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UNCERTAINTY QUANTIFICATION FOR THE HOMOGENEOUS LANDAU-FOKKER-PLANCK EQUATION VIA DETERMINISTIC PARTICLE GALERKIN METHODS

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Multiscale Modeling and Simulation 2025年第2期23卷 687-710页

作者： Bailo, Rafael Carrillo, Jose A. Medaglia, Andrea Zanella, Mattia Centre for Analysis Scientific Computing and Applications Eindhoven University of Technology Netherlands Mathematical Institute University of Oxford United Kingdom Department of Mathematics ` University of Pavia Italy

We design a deterministic particle method for the solution of the spatially homogeneous Landau equation with uncertainty. The deterministic particle approximation is based on the reformulation of the Landau equation as a formal gradient flow on the set of probability measures, whereas the propagation of uncertain quantities is computed by means of a stochastic Galerkin (sG) representation of each particle. This approach guarantees spectral accuracy in uncertainty space while preserving the fundamental structural properties of the model: the positivity of the solution, the conservation of invariant quantities, and the entropy dissipation. We provide a regularity result for the particle method in the random space. We perform the numerical validation of the particle method in a wealth of test cases. © 2025 Authors.

关键词： Stochastic systems

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INITIAL VALUE PROBLEMS FOR NONLINEAR NEUTRAL DELAY Ψ-CAPUTO FRACTIONAL HYBRID DIFFERENTIAL EQUATIONS WITH BOUNDED DELAYS

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Journal of Mathematical Sciences (United States) 2025年 1-11页

作者： Malghi, Hayat Taqbibt, Abdellah El Omari, M’hamed Qaffou, Abdelaziz Hilal, Khalid Department of Mathematics Applied Mathematics and Scientific Computing Laboratory Faculty of Sciences and Technology Beni Mellal Sultan Moulay Slimane University P.O. Box 523 Beni Mellal 23000 Morocco

In this work, we explore the existence of solutions to an initial value problem for nonlinear neutral delay Ψ-Caputo fractional hybrid differential equations with bounded delays. The existence results are established using fixed point theorems. To demonstrate the practical applicability of our findings, we provide an illustrative example. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

关键词： Fixed point Hybrid differential equation Neutral delay differential equation Ψ-Caputo derivative

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An algorithm based on homotopy perturbation theory and its mathematical analysis for singular nonlinear system of boundary value problems

An algorithm based on homotopy perturbation theory and its m...

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作者： Pathak, Priti Barnwal, Amit K. Sriwastav, Nikhil Singh, Randhir Singh, Mehakpreet Department of Mathematics and Scientific Computing Madan Mohan Malaviya University of Technology Gorakhpur India Department of Mathematics Birla Institute of Technology Mesra Mesra India Department of Mathematics and Statistics University of Limerick Limerick Ireland

In this paper, a promising technique based on the homotopy perturbation method for solving singular nonlinear system of boundary value problems (BVPs) is employed. The original singular system of BVPs is transformed into an equivalent system of integral equations to overcome the singular behavior at (Formula presented.), and then, the recursive series for the solution is established. The mathematical formulation is further supported by conducting the convergence analysis of the recursive schemes. The accuracy of the proposed method is tested by comparing against the current variational iteration method using many standard examples. It is shown that the proposed method is more accurate and efficient than the existing method such as Chebyshev operational matrix method and variational iteration method. © 2023 John Wiley & Sons, Ltd.

关键词： Nonlinear equations

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WeaponVision AI: a software for strengthening surveillance through deep learning in real-time automated weapon detection

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International Journal of Information Technology (Singapore) 2025年第3期17卷 1717-1727页

作者： Yadav, Pavinder Gupta, Nidhi Sharma, Pawan Kumar Department of Computer Applications National Institute of Technology Haryana Kurukshetra 136119 India Mathematics and Scientific Computing National Institute of Technology HP Hamirpur India

