Search Results (62)

This study presents the Fourier–Gegenbauer integral Galerkin (FGIG) method, a new numerical framework that uniquely integrates Fourier series and Gegenbauer polynomials to solve the one-dimensional advection–diffusion (AD) equation with spatially symmetric periodic boundary conditions, achieving exponential convergence and reduced computational cost compared to traditional methods. The FGIG method uniquely combines Fourier series for spatial periodicity and Gegenbauer polynomials for temporal integration within a Galerkin framework, resulting in highly accurate numerical and semi-analytical solutions. Unlike traditional approaches, this method eliminates the need for time-stepping procedures by reformulating the problem as a system of integral equations, reducing error accumulation over long-time simulations and improving computational efficiency. Key contributions include exponential convergence rates for smooth solutions, robustness under oscillatory conditions, and an inherently parallelizable structure, enabling scalable computation for large-scale problems. Additionally, the method introduces a barycentric formulation of the shifted Gegenbauer–Gauss (SGG) quadrature to ensure high accuracy and stability for relatively low Péclet numbers. This approach simplifies calculations of integrals, making the method faster and more reliable for diverse problems. Numerical experiments presented validate the method’s superior performance over traditional techniques, such as finite difference, finite element, and spline-based methods, achieving near-machine precision with significantly fewer mesh points. These results demonstrate its potential for extending to higher-dimensional problems and diverse applications in computational mathematics and engineering. The method’s fusion of spectral precision and integral reformulation marks a significant advancement in numerical PDE solvers, offering a scalable, high-fidelity alternative to conventional time-stepping techniques. Full article

The main goal of this study is to create a computational solver that analyzes the effects of magnetohydrodynamics (MHD) on heat radiation in Cu–water-based Prandtl nanofluid flow using artificial neural networks. Copper nanoparticles are utilized to boost the water-based fluid’s thermal effect. This study primarily focuses on heat transfer over a horizontal sheet, exploring different scenarios by varying key factors such as the magnetic field and thermal radiation properties. The mathematical model is formulated using partial differential equations (PDEs), which are then transformed into a corresponding set of ordinary differential equations (ODEs) through appropriate similarity transformations. The bvp4c solver is then used to simulate the numerical behavior. The effects of relevant parameters on the temperature, velocity, skin friction, and local Nusselt number profiles are examined. It is discovered that the parameters of the Prandtl fluid have a considerable impact. The local skin friction and the local Nusselt number are improved by increasing these parameters. The dataset is split into 70% training, 15% validation, and 15% testing. The ANN model successfully predicts skin friction and Nusselt number profiles, showing good agreement with numerical simulations. This hybrid framework offers a robust predictive approach for heat management systems in industrial applications. This study provides important insights for researchers and engineers aiming to comprehend flow characteristics and their behavior and to develop accurate predictive models. Full article

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields. Full article

Analog computing has re-emerged as a powerful tool for solving complex problems in various domains due to its energy efficiency and inherent parallelism. This paper summarizes recent advancements in analog computing, exploring discrete time and continuous time methods for solving combinatorial optimization problems, solving partial differential equations and systems of linear equations, accelerating machine learning (ML) inference, multi-beam beamforming, signal processing, quantum simulation, and statistical inference. We highlight CMOS implementations that leverage switched-capacitor, switched-current, and radio-frequency circuits, as well as non-CMOS implementations that leverage non-volatile memory, wave physics, and stochastic processes. These advancements demonstrate high-speed, energy-efficient computations for computational electromagnetics, finite-difference time-domain (FDTD) solvers, artificial intelligence (AI) inference engines, wireless systems, and related applications. Theoretical foundations, experimental validations, and potential future applications in high-performance computing and signal processing are also discussed. Full article

