Pub Date : 2025-01-13DOI: 10.1016/j.camwa.2025.01.008
Takuma Kimura , Teruya Minamoto , Mitsuhiro T. Nakao
We consider the constructive a priori error estimates for a full discrete approximation of a periodic solution for the heat equation. Our numerical scheme is based on the finite element semidiscretization in space direction combined with the Fourier expansion in time. We derive the optimal order explicit and error estimates which play an important role in the numerical verification method of exact solutions for nonlinear parabolic equations. Several numerical examples which confirm the theoretical results will be presented.
{"title":"Constructive error estimates for a full-discretized periodic solution of heat equation by spatial finite-element and time spectral method","authors":"Takuma Kimura , Teruya Minamoto , Mitsuhiro T. Nakao","doi":"10.1016/j.camwa.2025.01.008","DOIUrl":"10.1016/j.camwa.2025.01.008","url":null,"abstract":"<div><div>We consider the constructive a priori error estimates for a full discrete approximation of a periodic solution for the heat equation. Our numerical scheme is based on the finite element semidiscretization in space direction combined with the Fourier expansion in time. We derive the optimal order explicit <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> error estimates which play an important role in the numerical verification method of exact solutions for nonlinear parabolic equations. Several numerical examples which confirm the theoretical results will be presented.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 248-258"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986209","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 : 2025-01-13DOI: 10.1016/j.camwa.2025.01.002
Zhaohua Li , Guang-an Zou , Lina Ma , Xiaofeng Yang
This paper presents a linear, fully-decoupled discontinuous Galerkin (DG) method for the flow-coupled phase-field vesicle membrane model. The fully discrete scheme is implemented by combining several reliable numerical techniques. Firstly, we employ the DG method for spatial discretization, which does not require that the numerical solutions are continuous across different grid cells, enabling adaptive treatment of complex vesicle membrane shapes and fluid flows. Secondly, to cope with nonlinear and coupling terms, scalar auxiliary variables (SAV) and implicit-explicit approaches are used, which improve numerical stability and computational efficiency. Finally, for the momentum equation, the pressure-correction method is used to assure the decoupling of velocity and pressure. Through rigorous mathematical proofs, this paper demonstrates the unconditional energy stability of the proposed scheme and derives its optimal error estimation. Numerical examples also show the method's effectiveness in terms of precision, energy stability, and simulated vesicle deformation in thin channels. The results highlight the method's potential application in fluid-coupled phase-field vesicle membrane models, while also providing new methodologies and technological support for numerical simulation and computational research in related domains.
{"title":"An efficient discontinuous Galerkin method for the hydro-dynamically coupled phase-field vesicle membrane model","authors":"Zhaohua Li , Guang-an Zou , Lina Ma , Xiaofeng Yang","doi":"10.1016/j.camwa.2025.01.002","DOIUrl":"10.1016/j.camwa.2025.01.002","url":null,"abstract":"<div><div>This paper presents a linear, fully-decoupled discontinuous Galerkin (DG) method for the flow-coupled phase-field vesicle membrane model. The fully discrete scheme is implemented by combining several reliable numerical techniques. Firstly, we employ the DG method for spatial discretization, which does not require that the numerical solutions are continuous across different grid cells, enabling adaptive treatment of complex vesicle membrane shapes and fluid flows. Secondly, to cope with nonlinear and coupling terms, scalar auxiliary variables (SAV) and implicit-explicit approaches are used, which improve numerical stability and computational efficiency. Finally, for the momentum equation, the pressure-correction method is used to assure the decoupling of velocity and pressure. Through rigorous mathematical proofs, this paper demonstrates the unconditional energy stability of the proposed scheme and derives its optimal error estimation. Numerical examples also show the method's effectiveness in terms of precision, energy stability, and simulated vesicle deformation in thin channels. The results highlight the method's potential application in fluid-coupled phase-field vesicle membrane models, while also providing new methodologies and technological support for numerical simulation and computational research in related domains.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 259-286"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986210","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 : 2025-01-13DOI: 10.1016/j.camwa.2024.12.021
Peter Gangl , Richard Löscher , Olaf Steinbach
In this paper we discuss the numerical solution of elliptic distributed optimal control problems with state or control constraints when the control is considered in the energy norm. As in the unconstrained case we can relate the regularization parameter and the finite element mesh size in order to ensure an optimal balance between the error and the cost, and, on the discrete level, an optimal order of convergence which only depends on the regularity of the given target, also including discontinuous target functions. While in most cases, state or control constraints are discussed for the more common regularization, much less is known in the case of energy regularizations. But in this case, and for both control and state constraints, we can formulate first kind variational inequalities to determine the unknown state, from which we can compute the control in a post processing step. Related variational inequalities also appear in obstacle problems, and are well established both from a mathematical and a numerical analysis point of view. Numerical results confirm the applicability and accuracy of the proposed approach.
