Pub Date : 2025-02-01DOI: 10.1016/j.rinam.2025.100551
Yuelong Tang, Yuchun Hua, Yujun Zheng, Chao Wu
This paper studies a novel fully discrete mixed method for optimal control problems (OCPs) with parabolic equations and low regularity. The backward difference scheme and mixed finite elements (MFEs) are used for temporal and spatial discretization of state and adjoint state, respectively. Error estimates of all variables are derived through the introduction of specific auxiliary variables and the application of suitable regularity assumptions. The theoretical analysis is validated by two numerical examples.
{"title":"Fully discrete P02−P1 mixed elements for optimal control with parabolic equations and low regularity","authors":"Yuelong Tang, Yuchun Hua, Yujun Zheng, Chao Wu","doi":"10.1016/j.rinam.2025.100551","DOIUrl":"10.1016/j.rinam.2025.100551","url":null,"abstract":"<div><div>This paper studies a novel fully discrete mixed method for optimal control problems (OCPs) with parabolic equations and low regularity. The backward difference scheme and <span><math><mrow><msubsup><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>−</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> mixed finite elements (MFEs) are used for temporal and spatial discretization of state and adjoint state, respectively. Error estimates of all variables are derived through the introduction of specific auxiliary variables and the application of suitable regularity assumptions. The theoretical analysis is validated by two numerical examples.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100551"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487236","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}
Pub Date : 2025-02-01DOI: 10.1016/j.rinam.2025.100541
M. Hosseininia , M.H. Heydari , D. Baleanu , M. Bayram
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.
{"title":"A hybrid method based on the classical/piecewise Chebyshev cardinal functions for multi-dimensional fractional Rayleigh–Stokes equations","authors":"M. Hosseininia , M.H. Heydari , D. Baleanu , 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}
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), , and 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.
{"title":"A deep learning approach: Physics-informed neural networks for solving the 2D nonlinear Sine–Gordon equation","authors":"Alemayehu Tamirie Deresse , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.rinam.2025.100539
Nursyiva Irsalinda , Maharani A. Bakar , Fatimah Noor Harun , Sugiyarto Surono , Danang A. Pratama
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.
{"title":"A new hybrid approach for solving partial differential equations: Combining Physics-Informed Neural Networks with Cat-and-Mouse based Optimization","authors":"Nursyiva Irsalinda , Maharani A. Bakar , Fatimah Noor Harun , Sugiyarto Surono , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.rinam.2025.100552
Steven G. From , Suthakaran Ratnasingam
Some new upper and lower bounds for the upper incomplete gamma function are presented. Some of the bounds are given for all real and some are for only certain combinations of and . A number of different methods are used to obtain these new bounds. In particular, rational function and perturbations of rational function bounds are presented. Some numerical comparisons are made with previously proposed bounds.
{"title":"New upper and lower bounds for the upper incomplete gamma function","authors":"Steven G. From , Suthakaran Ratnasingam","doi":"10.1016/j.rinam.2025.100552","DOIUrl":"10.1016/j.rinam.2025.100552","url":null,"abstract":"<div><div>Some new upper and lower bounds for the upper incomplete gamma function <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> are presented. Some of the bounds are given for all real <span><math><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></math></span> and some are for only certain combinations of <span><math><mi>a</mi></math></span> and <span><math><mi>x</mi></math></span>. A number of different methods are used to obtain these new bounds. In particular, rational function and perturbations of rational function bounds are presented. Some numerical comparisons are made with previously proposed bounds.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100552"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529722","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}
Contraction type mappings are crucial for understanding fixed point theory under specific conditions. We propose generalized (Boyd–Wong) type A F 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 F — 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.
{"title":"Some fixed point results concerning various contractions in extended b- metric space endowed with a graph","authors":"Neeraj Kumar , Seema Mehra , Dania Santina , 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}
Pub Date : 2025-02-01DOI: 10.1016/j.rinam.2024.100527
Toai Luong
The Cahn–Hilliard equation is a widely used model for describing phase separation processes in a binary mixture. In this paper, we investigate the viscous Cahn–Hilliard equation with a degenerate, phase-dependent mobility. We define the concept of a weak solution and establish the existence of such a solution by taking limits of solutions to the viscous Cahn–Hilliard equation with positive mobility. Additionally, assuming that the initial data is positive, we demonstrate that the weak solution remains nonnegative and is not identically zero. Finally, we prove that the weak solution satisfies an energy dissipation inequality.