Background: Rock–blast design is a canonical inverse problem that joins elastodynamic partial differential equations (PDEs), fracture mechanics, and stochastic heterogeneity. Objective: Guided by the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) protocol, a systematic review of mathematical methods for geomechanically informed blast modelling and optimisation is provided. Methods: A Scopus–Web of Science search (2000–2025) retrieved 2415 records; semantic filtering and expert screening reduced the corpus to 97 studies. Topic modelling with Bidirectional Encoder Representations from Transformers Topic (BERTOPIC) and bibliometrics organised them into (i) finite-element and finite–discrete element simulations, including arbitrary Lagrangian–Eulerian (ALE) formulations; (ii) geomechanics-enhanced empirical laws; and (iii) machine-learning surrogates and multi-objective optimisers. Results: High-fidelity simulations delimit blast-induced damage with ≤0.2 m mean absolute error; extensions of the Kuznetsov–Ram equation cut median-size mean absolute percentage error (MAPE) from 27% to 15%; Gaussian-process and ensemble learners reach a coefficient of determination ($R^2>0.95$) while providing closed-form uncertainty; Pareto optimisers lower peak particle velocity (PPV) by up to 48% without productivity loss. Synthesis: Four themes emerge—surrogate-assisted PDE-constrained optimisation, probabilistic domain adaptation, Bayesian model fusion for digital-twin updating, and entropy-based energy metrics. Conclusions: Persisting challenges in scalable uncertainty quantification, coupled discrete–continuous fracture solvers, and rigorous fusion of physics-informed and data-driven models position blast design as a fertile test bed for advances in applied mathematics, numerical analysis, and machine-learning theory. Full article

This paper proposes a computational procedure to resolve the temporal fractional financial option pricing partial differential equation (PDE) using a localized meshless approach via the multiquadric radial basis function (RBF). Given that financial market information is best characterized within a martingale framework, the resulting option pricing model follows a modified Black–Sholes (BS) equation, requiring efficient numerical techniques for practical implementation. The key innovation in this study is the derivation of analytical weights for approximating first and second derivatives, ensuring improved numerical stability and accuracy. The construction of these weights is grounded in the second integration of a variant of the multiquadric RBF, which enhances smoothness and convergence properties. The performance of the presented solver is analyzed through computational tests, where the analytical weights exhibit superior accuracy and stability in comparison to conventional numerical weights. The results confirm that the new approach reduces absolute errors, demonstrating its effectiveness for financial option pricing problems. Full article

Physics-informed neural networks (PINNs) offer a mesh-free approach to solving partial differential equations (PDEs) with embedded physical constraints. Although PINNs have gained traction in various engineering fields, their adoption for railway bridge analysis remains under-explored. To address this gap, a systematic review was conducted across Scopus and Web of Science (2020–2025), filtering records by relevance, journal impact, and language. From an initial pool, 120 articles were selected and categorised into nine thematic clusters that encompass computational frameworks, hybrid integration with conventional solvers, and domain decomposition strategies. Through natural language processing (NLP) and trend mapping, this review evidences a growing but fragmented research landscape. PINNs demonstrate promising capabilities in load distribution modelling, structural health monitoring, and failure prediction, particularly under dynamic train loads on multi-span bridges. However, methodological gaps persist in large-scale simulations, plasticity modelling, and experimental validation. Future work should focus on scalable PINN architectures, refined modelling of inelastic behaviours, and real-time data assimilation, ensuring robustness and generalisability through interdisciplinary collaboration. Full article

In the current paper, an analysis of magnetohydrodynamic Williamson nanofluid boundary layer flow is presented, with multiple slips in a porous medium, using a newly designed human-brain-inspired Ricker wavelet neural network solver. The solver employs a hybrid approach that combines genetic algorithms, serving as a global search method, with sequential quadratic programming, which functions as a local optimization technique. The heat and mass transportation effects are examined through a stretchable surface with radiation, thermal, and velocity slip effects. The primary flow equations, originally expressed as partial differential equations (PDEs), are changed into a dimensionless nonlinear system of ordinary differential equations (ODEs) via similarity transformations. These ODEs are then numerically solved with the proposed computational approach. The current study has significant applications in a variety of practical engineering and industrial scenarios, including thermal energy systems, biomedical cooling devices, and enhanced oil recovery techniques, where the control and optimization of heat and mass transport in complex fluid environments are essential. The numerical outcomes gathered through the designed scheme are compared with reference results acquired through Adam’s numerical method in terms of graphs and tables of absolute errors. The rapid convergence, effectiveness, and stability of the suggested solver are analyzed using various statistical and performance operators. Full article

This study analyzes and evaluates the performance of various solvers and preconditioners for reservoir simulations of CO₂ injection and long-term storage using the model 11B of SPE CSP (Society of Petroleum Engineers, 11th Comparative Solution Project) and the MATLAB Reservoir Simulation Toolbox (MRST). The SPE CSP 11 model serves as a benchmark for testing numerical methods for solving partial differential equations (PDEs) in reservoir simulations. The research focuses on the Biconjugate Gradient Stabilized (BiCGSTAB) and Loose Generalized Minimum Residual (LGMRES) solver methods, as well as multiple preconditioning techniques designed to improve convergence rates and reduce computational effort for CO₂ storage. Extensive simulations were performed to compare the performance of different solver-preconditioner combinations under varying reservoir conditions, leveraging MRST’s flexible simulation capabilities. Key performance metrics, including iteration counts and computational time, were analyzed for the project. The results reveal trade-offs between computational efficiency and solution accuracy, providing valuable insights into the effectiveness of each approach. This study offers practical guidance for reservoir engineers and researchers seeking to analyze and optimize simulation workflows within MRST by identifying the strengths and limitations of specific solver-preconditioner combinations for complex linear systems. Full article