{"title":"Regularization and finite element error estimates for elliptic distributed optimal control problems with energy regularization and state or control constraints","authors":"Peter Gangl , Richard Löscher , Olaf Steinbach","doi":"10.1016/j.camwa.2024.12.021","DOIUrl":"10.1016/j.camwa.2024.12.021","url":null,"abstract":"<div><div>In this paper we discuss the numerical solution of elliptic distributed optimal control problems with state or control constraints when the control is considered in the energy norm. As in the unconstrained case we can relate the regularization parameter and the finite element mesh size in order to ensure an optimal balance between the error and the cost, and, on the discrete level, an optimal order of convergence which only depends on the regularity of the given target, also including discontinuous target functions. While in most cases, state or control constraints are discussed for the more common <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> regularization, much less is known in the case of energy regularizations. But in this case, and for both control and state constraints, we can formulate first kind variational inequalities to determine the unknown state, from which we can compute the control in a post processing step. Related variational inequalities also appear in obstacle problems, and are well established both from a mathematical and a numerical analysis point of view. Numerical results confirm the applicability and accuracy of the proposed approach.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 242-260"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986212","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 : 2025-01-13DOI: 10.1016/j.camwa.2024.12.025
M. Yousuf , M. Alshayqi , S.S. Alzahrani
By utilizing the power of the Fourier spectral approach and the restricted Padé rational approximations, we have devised two third-order numerical methods to investigate the complex phenomena that arise in multi-dimensional space fractional reaction-diffusion models. The Fourier spectral approach yields a fully diagonal representation of the fractional Laplacian with the ability to extend the methods to multi-dimensional cases with the same computational complexity as one-dimensional and makes it possible to attain spectral convergence. Third-order single-pole restricted Padé approximations of the matrix exponential are utilized in developing the time stepping methods. We also use sophisticated mathematical techniques, namely, discrete sine and cosine transforms, to improve the computational efficiency of the methods. Algorithms are derived from these methods for straight-forward implementation in one- and multidimensional models, accommodating both homogeneous Dirichlet and homogeneous Neumann boundary conditions. The third-order accuracy of these methods is proved analytically and demonstrated numerically. Linear error analysis of these methods is presented, stability regions of both methods are computed, and their graphs are plotted. The computational efficiency, reliability, and effectiveness of the presented methods are demonstrated through numerical experiments. The convergence results are computed to support the theoretical findings.
{"title":"Fourier spectral methods based on restricted Padé approximations for space fractional reaction-diffusion systems","authors":"M. Yousuf , M. Alshayqi , S.S. Alzahrani","doi":"10.1016/j.camwa.2024.12.025","DOIUrl":"10.1016/j.camwa.2024.12.025","url":null,"abstract":"<div><div>By utilizing the power of the Fourier spectral approach and the restricted Padé rational approximations, we have devised two third-order numerical methods to investigate the complex phenomena that arise in multi-dimensional space fractional reaction-diffusion models. The Fourier spectral approach yields a fully diagonal representation of the fractional Laplacian with the ability to extend the methods to multi-dimensional cases with the same computational complexity as one-dimensional and makes it possible to attain spectral convergence. Third-order single-pole restricted Padé approximations of the matrix exponential are utilized in developing the time stepping methods. We also use sophisticated mathematical techniques, namely, discrete sine and cosine transforms, to improve the computational efficiency of the methods. Algorithms are derived from these methods for straight-forward implementation in one- and multidimensional models, accommodating both homogeneous Dirichlet and homogeneous Neumann boundary conditions. The third-order accuracy of these methods is proved analytically and demonstrated numerically. Linear error analysis of these methods is presented, stability regions of both methods are computed, and their graphs are plotted. The computational efficiency, reliability, and effectiveness of the presented methods are demonstrated through numerical experiments. The convergence results are computed to support the theoretical findings.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 228-247"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986211","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 : 2025-01-10DOI: 10.1016/j.camwa.2025.01.003
Anderson L.A. de Araujo , Jose L. Boldrini , Bianca M.R. Calsavara , Maicon R. Correa
We consider the rigorous analysis and the numerical approximation of a mathematical model for geographical spreading of Aedes aegypti. The complete model is composed of a system of parabolic partial differential equations coupled with one ordinary differential equation and has control terms related to the effects of insecticide application and sterile male release. The existence and uniqueness of solutions for the model are proven, and an efficient numerical methodology for approximating the unique solution of the mathematical model is proposed. The proposed numerical approach is based on a time-splitting scheme combined with locally conservative finite element methods. This combination of a well-posed mathematical model with a robust and efficient numerical formulation provides a suitable tool for the simulation of different scenarios of the spreading of Aedes aegypti. Numerical experiments, including a convergence study and a series of simulations that illustrate how the numerical model can be used in the decision-making process of controlling Aedes aegypti populations through the release of sterile male mosquitoes, assessing the responses to different inputs such as the total of sterile males released, the period for the release and locations for the intervention.