{"title":"Nonnegative weak solution to the degenerate viscous Cahn–Hilliard equation","authors":"Toai Luong","doi":"10.1016/j.rinam.2024.100527","DOIUrl":"10.1016/j.rinam.2024.100527","url":null,"abstract":"<div><div>The Cahn–Hilliard equation is a widely used model for describing phase separation processes in a binary mixture. In this paper, we investigate the viscous Cahn–Hilliard equation with a degenerate, phase-dependent mobility. We define the concept of a weak solution and establish the existence of such a solution by taking limits of solutions to the viscous Cahn–Hilliard equation with positive mobility. Additionally, assuming that the initial data is positive, we demonstrate that the weak solution remains nonnegative and is not identically zero. Finally, we prove that the weak solution satisfies an energy dissipation inequality.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100527"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150062","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}
Pub Date : 2025-02-01DOI: 10.1016/j.rinam.2024.100531
Abdullah Shah , Maaz ur Rehman , Jamilu Sabi’u , Muhammad Sohaib , Khaled M. Furati
This paper introduces a new version of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, characterized as a scaled memoryless, projection-based, and derivative-free method for finding approximate solutions of monotone nonlinear equations with convex constraints. The optimal value of the scaling parameter is achieved by minimizing the BFGS update matrix. The theoretical analysis is performed to demonstrate the global convergence of the approach. Numerical analysis and comparisons with prior results indicate that the proposed approach has superior performance for CPU time, iteration count, and function evaluations. The new algorithm is used to solve the motion control issue of a two-jointed coplanar robot manipulator.
{"title":"A new scaled BFGS method for convex constraints monotone systems: Applications in motion control","authors":"Abdullah Shah , Maaz ur Rehman , Jamilu Sabi’u , Muhammad Sohaib , Khaled M. Furati","doi":"10.1016/j.rinam.2024.100531","DOIUrl":"10.1016/j.rinam.2024.100531","url":null,"abstract":"<div><div>This paper introduces a new version of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, characterized as a scaled memoryless, projection-based, and derivative-free method for finding approximate solutions of monotone nonlinear equations with convex constraints. The optimal value of the scaling parameter is achieved by minimizing the BFGS update matrix. The theoretical analysis is performed to demonstrate the global convergence of the approach. Numerical analysis and comparisons with prior results indicate that the proposed approach has superior performance for CPU time, iteration count, and function evaluations. The new algorithm is used to solve the motion control issue of a two-jointed coplanar robot manipulator.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100531"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143098506","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}
Pub Date : 2025-02-01DOI: 10.1016/j.rinam.2024.100523
Ángela Jiménez-Casas
In this paper we analyze a generalization of the semilinear phase field model from G. Caginalp (1986, 1991) and A. Jiménez-Casas-A. Rodriguez-Bernal (1996, 2005), where we consider a singular term concentrated in a neighborhood of , the boundary of domain. The neighborhood shrinks to as a parameter approaches zero.
We prove that this family of solutions, of the new semilinear phase field model, converges in suitable spaces when this parameter tends to zero, to the solutions of a semilinear phase field problem where the concentrating potential are transformed into an extra flux condition on .
{"title":"Phase-field model with concentrating-potential terms on the boundary","authors":"Ángela Jiménez-Casas","doi":"10.1016/j.rinam.2024.100523","DOIUrl":"10.1016/j.rinam.2024.100523","url":null,"abstract":"<div><div>In this paper we analyze a generalization of the semilinear phase field model from G. Caginalp (1986, 1991) and A. Jiménez-Casas-A. Rodriguez-Bernal (1996, 2005), where we consider a singular term concentrated in a neighborhood of <span><math><mi>Γ</mi></math></span>, the boundary of domain. The neighborhood shrinks to <span><math><mi>Γ</mi></math></span> as a parameter <span><math><mi>ϵ</mi></math></span> approaches zero.</div><div>We prove that this family of solutions, of the new semilinear phase field model, converges in suitable spaces when this parameter tends to zero, to the solutions of a semilinear phase field problem where the concentrating potential are transformed into an extra flux condition on <span><math><mi>Γ</mi></math></span>.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100523"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149998","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}
Pub Date : 2025-02-01DOI: 10.1016/j.rinam.2025.100540
Richard A. Zalik
Using properties of the Fourier transform we prove that if a Hartree–Fock molecular spatial orbital is in , then it decays to zero as its argument diverges to infinity. The proof is rigorous, elementary and short. Our result implies that occupied orbitals with positive eigenvalues will decay to zero provided they are in .
{"title":"Decay properties of spatial molecular orbitals","authors":"Richard A. Zalik","doi":"10.1016/j.rinam.2025.100540","DOIUrl":"10.1016/j.rinam.2025.100540","url":null,"abstract":"<div><div>Using properties of the Fourier transform we prove that if a Hartree–Fock molecular spatial orbital is in <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>, then it decays to zero as its argument diverges to infinity. The proof is rigorous, elementary and short. Our result implies that occupied orbitals with positive eigenvalues will decay to zero provided they are in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"25 ","pages":"Article 100540"},"PeriodicalIF":1.4,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150011","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号