We conducted a comprehensive comparative study of numerical solvers for the generalized Korteweg–de Vries (gKdV) equation, focusing on classical Fourier-based Crank–Nicolson methods and physics-informed neural networks (PINNs). Our work benchmarks these approaches across nonlinear regimes—including the cubic case ($\nu =3$)—and diverse initial conditions such as solitons, smooth pulses, discontinuities, and noisy profiles. In addition to pure PINN and spectral models, we propose a novel hybrid PINN–spectral method incorporating a regularization term based on Fourier reference solutions, leading to improved accuracy and stability. Numerical experiments show that while spectral methods achieve superior efficiency in structured domains, PINNs provide flexible, mesh-free alternatives for data-driven and irregular setups. The hybrid model achieves lower relative $L^2$ error and better captures soliton interactions. Our results demonstrate the complementary strengths of spectral and machine learning methods for nonlinear dispersive PDEs. Full article

Land cover dynamics play a critical role in understanding environmental changes, but accurately modeling these dynamics remains a challenge due to the complex interactions between temporal and spatial factors. In this study, we propose a novel temporal–spatial partial differential equation (TS-PDE) modeling method combining sparse regression to uncover the governing equations behind long-term satellite image time series. By integrating temporal and spatial differential terms, the TS-PDE framework captures the intricate interactivity of these factors, overcoming the limitations of traditional pixel-wise prediction methods. Our approach leverages $1\times 1$ convolutional kernels within a convolutional neural network (CNN) solver to approximate derivatives, enabling the discovery of interpretable equations that generalize across temporal–spatial domains. Using MODIS and Planet satellite data, we demonstrate the effectiveness of the TS-PDE method in predicting the value of the normalized difference vegetation index (NDVI) and interpreting the physical significance of the derived equations. The numerical results show that the model achieves good performance, with mean structural similarity index (SSIM) values exceeding 0.82, mean peak signal-to-noise ratio (PSNR) values ranging from 28.5 to 32.8, and mean mean squared error (MSE) values approximating $9\times {10}^{-4}$ for low-resolution MODIS images. For high-resolution Planet images, this study emphasizes the efficacy of TS-PDE in terms of PSNR, SSIM, and MSE metrics, with all datasets exhibiting an average SSIM value of over 0.81, an average PSNR maximum of 30.9, and an average MSE of less than 0.0042. The experimental findings demonstrate the capability of TS-PDE in deriving governing equations and providing effective predictions for the regional-scale dynamics of these time series images. The findings of this study provide potential insights into the mathematical modeling of land cover dynamics. Full article

The Heston-Hull-White (HHW) model is a generalization of the classical Heston approach that incorporates stochastic interest rates, making it a more accurate representation of financial markets. In this work, we investigate a computational procedure via a three-dimensional partial differential equation (PDE) to solve option pricing problems under the HHW framework. We propose a local radial basis function–finite difference (RBF–FD) framework under the integration of a new variant of the multiquadric function for efficiently resolving the model. Our study highlights the error analysis of the proposed weights for the first and second derivatives of a suitable function and demonstrates the effectiveness of the RBF–FD approach for this high-dimensional financial model. Full article

