Results in Applied Mathematics最新文献

{"title":"The numerical solution of an Abel integral equation by the optimal quadrature formula","authors":"Abdullo Hayotov ,&nbsp;Samandar Babaev ,&nbsp;Bobomurod Boytillayev","doi":"10.1016/j.rinam.2025.100542","DOIUrl":"10.1016/j.rinam.2025.100542","url":null,"abstract":"<div><div>In this study, a novel and efficient approach utilizing optimal quadrature formulas is introduced to derive approximate solutions for generalizing Abel’s integral equations. The method, characterized by high accuracy and simplicity, involves constructing optimal quadrature formulas in the sense of Sard and providing error estimates within the Hilbert space of differentiable functions. The squared norm of the error functional for the quadrature formula in the space <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow><mrow><mrow><mo>(</mo><mn>2</mn><mo>.</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> is computed. To minimize this error, a system of linear equations regarding the formula’s coefficients is derived, leading to a unique solution. Then the explicit expressions for these optimal coefficients are obtained. The validity of the approach is demonstrated by solving several integral equations, with approximation errors presented in the corresponding tables.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100542"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two step inertial Tseng method for solving monotone variational inclusion problem","authors":"Lehlogonolo Mokaba ,&nbsp;Hammed Anuoluwapo Abass ,&nbsp;Abubakar Adamu","doi":"10.1016/j.rinam.2025.100545","DOIUrl":"10.1016/j.rinam.2025.100545","url":null,"abstract":"<div><div>In this paper, we examine the monotone variational inclusion problem with a maximal monotone operator and a Lipschitz continuous monotone operator. We propose two different iterative algorithms for solving the monotone variational inclusion problem, utilizing a new self-adaptive step size and a two-step inertial technique. Under the assumption that the solution set of the monotone variational inclusion problem is nonempty, we prove weak and strong convergence theorems concerning the sequences generated by our proposed algorithms. The convergence is guaranteed under some mild assumptions. Some numerical experiments are presented to demonstrate the performance of our iterative algorithms in comparison with recent results in the literature.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100545"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence of multiple weak solutions to a weighted quasilinear elliptic equation","authors":"Khaled Kefi","doi":"10.1016/j.rinam.2024.100536","DOIUrl":"10.1016/j.rinam.2024.100536","url":null,"abstract":"<div><div>In this study, we explore the existence of solutions to certain quasilinear degenerate elliptic equations that involve Hardy singular coefficients. Using variational techniques and critical point theorems, we establish new criteria for the existence of at least three weak solutions, under the assumption that the nonlinearity meets appropriate conditions.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100536"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A generalization of the second Pappus–Guldin theorem","authors":"Harald Schmid","doi":"10.1016/j.rinam.2025.100537","DOIUrl":"10.1016/j.rinam.2025.100537","url":null,"abstract":"<div><div>This paper deals with the question of how to calculate the volume of a body in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> when it is cut into slices perpendicular to a given curve. The answer is provided by a formula that can be considered as a generalized version of the second Pappus–Guldin theorem. It turns out that the computation becomes very simple if the curve passes directly through the centroids of the perpendicular cross-sections. In this context, the question arises whether a curve with this centroid property exists. We investigate this problem for a convex body <span><math><mi>K</mi></math></span> by using the volume distance and certain features of the so-called floating bodies of <span><math><mi>K</mi></math></span>. As an example, we further determine the non-trivial centroid curves of a triaxial ellipsoid, and finally we apply our results to derive a rather simple formula for determining the centroid of a bent rod.