Pub Date : 2025-02-21DOI: 10.1016/j.cam.2025.116565
Pshtiwan Othman Mohammed
In this paper, we first introduce a discrete Mittag-Leffler function of delta type. Using the Laplace transformation, some properties of the new special function are obtained. Second, we use this function to define new discrete fractional operators, namely AB fractional differences and sums, based on the Riemann–Liouville operators. We also applied the Laplace transformation on the new special functions and the related discrete operators. Finally, we propose and implement the mean value technique of discrete fractional calculus and demonstrate the advantages in terms of AB fractional differences.
在本文中，我们首先介绍了一种离散的德尔塔型米塔格-勒弗勒函数。利用拉普拉斯变换，我们得到了新的特殊函数的一些性质。其次，我们利用该函数定义了新的离散分式算子，即基于黎曼-刘维尔算子的 AB 分式差与和。我们还在新的特殊函数和相关离散算子上应用了拉普拉斯变换。最后，我们提出并实现了离散分数微积分的均值技术，并展示了 AB 分数差的优势。
{"title":"On the delta Mittag-Leffler functions and its application in monotonic analysis","authors":"Pshtiwan Othman Mohammed","doi":"10.1016/j.cam.2025.116565","DOIUrl":"10.1016/j.cam.2025.116565","url":null,"abstract":"<div><div>In this paper, we first introduce a discrete Mittag-Leffler function of delta type. Using the Laplace transformation, some properties of the new special function are obtained. Second, we use this function to define new discrete fractional operators, namely AB fractional differences and sums, based on the Riemann–Liouville operators. We also applied the Laplace transformation on the new special functions and the related discrete operators. Finally, we propose and implement the mean value technique of discrete fractional calculus and demonstrate the advantages in terms of AB fractional differences.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116565"},"PeriodicalIF":2.1,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143465468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-21DOI: 10.1016/j.cam.2025.116568
E.O. Asante-Asamani , A. Kleefeld , B.A. Wade
A fourth-order exponential time differencing (ETD) Runge–Kutta scheme with dimensional splitting is developed to solve multidimensional non-linear systems of reaction–diffusion equations (RDE). By approximating the matrix exponential in the scheme with the A-acceptable Padé (2,2) rational function, the resulting scheme (ETDRK4P22-IF) is verified empirically to be fourth-order accurate for several RDE. The scheme is shown to be more efficient than competing fourth-order ETD and IMEX schemes, achieving up to 20X speed-up in CPU time. Inclusion of up to three pre-smoothing steps of a lower order L-stable scheme facilitates efficient damping of spurious oscillations arising from problems with non-smooth initial/boundary conditions.
{"title":"A fourth-order exponential time differencing scheme with dimensional splitting for non-linear reaction–diffusion systems","authors":"E.O. Asante-Asamani , A. Kleefeld , B.A. Wade","doi":"10.1016/j.cam.2025.116568","DOIUrl":"10.1016/j.cam.2025.116568","url":null,"abstract":"<div><div>A fourth-order exponential time differencing (ETD) Runge–Kutta scheme with dimensional splitting is developed to solve multidimensional non-linear systems of reaction–diffusion equations (RDE). By approximating the matrix exponential in the scheme with the A-acceptable Padé (2,2) rational function, the resulting scheme (ETDRK4P22-IF) is verified empirically to be fourth-order accurate for several RDE. The scheme is shown to be more efficient than competing fourth-order ETD and IMEX schemes, achieving up to 20X speed-up in CPU time. Inclusion of up to three pre-smoothing steps of a lower order L-stable scheme facilitates efficient damping of spurious oscillations arising from problems with non-smooth initial/boundary conditions.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116568"},"PeriodicalIF":2.1,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we present a linear second-order single time-stepping finite difference scheme for solving the Allen–Cahn equation. The temporal integration is realized by combining the predictor-correction fashion of the Crank–Nicolson scheme with a linear stabilization technique, where central finite differences are employed for spatial discretization. In contrast to the BDF2 scheme, the proposed method operates without any extrapolation strategies, avoiding the need to compute the ratio of adjacent time steps during each time iteration. The discrete Maximum bound principle (MBP) is proven under the mild constraints on the time step size. The convergence analysis in and norms is also presented as well as the energy stability. Several typical 2D and 3D numerical experiments are carried out to verify the theoretical results and demonstrate the efficiency of the proposed scheme.