This work presents a physics-guided parameter estimation framework for cold spray additive manufacturing (CSAM), focusing on simulating and validating deposit profiles across diverse process conditions. The proposed model employs a two-zone flow representation: quasi-constant velocity near the nozzle exit followed by an exponentially decaying free jet to capture particle acceleration and impact dynamics. The framework employs a comprehensive approach by numerically integrating drag-dominated particle trajectories to predict deposit formation with high accuracy. This physics-based framework incorporates both operational and geometric parameters to ensure robust prediction capabilities. Operational parameters include spray angle, standoff distance, traverse speed, and powder feed rate, while geometric factors encompass nozzle design characteristics such as exit diameter and divergence angle. Validation is performed using 36 experimentally measured profiles of commercially pure titanium powder. The simulator shows excellent agreement with the experimental data, achieving a global root mean square error (RMSE) of 0.048 mm and a coefficient of determination $R^2=0.991$, improving the mean absolute error by more than 40% relative to a neural network-based approach. Sensitivity analyses reveal that nozzle geometry, feed rate, and critical velocity strongly modulate the amplitude and shape of the deposit. Notably, decreasing the nozzle exit diameter or divergence angle significantly increases local deposition rates, while increasing the standoff distance dampens particle velocities, thereby reducing deposit height. Although the partial differential equation (PDE)-based framework entails a moderate increase in computational time—about 50 s per run, roughly 2.5 times longer than simpler empirical models—this remains practical for most process design and optimization tasks. Beyond its accuracy, the PDE-based simulation framework’s principal advantage lies in its minimal reliance on sampling data. It can readily be adapted to new materials or untested process parameters, making it a powerful predictive tool in cold spray process design. This study underscores the simulator’s potential for guiding parameter selection, improving process reliability and offering deeper physical insights into cold spray deposit formation. Full article

Quantum computing has the potential to solve certain compute-intensive problems faster than classical computing by leveraging the quantum mechanical properties of superposition and entanglement. This capability can be particularly useful for solving Partial Differential Equations (PDEs), which are challenging to solve even for High-Performance Computing (HPC) systems, especially for multidimensional PDEs. This led researchers to investigate the usage of Quantum-Centric High-Performance Computing (QC-HPC) to solve multidimensional PDEs for various applications. However, the current quantum computing-based PDE-solvers, especially those based on Variational Quantum Algorithms (VQAs) suffer from limitations, such as low accuracy, long execution times, and limited scalability. In this work, we propose an innovative algorithm to solve multidimensional PDEs with two variants. The first variant uses Finite Difference Method (FDM), Classical-to-Quantum (C2Q) encoding, and numerical instantiation, whereas the second variant utilizes FDM, C2Q encoding, and Column-by-Column Decomposition (CCD). We evaluated the proposed algorithm using the Poisson equation as a case study and validated it through experiments conducted on noise-free and noisy simulators, as well as hardware emulators and real quantum hardware from IBM. Our results show higher accuracy, improved scalability, and faster execution times in comparison to variational-based PDE-solvers, demonstrating the advantage of our approach for solving multidimensional PDEs. Full article

Nanofluids, with their enhanced thermal properties, provide innovative solutions for improving heat transfer efficiency in renewable energy systems. This study investigates a numerical simulation of bioconvective flow and heat transfer in a Williamson nanofluid over a stretching wedge, incorporating the effects of chemical reactions and hydrogen diffusion. The system also includes motile microorganisms, which induce bioconvection, a phenomenon where microorganisms’ collective motion creates a convective flow that enhances mass and heat transport processes. This mechanism is crucial for improving the distribution of nanoparticles and maintaining the stability of the nanofluid. The unique rheological behavior of Williamson fluid, extensively utilized in hydrometallurgical and chemical processing industries, significantly influences thermal and mass transport characteristics. The governing nonlinear partial differential equations (PDEs), derived from conservation laws and boundary conditions, are converted into dimensionless ordinary differential equations (ODEs) using similarity transformations. MATLAB’s bvp4c solver is employed to numerically analyze these equations. The outcomes highlight the complex interplay between fluid parameters and flow characteristics. An increase in the Williamson nanofluid parameters leads to a reduction in fluid velocity, with solutions observed for the skin friction coefficient. Higher thermophoresis and Williamson nanofluid parameters elevate the fluid temperature, enhancing heat transfer efficiency. Conversely, a larger Schmidt number boosts fluid concentration, while stronger chemical reaction effects reduce it. These results are generated by fixing parametric values as $0.1<\varpi <1.5, 0.1<\mathit{Nr}<3.0, 0.2<\mathrm{Pr}<0.5, 0.1<\mathit{Sc}<0.4, \mathrm{and} 0.1<\mathit{Pe}<1.5.$ This work provides valuable insights into the dynamics of Williamson nanofluids and their potential for thermal management in renewable energy systems. The combined impact of bioconvection, chemical reactions, and advanced rheological properties underscores the suitability of these nanofluids for applications in solar thermal, geothermal, and other energy technologies requiring precise heat and mass transfer control. This paper is also focused on their applications in solar thermal collectors, geothermal systems, and thermal energy storage, highlighting advanced experimental and computational approaches to address key challenges in renewable energy technologies. Full article