During situations involving dangerous activities, such as armed robbery in public areas, surveillance systems often exhibit delays or inefficiencies in their prompt responding. To obviate the necessity for human involvement, there is a requirement for technology capable of autonomously detecting harmful objects, like firearms in surveillance footages. This article presents WeaponVision AI, an advanced software system that has the ability to accurately identify weapons in live feeds, recorded videos, and images. Moreover, this software has the capability to detect guns even under weak lighting circumstances. The deep learning architecture based on modified YOLOv7 was trained on a vast dataset assortment of 79,558 images of weapons, in developing the WeaponVision AI. The model exhibited satisfactory results following the training phase, achieving significant outcome metrics: a precision rate of 91.75% and a mean average precision of 92.15%. The efficacy of WeaponVision AI is showcased through its ability to accurately identify weapons across diverse environmental and visual conditions. © Bharati Vidyapeeth's institute of Computer Applications and Management 2025.

关键词： Computer vision Deep learning Machine learning Security and surveillance Surveillance system Weapon detection

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Mixed finite element methods for biharmonic obstacle problems

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SeMA Journal 2025年 1-10页

作者： Piersanti, Paolo Sun, Tianyu School of Science and Engineering The Chinese University of Hong Kong (Shenzhen) 2001 Longxiang Blvd. Longgang District Shenzhen China Department of Mathematics and Institute for Scientific Computing and Applied Mathematics Indiana University Bloomington 729 East Third Street Bloomington IN United States

This paper is devoted to the study of an alternative Finite Element Method to the one originally proposed by Ciarlet & Raviart, and then complemented by Ciarlet & Glowinski, for studying the numerical approximation of the solution of the biharmonic problem. In addition to providing an alternative way of approximating the solution of the biharmonic problem, the method here presented is valid also in the context where the solution to the biharmonic problem is subjected to a constraint. The theoretical results we derived are complemented by a series of numerical experiments. © The Author(s), under exclusive licence to Sociedad Española de Matemática Aplicada 2025.

关键词： Elasticity theory Finite element method Obstacle problems Penalty method Variational inequalities

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A deep learning approach for recognizing and solving handwritten mathematical equations

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Neural computing and Applications 2025年第14期37卷 8759-8772页

作者： Yadav, Pavinder Shantilal, Sankhala Bhavik Kumar, Vipin Sihag, Parveen Sharma, Pawan Kumar Rana, Pankaj Department of Mathematics and Scientific Computing National Institute of Technology HP Hamirpur India Department of Civil Engineering Chandigarh University Punjab Mohali India Department of Physical Sciences Baba Mastnath University Asthal Bohar Haryana Rohtak India

Recognizing handwritten equations is considerably more challenging than identifying printed ones, primarily due to variations in writing styles. Even though there have been recent improvements in handwriting recognition for things like equation solvers and automation, it is still hard for machines to understand a handwritten mathematical expression (HME). This paper presents a deep learning model for recognizing and solving handwritten equations, as well as its graphical user interface (GUI). A manually constructed large dataset comprising approximately 66k images of digits, mathematical symbols, and characters is utilized, with a fine-tuned InceptionV3 model employed for image classification and segmentation. The system is designed to extract handwritten equations from an image, perform character recognition, and generate the roots of the polynomial equation up to degree four. According to the findings, the trained model achieves remarkable accuracy in recognizing and solving handwritten equations, with over 94% accuracy on the training set. By eliminating the possibility of human error, automatic handwritten equation solvers provide more accurate results than manual methods. © The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature 2025.

关键词： Convolutional neural networks

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LIE SYMMETRY ANALYSIS, PARAMETRIC REDUCTION AND CONSERVATION LAWS OF (3+1) DIMENSIONAL NONLINEAR DISPERSIVE SOLITON EQUATION

arXiv

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arXiv 2025年

作者： Singhal, Anshika Joshi, Urvashi Arora, Rajan Department of Applied Mathematics and Scientific Computing Indian Institute of Technology Roorkee India

The core focus of this research work is to obtain invariant solutions and conservation laws of the (3+1)-dimensional Zakharov–Kuznetsov (ZK) equation which describes the phenomenon of wave stability and solitons propagation. ZK equation is significant in fluid dynamics, nonlinear optics, and acoustics. ZK equation plays an important role in understanding the behavior of nonlinear, dispersive waves especially in the presence of the magnetic field. Lie symmetry analysis has been applied to the (3+1)-dimensional ZK equation to derive invariant solutions. These solutions reveal how waves retain their shape as they travel, how they interact in space, and the impact of magnetic fields on wave propagation. Using the Lie symmetry, the (3+1) dimensional Zakharov-Kuznetsov equation is reduced to various ordinary differential equations. It also concludes with the conservation laws and non-linear self-adjointness property. Additionally, we align our findings with real-life problems, namely plasma physics, electromagnetic wave propagation, and water wave dynamics, providing the applicability of our theoretical results. Copyright © 2025, The Authors. All rights reserved.