{"title":"An efficient Crank–Nicolson scheme with preservation of the maximum bound principle for the high-dimensional Allen–Cahn equation","authors":"Yabin Hou , Jingwei Li , Yuanyang Qiao , Xinlong Feng","doi":"10.1016/j.cam.2025.116586","DOIUrl":"10.1016/j.cam.2025.116586","url":null,"abstract":"<div><div>In this study, we present a linear second-order single time-stepping finite difference scheme for solving the Allen–Cahn equation. The temporal integration is realized by combining the predictor-correction fashion of the Crank–Nicolson scheme with a linear stabilization technique, where central finite differences are employed for spatial discretization. In contrast to the BDF2 scheme, the proposed method operates without any extrapolation strategies, avoiding the need to compute the ratio of adjacent time steps during each time iteration. The discrete Maximum bound principle (MBP) is proven under the mild constraints on the time step size. The convergence analysis in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> norms is also presented as well as the energy stability. Several typical 2D and 3D numerical experiments are carried out to verify the theoretical results and demonstrate the efficiency of the proposed scheme.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116586"},"PeriodicalIF":2.1,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-21DOI: 10.1016/j.cam.2025.116567
Zhan Liu, Yi Sun, Yong Li, Yuanmeng Li
With the development of network technology and the rise of big data, non-probability sampling has wider applications in practice. However, it brings a challenge to make inference from non-probability samples since the inclusion probabilities of non-probability samples are unknown. The propensity score approach, superpopulation model approach, doubly robust estimation are three main methods to infer the population from non-probability samples. However, the first two methods are sensitive to the misspecified models. Thus, they cannot generate desirable performances when deal with heterogeneous non-probability samples. In this paper, a doubly robust estimation method for non-probability samples with heterogeneous data is proposed. A heterogeneous superpopulation model is fitted based on a heterogeneous non-probability sample and used to construct a doubly robust estimator for the population mean. Specifically, the inverse estimated inclusion probabilities of the non-probability sample are added into the estimating equation as weights in model parameter estimation. The simulation results confirm that the proposed method outperforms the other contrastive methods in terms of bias, standard deviation and mean square error. Its application is illustrated with the Pew Research Center dataset and the Behavioral Risk Factor Surveillance System dataset, which is consistent with the simulation results.
{"title":"Doubly robust estimation for non-probability samples with heterogeneity","authors":"Zhan Liu, Yi Sun, Yong Li, Yuanmeng Li","doi":"10.1016/j.cam.2025.116567","DOIUrl":"10.1016/j.cam.2025.116567","url":null,"abstract":"<div><div>With the development of network technology and the rise of big data, non-probability sampling has wider applications in practice. However, it brings a challenge to make inference from non-probability samples since the inclusion probabilities of non-probability samples are unknown. The propensity score approach, superpopulation model approach, doubly robust estimation are three main methods to infer the population from non-probability samples. However, the first two methods are sensitive to the misspecified models. Thus, they cannot generate desirable performances when deal with heterogeneous non-probability samples. In this paper, a doubly robust estimation method for non-probability samples with heterogeneous data is proposed. A heterogeneous superpopulation model is fitted based on a heterogeneous non-probability sample and used to construct a doubly robust estimator for the population mean. Specifically, the inverse estimated inclusion probabilities of the non-probability sample are added into the estimating equation as weights in model parameter estimation. The simulation results confirm that the proposed method outperforms the other contrastive methods in terms of bias, standard deviation and mean square error. Its application is illustrated with the Pew Research Center dataset and the Behavioral Risk Factor Surveillance System dataset, which is consistent with the simulation results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116567"},"PeriodicalIF":2.1,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-20DOI: 10.1016/j.cam.2025.116583
Warren L. Davis IV , Max L. Carlson , Irina K. Tezaur , Diana L. Bull , Kara J. Peterson , Laura P. Swiler
Recent years have seen a growing concern about climate change and its impacts. While Earth System Models (ESMs) can be invaluable tools for studying the impacts of climate change, the complex coupling processes encoded in ESMs and the large amounts of data produced by these models, together with the high internal variability of the Earth system, can obscure important source-to-impact relationships. This paper presents a novel and efficient unsupervised data-driven approach for detecting statistically-significant impacts and tracing spatio-temporal source-impact pathways in the climate through a unique combination of ideas from anomaly detection, clustering and Natural Language Processing (NLP). Using as an exemplar the 1991 eruption of Mount Pinatubo in the Philippines, we demonstrate that the proposed approach is capable of detecting known post-eruption impacts/events. We additionally describe a methodology for extracting meaningful sequences of post-eruption impacts/events by using NLP to efficiently mine frequent multivariate cluster evolutions, which can be used to confirm or discover the chain of physical processes between a climate source and its impact(s).