{"title":"Analysis and numerical approximation of a mathematical model for Aedes aegypti populations","authors":"Anderson L.A. de Araujo , Jose L. Boldrini , Bianca M.R. Calsavara , Maicon R. Correa","doi":"10.1016/j.camwa.2025.01.003","DOIUrl":"10.1016/j.camwa.2025.01.003","url":null,"abstract":"<div><div>We consider the rigorous analysis and the numerical approximation of a mathematical model for geographical spreading of <em>Aedes aegypti</em>. The complete model is composed of a system of parabolic partial differential equations coupled with one ordinary differential equation and has control terms related to the effects of insecticide application and sterile male release. The existence and uniqueness of solutions for the model are proven, and an efficient numerical methodology for approximating the unique solution of the mathematical model is proposed. The proposed numerical approach is based on a time-splitting scheme combined with locally conservative finite element methods. This combination of a well-posed mathematical model with a robust and efficient numerical formulation provides a suitable tool for the simulation of different scenarios of the spreading of <em>Aedes aegypti</em>. Numerical experiments, including a convergence study and a series of simulations that illustrate how the numerical model can be used in the decision-making process of controlling <em>Aedes aegypti</em> populations through the release of sterile male mosquitoes, assessing the responses to different inputs such as the total of sterile males released, the period for the release and locations for the intervention.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"180 ","pages":"Pages 214-241"},"PeriodicalIF":2.9,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986228","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 : 2025-01-10DOI: 10.1016/j.camwa.2025.01.007
Bo Gao, Ruoxia Yao, Yan Li
In recent years, physics-informed neural networks (PINNs) have garnered widespread attentions for the ability of solving nonlinear partial differential equations (PDE) using neural networks. The paper regards PINNs as multitask learning and proposes an adaptive loss weighting algorithm in physics-informed neural networks (APINNs). APINNs could balance the magnitudes of different loss functions during the training process to ensure a balanced contribution of parameters with different magnitudes to loss functions, thereby training solutions that satisfy initial boundary conditions and physical equations. Based on the original PINNs and APINNs, we respectively simulated the solitary wave solution of the Benjamin-Ono equation, the breather wave solution of the Sine-Gordon equation and the breather wave solution of the Mukherjee-Kundu equation. In the experiment of solving the solitary wave solution of the Benjamin-Ono equation, the minimum predict error of PINNs is about 60%, while the minimum predict error of APINNs is about 1%. As for solving breather wave solution, for the Sine-Gordon equation the minimum predict error of PINNs is around 10%, while the minimum predict error of APINNs is around 2%; and for the Mukherjee-Kundu equation, the minimum predict error of PINNs is about 30%, while the minimum predict error of APINNs is about 10%. The experimental results show that compared with PINNs, the predict solutions trained by APINNs have smaller errors.
