa R. Stepanova, Alina Az. Gizzatullina, Arsen M. Khabibullin

“…It is known that the motion of two-phase hydrocarbon systems in deformable reservoirs is represented by complex nonlinear partial differential equations. An analytical solution of such equations is possible only with the use of special approaches. …”
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Understanding biofilm--phage interactions in cystic fibrosis patients using mathematical frameworks by Emerenini Blessing O., Hartung Doris, Reyes Grimaldo Ricardo N. G., Canner Claire, Williams Maya, Agyingi Ephraim, Osgood Robert

“…To evaluate the effectiveness of phage therapy against biofilm bacteria, we create a phage-biofilm model with ordinary differential equations and a stochastic model. We considered three possible cases for the stability of disease-free equilibrium; by model simulations and parameter alterations in both models, we investigated the effect of bacteria–phage interactions alongside biofilm bacteria cell detachment from the biofilm phase to the planktonic phase, and how these will affect the efficiency of phage therapy against bacteria. …”
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Computational analysis for efficient thermal transportation of ternary hybrid nanofluid flow across a stretching sheet with Cattaneo-Christov heat flux model by Wei Li, Shan Ali Khan, Muhammad Shafqat, Qamar Abbas, Taseer Muhammad, Muhammad Imran

“…The phenomenon were designed as partial differential equations. These are condensed to system of ODEs via similarity transformations. …”
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Presenting two new methods for solving linear interval optimal control problems using the resilience approach of control signal by Elnaz Hosseini, Mehdi Allahdadi, Samaneh Soradi-Zeid

“…Purpose: In this paper, two new methods are presented to find the optimal control of systems described by linear differential equations that have interval coefficients. In these methods, the given interval optimal control problem is transformed into a non-interval (deterministic) optimal control problem so that it is possible to use the optimal control theory to solve it. …”
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Exploring novel solitary wave phenomena in Klein–Gordon equation using $$\phi ^{6}$$ ϕ 6 model expansion method by Yasir A. Madani, Khidir Shaib Mohamed, Sadia Yasin, Sehrish Ramzan, Khaled Aldwoah, Mohammed Hassan

“…The $$\phi ^{6}$$ ϕ 6 model expansion technique is exceptionally adaptable and may be utilised for a wide array of nonlinear partial differential equations. Despite its versatility, the technique may not be applicable to all nonlinear PDEs, especially those that do not meet the specified requirements or structures manageable by this technique. …”
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Randomized radial basis function neural network for solving multiscale elliptic equations by Yuhang Wu, Ziyuan Liu, Wenjun Sun, Xu Qian

“…Ordinary deep neural network (DNN)-based methods frequently encounter difficulties when tackling multiscale and high-frequency partial differential equations. To overcome these obstacles and improve computational accuracy and efficiency, this paper presents the Randomized Radial Basis Function Neural Network (RRNN), an innovative approach explicitly crafted for solving multiscale elliptic equations. …”
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A holistic physics-informed neural network solution for precise destruction of breast tumors using focused ultrasound on a realistic breast model by Salman Lari, Hossein Rajabzadeh, Mohammad Kohandel, Hyock Ju Kwon

“…Additionally, employing PINN for estimating partial differential equations (PDE) solutions can notably decrease the enormous number of discretized elements needed. …”
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Direct and Indirect Protection with Pediatric Quadrivalent Live-Attenuated Influenza Vaccination in Europe Estimated by a Dynamic Transmission Model by Laetitia Gerlier, Judith Hackett, Richard Lawson, Sofia Dos Santos Mendes, Catherine Weil-Olivier, Markus Schwehm, Martin Eichner

“…**Methods:** A deterministic, age-structured, dynamic model was used to simulate influenza transmission across 14 European countries, comparing current vaccination coverage using a quadrivalent inactivated vaccine (QIV) to a scenario whereby vaccination coverage was extended to 50% of 2–17 year-old children, using QLAIV. Differential equations described demographic changes, exposure to infectious individuals, recovery and immunity dynamics. …”
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Modeling the co-infection of HTLV-2 and HIV-1 in vivo by A. M. Elaiw, E. A. Almohaimeed, A. D. Hobiny

“…Six ordinary differential equations made up the model, which depicted the interactions between uninfected CD4+ T cells, HIV-1-infected CD4+ T cells, HIV-1 particles, uninfected CD8+ T cells, HTLV-2-infected CD8+ T cells, and HIV-1-specific B cells. …”
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Magnetohydrodynamic Peristaltic Propulsion of Casson Nanofluids With Slip Effects Over Heterogeneous Rough Channel by Hanumesh Vaidya, Fateh Mebarek‐Oudina, Rakesh Kumar, C. Rajashekhar, Kerehalli Vinayaka Prasad, Sangeeta Kalal, Kottakkaran Sooppy Nisar

“…A novel rough non‐uniform model is effectively governed by a set of nonlinear coupled governing partial differential equations, which are simplified under long wavelength and creeping flow approximations. …”
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Extrapolation of cavitation and hydrodynamic pressure in lubricated contacts: a physics-informed neural network approach by Faras Brumand-Poor, Freddy Kokou Azanledji, Nils Plückhahn, Florian Barlog, Lukas Boden, Katharina Schmitz

“…In particular, physics-informed neural networks (PINNs) are as effective hybrid solvers that combine data-driven and physics-based methods to solve the partial differential equations that drive EHL simulations. By integrating physical laws into the parameter optimization of the neural network (NN), PINNs provide accurate and fast solutions. …”
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Accelerating phase field simulations through a hybrid adaptive Fourier neural operator with U-net backbone by Christophe Bonneville, Nathan Bieberdorf, Arun Hegde, Mark Asta, Habib N. Najm, Laurent Capolungo, Cosmin Safta

“…However, the LMD governing equations in these models often involve coupled non-linear partial differential equations (PDE), which are challenging to solve numerically. …”
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Numerical Analysis on the Storage of Nuclear Waste in Gas-Saturated Deep Coal Seam by Teng Teng, Yuming Wang, Xiaoyan Zhu, Xiangyang Zhang, Sihai Yi, Guowei Fan, Bin Liu

“…As the second step, a finite element numerical model and numerical simulation are developed to analyze the storage of nuclear waste in a gas-saturated deep coal seam based on the partial differential equations (PDE) solver of COMSOL Multiphysics with MATLAB. …”
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A computational role of blood nanofluid induced by a stenosed artery with porous medium and thermophoretic particle deposition effects by Shivalila Hangaragi, N. Neelima, N. Beemkumar, Ankur Kulshreshta, Umair Khan, Noreen Sher Akbar, Mohammad Kanan, Mona Mahmoud

“…Based on these assumptions, the governing equations are formulated and transformed into a system of ordinary differential equations using appropriate similarity transformations. …”
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Community behavior for mathematical model of coronavirus disease 2019 (COVID-19) by M. Ramli, M. Mukramati, M. Ikhwan, H. Hafnani

“…The model-generated equation system is a nonlinear system of differential equations. The developed model is examined by determining the equilibrium point and the basic reproduction number.FINDINGS: The model resulted an asymptotically stable disease-free equilibrium and endemic equilibrium. …”
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