{"title":"Spatio-temporal multivariate cluster evolution analysis for detecting and tracking climate impacts","authors":"Warren L. Davis IV , Max L. Carlson , Irina K. Tezaur , Diana L. Bull , Kara J. Peterson , Laura P. Swiler","doi":"10.1016/j.cam.2025.116583","DOIUrl":"10.1016/j.cam.2025.116583","url":null,"abstract":"<div><div>Recent years have seen a growing concern about climate change and its impacts. While Earth System Models (ESMs) can be invaluable tools for studying the impacts of climate change, the complex coupling processes encoded in ESMs and the large amounts of data produced by these models, together with the high internal variability of the Earth system, can obscure important source-to-impact relationships. This paper presents a novel and efficient unsupervised data-driven approach for detecting statistically-significant impacts and tracing spatio-temporal source-impact pathways in the climate through a unique combination of ideas from anomaly detection, clustering and Natural Language Processing (NLP). Using as an exemplar the 1991 eruption of Mount Pinatubo in the Philippines, we demonstrate that the proposed approach is capable of detecting known post-eruption impacts/events. We additionally describe a methodology for extracting meaningful sequences of post-eruption impacts/events by using NLP to efficiently mine frequent multivariate cluster evolutions, which can be used to confirm or discover the chain of physical processes between a climate source and its impact(s).</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116583"},"PeriodicalIF":2.1,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work examines a stochastic volatility model with double-exponential jumps in the context of option pricing. The model has been considered in previous research articles, but no thorough analysis had been conducted to study its quality of calibration and pricing capabilities thus far. We provide evidence that this model outperforms challenger models possessing similar features (stochastic volatility and jumps), especially in the fit of the short term implied volatility smile, and that it is particularly tractable for the pricing of exotic options from different generations. The article utilizes Fourier pricing techniques (the PROJ method and its refinements) for different types of claims and several generations of exotics (Asian options, cliquets, barrier options, and options on realized variance), and all source codes are made publicly available to facilitate adoption and future research. The results indicate that this model is highly promising, thanks to the asymmetry of the jumps distribution allowing it to capture richer dynamics than a normal jump size distribution. The parameters all have meaningful econometrics interpretations that are important for adoption by risk-managers.
{"title":"Calibration and option pricing with stochastic volatility and double exponential jumps","authors":"Gaetano Agazzotti , Jean-Philippe Aguilar , Claudio Aglieri Rinella , Justin Lars Kirkby","doi":"10.1016/j.cam.2025.116563","DOIUrl":"10.1016/j.cam.2025.116563","url":null,"abstract":"<div><div>This work examines a stochastic volatility model with double-exponential jumps in the context of option pricing. The model has been considered in previous research articles, but no thorough analysis had been conducted to study its quality of calibration and pricing capabilities thus far. We provide evidence that this model outperforms challenger models possessing similar features (stochastic volatility and jumps), especially in the fit of the short term implied volatility smile, and that it is particularly tractable for the pricing of exotic options from different generations. The article utilizes Fourier pricing techniques (the PROJ method and its refinements) for different types of claims and several generations of exotics (Asian options, cliquets, barrier options, and options on realized variance), and all source codes are made publicly available to facilitate adoption and future research. The results indicate that this model is highly promising, thanks to the asymmetry of the jumps distribution allowing it to capture richer dynamics than a normal jump size distribution. The parameters all have meaningful econometrics interpretations that are important for adoption by risk-managers.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116563"},"PeriodicalIF":2.1,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143465464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-20DOI: 10.1016/j.cam.2025.116576
Yanping Chen , Wanxiang Liu , Yang Wang , Huaming Yi
A family of compact and linearly implicit Galerkin method is proposed for the general nonlinear parabolic equation based on second-order weighted implicit–explicit schemes in time and the lowest-order nonconforming virtual element discretization in space. The proposed method achieves second-order global accuracy in the temporal directions, and no additional initial iterations are required. To address consistency errors arising from nonconforming virtual element space, we construct two novel elliptic projection operators and rigorously prove the convergence and boundedness of the projection solutions. With the help of a novel elliptic projection operator and the temporal–spatial error splitting technique, we establish the boundedness and unconditional optimal error estimate of the fully discrete solution. Several numerical experiments are presented to validate our theoretical discoveries.