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100537"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence of the mathematical prediction equation and its convergence condition and its simplified derivation","authors":"Xuzan Gu,&nbsp;Zhibin Wang","doi":"10.1016/j.rinam.2025.100543","DOIUrl":"10.1016/j.rinam.2025.100543","url":null,"abstract":"<div><div>The mathematical prediction equation (MPE) offers a mathematical definition for wave-motion field prediction in physical systems. It applies to fluid mechanics and numerical weather forecasting models. MPE is an infinite series ensuring space-time accuracy in the Taylor expansion, and its convergence condition (convergence domain) must be established. The provided simplified derivation and inference of MPE include an illustrative Fourier wave-motion equation analysis. The derivation also establishes mathematical relationships between the Euler constant equation and Newton's displacement, velocity, and acceleration. MPE accurately predicts fluctuations of known variabilities for any order. It should serve as the mathematical foundation for fluid numerical prediction models. Moreover, combining MPE and its convergence condition with the cubic spline function yielded closed prediction equations for ideal gas pressure, temperature, and flow fields with second-order space-time accuracy. Further research directions include demonstrating the validity of MPE rigorously by the mathematical community and exploring MPE applications in quantum fields and discussing the mathematical definitions and differences between Newton and Einstein displacements.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100543"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"FastLRNR and Sparse Physics Informed Backpropagation","authors":"Woojin Cho ,&nbsp;Kookjin Lee ,&nbsp;Noseong Park ,&nbsp;Donsub Rim ,&nbsp;Gerrit Welper","doi":"10.1016/j.rinam.2025.100547","DOIUrl":"10.1016/j.rinam.2025.100547","url":null,"abstract":"<div><div>We introduce <u>S</u>parse <u>P</u>hysics <u>In</u>formed Back<u>prop</u>agation (SPInProp), a new class of methods for accelerating backpropagation for a specialized neural network architecture called Low Rank Neural Representation (LRNR). The approach exploits the low rank structure within LRNR and constructs a reduced neural network approximation that is much smaller in size. We call the smaller network FastLRNR. We show that backpropagation of FastLRNR can be substituted for that of LRNR, enabling a significant reduction in complexity. We apply SPInProp to a physics informed neural networks framework and demonstrate how the solution of parametrized partial differential equations is accelerated.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100547"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A hybrid method based on the classical/piecewise Chebyshev cardinal functions for multi-dimensional fractional Rayleigh–Stokes equations","authors":"M. Hosseininia ,&nbsp;M.H. Heydari ,&nbsp;D. Baleanu ,&nbsp;M. Bayram","doi":"10.1016/j.rinam.2025.100541","DOIUrl":"10.1016/j.rinam.2025.100541","url":null,"abstract":"<div><div>This study presents a numerical hybrid strategy for deriving approximate solutions to the one- and two-dimensional fractional Rayleigh–Stokes equations involving the Caputo derivative. This scheme mutually utilizes the classical and piecewise Chebyshev cardinal functions as basis functions. To this end, the operational matrices of the ordinary integral and fractional derivative of the piecewise Chebyshev cardinal functions, along with the ordinary and partial derivatives of the one- and two-variable Chebyshev cardinal functions, are derived. To create the desired approach by considering a hybrid expansion of the solution of the problem using the Chebyshev cardinal functions (for the spatial variable) and piecewise Chebyshev cardinal functions (for the temporal variable), and employing the aforementioned operational matrices, solving the problem under consideration turns into solving an algebraic system of linear equations. The convergence analysis of the established method is examined both theoretically and numerically. The accuracy and validity of the developed scheme are examined by solving several numerical examples.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100541"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A deep learning approach: Physics-informed neural networks for solving the 2D nonlinear Sine–Gordon equation","authors":"Alemayehu Tamirie Deresse ,&nbsp;Tamirat Temesgen Dufera","doi":"10.1016/j.rinam.2024.100532","DOIUrl":"10.1016/j.rinam.2024.100532","url":null,"abstract":"<div><div>In the current work, we apply a physics-informed neural networks (PINNs), a machine learning approach, for solving the non-linear hyperbolic sine–Gordon problem with two space dimensions. To include all the physical information of a PDE in to the learning process, we considered a multi-objective loss function that takes into account the problem PDE residual, the initial condition residual, and the boundary condition residual. The problem was approximated using PINNs employing a variety of artificial neural network topologies, one of which being feedforward deep neural networks, a densely connected network. To establish the effectiveness, soundness, and