关键词： Electromagnetic wave propagation in plasma

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ON INITIAL VALUE PROBLEMS FOR THE FRACTIONAL DIFFUSION EQUATION WITH DZHRBASHYAN–NERSESYAN OPERATORS

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Journal of Mathematical Sciences (United States) 2025年 1-12页

作者： Bogatyreva, Fatima T. Department of Fractional Calculus Institute of Applied Mathematics and Automation of Kabardino-Balkarian Scientific Center RAS Shortanova St. 89-A Nalchik 360017 Russian Federation

We study initial value problems for a diffusion-wave equation with fractional order differentiation operator in time. The fractional differentiation operator is defined as a linear combination of two Dzhrbashyan–Nersesyan operators associated with two-parameter sequences. It is shown that the distribution of parameters in the sequences defining these operators influences the form and number of initial data and induces different initial value problems in general. In this paper, these problems are described, and explicit representations of their solutions are constructed. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

关键词： Dzhrbashyan–Nersesyan operators Fractional diffusion equation Well-posed problems

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Analysis of magnetogasdynamic shock wave in a self-gravitating non-ideal dusty gas using Lie group theory

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Physics of Fluids 2025年第4期37卷

作者： Bhardwaj, Akshita Singh, Deepika Arora, Rajan Department of Applied Mathematics and Scientific Computing Indian Institute of Technology Roorkee Roorkee India Department of Mathematics Netaji Subhas University of Technology New Delhi India

This study delves into the behavior of a one-dimensional, unsteady flow following a shock wave in a non-ideal dusty gas, including an axial magnetic field and gravitational field. The Lie group theoretic method has been used to discuss all possible cases of the self-similarity solutions, for the considered problem in the medium of uniform density. This model has implications for astrophysical and laboratory plasma systems. The arbitrary constants in the Lie group generators yield solutions with power-law and exponential-law shock paths. The graphs of the reduced flow variables have been analyzed to study how they vary behind the shock front concerning the physical parameters like non-ideal parameter ( b ¯ ) , adiabatic exponent ( γ ) , gravitational parameter ( g 0 ) , the ratio of the density of solid particles to the initial density of the gas ( η 0 ) , Alfvén-Mach number ( M a − 2 ) , and geometric variation index ( ν ) . We investigate a detailed comparison between adiabatic and isothermal flow conditions. In isothermal case, the increase in fluid velocity and magnetic field suggests a comparatively accelerated and dynamic flow. Meanwhile, the decrease in density and pressure implies a more expansion and less compression state compared to the adiabatic case. The research concludes with an analysis of flow variable behavior under different physical conditions. © 2025 Author(s).

关键词： Lie groups

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Modified Dai-Liao Spectral Conjugate Gradient Method with Application to Signal Processing

arXiv

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arXiv 2025年

作者： Sahu, D.R. Sharma, Shikher Gautam, Pankaj Department of Mathematics Banaras Hindu University India Department of Applied Mathematics and Scientific Computing Indian Institute of Technology Roorkee India

In this article, we present a modified variant of the Dai-Liao spectral conjugate gradient method, developed through an analysis of eigenvalues and inspired by a modified secant condition. We show that the proposed method is globally convergent for general nonlinear functions under standard assumptions. By incorporating the new secant condition and a quasi-Newton direction, we introduce updated spectral parameters. These changes ensure that the resulting search direction satisfies the sufficient descent property without relying on any line search. Numerical experiments show that the proposed algorithm performs better than several existing methods in terms of convergence speed and computational efficiency. Its effectiveness is further demonstrated through an application in signal processing. © 2025, CC BY.

关键词： Conjugate gradient method

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