{"title":"Optimal convergence analysis of the lowest-order nonconforming virtual element method for general nonlinear parabolic equations","authors":"Yanping Chen , Wanxiang Liu , Yang Wang , Huaming Yi","doi":"10.1016/j.cam.2025.116576","DOIUrl":"10.1016/j.cam.2025.116576","url":null,"abstract":"<div><div>A family of compact and linearly implicit Galerkin method is proposed for the general nonlinear parabolic equation based on second-order weighted implicit–explicit schemes in time and the lowest-order nonconforming virtual element discretization in space. The proposed method achieves second-order global accuracy in the temporal directions, and no additional initial iterations are required. To address consistency errors arising from nonconforming virtual element space, we construct two novel elliptic projection operators and rigorously prove the convergence and boundedness of the projection solutions. With the help of a novel elliptic projection operator and the temporal–spatial error splitting technique, we establish the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> boundedness and unconditional optimal error estimate of the fully discrete solution. Several numerical experiments are presented to validate our theoretical discoveries.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116576"},"PeriodicalIF":2.1,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143465467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-20DOI: 10.1016/j.cam.2025.116584
Christopher R. Wentland , Francesco Rizzi , Joshua L. Barnett , Irina K. Tezaur
This paper presents and evaluates a framework for the coupling of subdomain-local projection-based reduced order models (PROMs) using the Schwarz alternating method following a domain decomposition (DD) of the spatial domain on which a given problem of interest is posed. In this approach, the solution on the full domain is obtained via an iterative process in which a sequence of subdomain-local problems are solved, with information propagating between subdomains through transmission boundary conditions (BCs). We explore several new directions involving the Schwarz alternating method aimed at maximizing the method’s efficiency and flexibility, and demonstrate it on three challenging two-dimensional nonlinear hyperbolic problems: the shallow water equations, Burgers’ equation, and the compressible Euler equations. We demonstrate that, for a cell-centered finite volume discretization and a non-overlapping DD, it is possible to obtain a stable and accurate coupled model utilizing Dirichlet–Dirichlet (rather than Robin–Robin or alternating Dirichlet–Neumann) transmission BCs on the subdomain boundaries. We additionally explore the impact of boundary sampling when utilizing the Schwarz alternating method to couple subdomain-local hyper-reduced PROMs. Our numerical results suggest that the proposed methodology has the potential to improve PROM accuracy by enabling the spatial localization of these models via domain decomposition, and achieve up to two orders of magnitude speedup over equivalent coupled full order model solutions and moderate speedups over analogous monolithic solutions.