practical implications of the suggested technique, we provide three computational illustrations from the nonlinear two-dimensional sine–Gordon equations. We trained the PINNs model and run various tests using Python software as a computational tool. We gave the theoretical error bounds of the proposed approach in approximating the NLSGE. We evaluated the accuracy of the model by comparing it to other standard numerical methods in the literature through root mean square error (RMSE), <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span>, and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> relative errors. The findings suggested that the offered PINN approach is more effective and accurate than the other numerical methods. The method can be directly applied to any problem that involves different boundary conditions without requiring linearization, perturbation, or interpolation techniques. Thus, for the purpose of solving the nonlinear hyperbolic sine–Gordon equation in two dimensions and other difficult nonlinear physical issues across several fields, the PINN model provides an appropriate programming machine learning technique that is both accurate and efficient.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100532"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new hybrid approach for solving partial differential equations: Combining Physics-Informed Neural Networks with Cat-and-Mouse based Optimization","authors":"Nursyiva Irsalinda ,&nbsp;Maharani A. Bakar ,&nbsp;Fatimah Noor Harun ,&nbsp;Sugiyarto Surono ,&nbsp;Danang A. Pratama","doi":"10.1016/j.rinam.2025.100539","DOIUrl":"10.1016/j.rinam.2025.100539","url":null,"abstract":"<div><div>Partial differential equations (PDEs) are essential for modeling a wide range of physical phenomena. Physics-Informed Neural Networks (PINNs) offer a promising numerical framework for solving PDEs, but their performance often depends on the choice of optimization strategy and network configuration. In this study, we propose a hybrid PINN with a Cat and Mouse-based Optimizer (CMBO) to enhance optimization effectiveness and improve accuracy across elliptic, parabolic, and hyperbolic PDEs. CMBO utilizes a cat and mouse interaction mechanism to effectively balance exploration and exploitation, improving parameter initialization and guiding the optimization process toward favorable regions of the parameter space. Extensive experiments were conducted under varying scenarios, including different numbers of hidden layers (3, 5, 7) and neurons per layer (10, 30, 50). The proposed PINN-CMBO was systematically evaluated against state-of-the-art optimization methods, including PINN Adam, PINN L-BFGS, PINN Adam L-BFGS, and PINN PSO, across a diverse set of PDE categories. Experimental results revealed that PINN CMBO consistently achieved superior performance, recording the lowest loss values among all methods within fewer iteration. For parabolic and hyperbolic PDEs, PINN CMBO achieved an impressive minimum loss value, significantly outperforming PINN Adam, PINN L-BFGS, PINN Adam L-BFGS, and PINN PSO. Similar improvements were observed in elliptic and parabolic PDEs, where PINN-CMBO demonstrated unparalleled accuracy and stability across all tested network configurations. The integration of CMBO into PINN enabled efficient parameter initialization, driving a substantial reduction in the loss function compared to conventional PINN approaches. By guiding the training process toward optimal regions of the parameter space, PINN-CMBO not only accelerates convergence but also enhances overall performance. These findings establish PINN-CMBO as a highly effective framework for solving complex PDE problems, surpassing existing methods in terms of accuracy and stability.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100539"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some fixed point results concerning various contractions in extended b- metric space endowed with a graph","authors":"Neeraj Kumar ,&nbsp;Seema Mehra ,&nbsp;Dania Santina ,&nbsp;Nabil Mlaiki","doi":"10.1016/j.rinam.2024.100524","DOIUrl":"10.1016/j.rinam.2024.100524","url":null,"abstract":"<div><div>Contraction type mappings are crucial for understanding fixed point theory under specific conditions. We propose generalized (Boyd–Wong) type A <strong>F</strong> and (S - N) rational type contractions in an enlarged b-metric space which are represented by a graphically. Also, we gave a contrast of generalized (Boyd–Wong) type A <strong>F</strong> — contraction in 2D and 3D. We use appropriate illustrations to demonstrate the validity and primacy of our outcomes. Additionally, we use our derived conclusions to solve the Fredholm integral problem.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100524"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}