{"title":"Physics-informed neural networks with adaptive loss weighting algorithm for solving partial differential equations","authors":"Bo Gao, Ruoxia Yao, Yan Li","doi":"10.1016/j.camwa.2025.01.007","DOIUrl":"10.1016/j.camwa.2025.01.007","url":null,"abstract":"<div><div>In recent years, physics-informed neural networks (PINNs) have garnered widespread attentions for the ability of solving nonlinear partial differential equations (PDE) using neural networks. The paper regards PINNs as multitask learning and proposes an adaptive loss weighting algorithm in physics-informed neural networks (APINNs). APINNs could balance the magnitudes of different loss functions during the training process to ensure a balanced contribution of parameters with different magnitudes to loss functions, thereby training solutions that satisfy initial boundary conditions and physical equations. Based on the original PINNs and APINNs, we respectively simulated the solitary wave solution of the Benjamin-Ono equation, the breather wave solution of the Sine-Gordon equation and the breather wave solution of the Mukherjee-Kundu equation. In the experiment of solving the solitary wave solution of the Benjamin-Ono equation, the minimum predict error of PINNs is about 60%, while the minimum predict error of APINNs is about 1%. As for solving breather wave solution, for the Sine-Gordon equation the minimum predict error of PINNs is around 10%, while the minimum predict error of APINNs is around 2%; and for the Mukherjee-Kundu equation, the minimum predict error of PINNs is about 30%, while the minimum predict error of APINNs is about 10%. The experimental results show that compared with PINNs, the predict solutions trained by APINNs have smaller errors.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 216-227"},"PeriodicalIF":2.9,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-10DOI: 10.1016/j.camwa.2024.12.020
Amy de Castro , Hyesuk Lee , Margaret M. Wiecek
This work presents a strongly coupled partitioned method for fluid-structure interaction (FSI) problems based on a monolithic formulation of the system which employs a Lagrange multiplier. We prove that both the semi-discrete and fully discrete formulations are well-posed. To derive a partitioned scheme, a Schur complement equation, which implicitly expresses the Lagrange multiplier and the fluid pressure in terms of the fluid velocity and structural displacement, is constructed based on the monolithic FSI system. Solving the Schur complement system at each time step allows for the decoupling of the fluid and structure subproblems, making the method non-iterative between subdomains. We investigate bounds for the condition number of the Schur complement matrix and present initial numerical results to demonstrate the performance of our approach, which attains the expected convergence rates.
{"title":"A Lagrange multiplier method for fluid-structure interaction: Well-posedness and domain decomposition","authors":"Amy de Castro , Hyesuk Lee , Margaret M. Wiecek","doi":"10.1016/j.camwa.2024.12.020","DOIUrl":"10.1016/j.camwa.2024.12.020","url":null,"abstract":"<div><div>This work presents a strongly coupled partitioned method for fluid-structure interaction (FSI) problems based on a monolithic formulation of the system which employs a Lagrange multiplier. We prove that both the semi-discrete and fully discrete formulations are well-posed. To derive a partitioned scheme, a Schur complement equation, which implicitly expresses the Lagrange multiplier and the fluid pressure in terms of the fluid velocity and structural displacement, is constructed based on the monolithic FSI system. Solving the Schur complement system at each time step allows for the decoupling of the fluid and structure subproblems, making the method non-iterative between subdomains. We investigate bounds for the condition number of the Schur complement matrix and present initial numerical results to demonstrate the performance of our approach, which attains the expected convergence rates.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 193-215"},"PeriodicalIF":2.9,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986226","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 : 2025-01-09DOI: 10.1016/j.camwa.2025.01.006
David Amilo , Khadijeh Sadri , Muhammad Farman , Evren Hincal , Kottakkaran Sooppy Nisar
In this paper, we developed a mathematical model for pancreatic cancer progression using a system of nonlinear partial differential equations (PDEs) with time delays, capturing disease dynamics in the human body. The model represents six key cell populations involved in pancreatic cancer: cancer cells (C), pancreatic stellate cells (P), stromal cells (S), extracellular matrix-degrading enzymes (E), tumor-associated macrophages (N), and immunosuppressive cells (I). For biological feasibility, we established model existence and uniqueness via the method of continuity and Banach's contraction principle, with global stability verified through the Lyapunov method. Sensitivity analysis identified critical factors such as cancer cell division, stromal cell activation, and immune cell infiltration, as targets for effective treatment. Optimal control and PID strategies demonstrated potential in limiting cancer proliferation and reprogramming the tumor microenvironment, while simulations highlighted the need for timely and sustained interventions. The results emphasize the importance of early surgery and immunomodulation in maximizing treatment efficacy, offering new insights into personalized and adaptive approaches to improve patient outcomes in pancreatic cancer treatment.