{"title":"The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models","authors":"Christopher R. Wentland , Francesco Rizzi , Joshua L. Barnett , Irina K. Tezaur","doi":"10.1016/j.cam.2025.116584","DOIUrl":"10.1016/j.cam.2025.116584","url":null,"abstract":"<div><div>This paper presents and evaluates a framework for the coupling of subdomain-local projection-based reduced order models (PROMs) using the Schwarz alternating method following a domain decomposition (DD) of the spatial domain on which a given problem of interest is posed. In this approach, the solution on the full domain is obtained via an iterative process in which a sequence of subdomain-local problems are solved, with information propagating between subdomains through transmission boundary conditions (BCs). We explore several new directions involving the Schwarz alternating method aimed at maximizing the method’s efficiency and flexibility, and demonstrate it on three challenging two-dimensional nonlinear hyperbolic problems: the shallow water equations, Burgers’ equation, and the compressible Euler equations. We demonstrate that, for a cell-centered finite volume discretization and a non-overlapping DD, it is possible to obtain a stable and accurate coupled model utilizing Dirichlet–Dirichlet (rather than Robin–Robin or alternating Dirichlet–Neumann) transmission BCs on the subdomain boundaries. We additionally explore the impact of boundary sampling when utilizing the Schwarz alternating method to couple subdomain-local hyper-reduced PROMs. Our numerical results suggest that the proposed methodology has the potential to improve PROM accuracy by enabling the spatial localization of these models via domain decomposition, and achieve up to two orders of magnitude speedup over equivalent coupled full order model solutions and moderate speedups over analogous monolithic solutions.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116584"},"PeriodicalIF":2.1,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143508325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-20DOI: 10.1016/j.cam.2025.116589
Pavel Karban, David Pánek, Jan Kaska
Agros is an open-source software package designed to simulate and analyze physical fields, including electromagnetic, thermal, and mechanical phenomena. The software offers a versatile and intuitive graphical user interface (GUI) and supports multi-physics simulations, making it a valuable tool for academic research and industrial applications. This article provides an overview of agros development, focusing on recent updates that enhance its functionality, modularity, and usability. We discuss agros integration with the deal.II library and Python for advanced customization, which expands the capabilities of its core engine and postprocessor.
{"title":"Open-source platform for simulation of physical fields: Agros","authors":"Pavel Karban, David Pánek, Jan Kaska","doi":"10.1016/j.cam.2025.116589","DOIUrl":"10.1016/j.cam.2025.116589","url":null,"abstract":"<div><div>Agros is an open-source software package designed to simulate and analyze physical fields, including electromagnetic, thermal, and mechanical phenomena. The software offers a versatile and intuitive graphical user interface (GUI) and supports multi-physics simulations, making it a valuable tool for academic research and industrial applications. This article provides an overview of agros development, focusing on recent updates that enhance its functionality, modularity, and usability. We discuss agros integration with the deal.II library and Python for advanced customization, which expands the capabilities of its core engine and postprocessor.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116589"},"PeriodicalIF":2.1,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143465466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-20DOI: 10.1016/j.cam.2025.116585
Min Wang , Zhimin Zhang
The Fredholm–Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical methods. To address this, we propose a novel class of mapped Hermite functions, which are constructed by applying a mapping to Hermite polynomials. We establish fundamental approximation theory for the orthogonal functions. We propose MHFs-spectral collocation method and MHFs-smoothing transformation method to solve the two-point weakly singular FHIEs, respectively. Error analysis and numerical results demonstrate that our methods, based on the new orthogonal functions, are particularly effective for handling problems with weak singularities at two endpoints, yielding exponential convergence rate. We position this work as the first to directly study the mapped spectral method for multi-point singularity problems, to the best of our knowledge.
{"title":"Mapped Hermite functions and their applications to two-dimensional weakly singular Fredholm–Hammerstein integral equations","authors":"Min Wang , Zhimin Zhang","doi":"10.1016/j.cam.2025.116585","DOIUrl":"10.1016/j.cam.2025.116585","url":null,"abstract":"<div><div>The Fredholm–Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical methods. To address this, we propose a novel class of mapped Hermite functions, which are constructed by applying a mapping to Hermite polynomials. We establish fundamental approximation theory for the orthogonal functions. We propose MHFs-spectral collocation method and MHFs-smoothing transformation method to solve the two-point weakly singular FHIEs, respectively. Error analysis and numerical results demonstrate that our methods, based on the new orthogonal functions, are particularly effective for handling problems with weak singularities at two endpoints, yielding exponential convergence rate. We position this work as the first to directly study the mapped spectral method for multi-point singularity problems, to the best of our knowledge.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"465 ","pages":"Article 116585"},"PeriodicalIF":2.1,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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