{"title":"New insights into disease dynamics and treatment interventions with PID controller-based therapeutic strategies for pancreatic cancer","authors":"David Amilo , Khadijeh Sadri , Muhammad Farman , Evren Hincal , Kottakkaran Sooppy Nisar","doi":"10.1016/j.camwa.2025.01.006","DOIUrl":"10.1016/j.camwa.2025.01.006","url":null,"abstract":"<div><div>In this paper, we developed a mathematical model for pancreatic cancer progression using a system of nonlinear partial differential equations (PDEs) with time delays, capturing disease dynamics in the human body. The model represents six key cell populations involved in pancreatic cancer: cancer cells (<em>C</em>), pancreatic stellate cells (<em>P</em>), stromal cells (<em>S</em>), extracellular matrix-degrading enzymes (<em>E</em>), tumor-associated macrophages (<em>N</em>), and immunosuppressive cells (<em>I</em>). For biological feasibility, we established model existence and uniqueness via the method of continuity and Banach's contraction principle, with global stability verified through the Lyapunov method. Sensitivity analysis identified critical factors such as cancer cell division, stromal cell activation, and immune cell infiltration, as targets for effective treatment. Optimal control and PID strategies demonstrated potential in limiting cancer proliferation and reprogramming the tumor microenvironment, while simulations highlighted the need for timely and sustained interventions. The results emphasize the importance of early surgery and immunomodulation in maximizing treatment efficacy, offering new insights into personalized and adaptive approaches to improve patient outcomes in pancreatic cancer treatment.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 133-162"},"PeriodicalIF":2.9,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986227","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 : 2025-01-09DOI: 10.1016/j.camwa.2025.01.005
Óscar López Pouso , Javier Segura
This paper is dedicated to the mathematical analysis of difference schemes for discretizing the angular diffusion operator present in the azimuth–independent Fokker–Planck equation. The study establishes sets of sufficient conditions to ensure that the schemes achieve convergence of order 2, and provides insights into the rationale behind certain widely recognized discrete ordinates methods. In the process, interesting properties regarding Gaussian nodes and weights, which until now have remained unnoticed by mathematicians, naturally emerge.
{"title":"Analysis of difference schemes for the Fokker–Planck angular diffusion operator","authors":"Óscar López Pouso , Javier Segura","doi":"10.1016/j.camwa.2025.01.005","DOIUrl":"10.1016/j.camwa.2025.01.005","url":null,"abstract":"<div><div>This paper is dedicated to the mathematical analysis of difference schemes for discretizing the angular diffusion operator present in the azimuth–independent Fokker–Planck equation. The study establishes sets of sufficient conditions to ensure that the schemes achieve convergence of order 2, and provides insights into the rationale behind certain widely recognized discrete ordinates methods. In the process, interesting properties regarding Gaussian nodes and weights, which until now have remained unnoticed by mathematicians, naturally emerge.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 84-110"},"PeriodicalIF":2.9,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986229","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 : 2025-01-09DOI: 10.1016/j.camwa.2024.12.022
Chunyu Yan
We introduce a novel numerical method via a class of radial basis function-produced finite difference solvers, applicable to both interpolation and partial differential equation (PDE) problems. The method leverages integrals of the Gaussian kernel, introducing new weights for problem-solving. Analytical solutions to approximate the derivatives of a function are derived and computed on a stencil with both non-uniform and uniform distances. Our observations indicate that the analytical weights exhibit greater stability compared to the numerical weights when addressing problems. In the final step, we use the derived formulations to solve a multi-dimensional option pricing problem in finance. The results demonstrate that our proposed numerical method outperforms in terms of numerical accuracy across grids of different sizes. Given the multi-dimensional nature of the dealing model, which involves handling a basket of assets, our approach becomes particularly relevant for assessing and managing financial risks.
{"title":"Exploring Gaussian radial basis function integrals for weight generation with application in financial option pricing","authors":"Chunyu Yan","doi":"10.1016/j.camwa.2024.12.022","DOIUrl":"10.1016/j.camwa.2024.12.022","url":null,"abstract":"<div><div>We introduce a novel numerical method via a class of radial basis function-produced finite difference solvers, applicable to both interpolation and partial differential equation (PDE) problems. The method leverages integrals of the Gaussian kernel, introducing new weights for problem-solving. Analytical solutions to approximate the derivatives of a function are derived and computed on a stencil with both non-uniform and uniform distances. Our observations indicate that the analytical weights exhibit greater stability compared to the numerical weights when addressing problems. In the final step, we use the derived formulations to solve a multi-dimensional option pricing problem in finance. The results demonstrate that our proposed numerical method outperforms in terms of numerical accuracy across grids of different sizes. Given the multi-dimensional nature of the dealing model, which involves handling a basket of assets, our approach becomes particularly relevant for assessing and managing financial risks.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"181 ","pages":"Pages 71-83"},"PeriodicalIF":2.9,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986232","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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