Pub Date : 2026-02-01Epub Date: 2026-01-28DOI: 10.1016/j.rinam.2026.100683
Anna Piterskaya, Mikael Mortensen
The research adopts the Galerkin and Petrov–Galerkin spectral methods to analyze the effects of implicit–explicit Runge–Kutta (IMEX RK) time schemes on the stability, accuracy, precision, and efficiency of computations when used in numerical simulations of the diffusion, Burgers’, and magneto-hydrodynamic (MHD) equations. The maximum time steps suitable for the Galerkin and Petrov–Galerkin spectral methods are determined based on an analysis of the diffusion equation. It is shown that for the Petrov–Galerkin spectral method it is possible to use a time step size that is nearly three times as large as that of the Galerkin spectral method. The results of the relative error analysis of Burgers’ equation show that the IMEX RK schemes become more stable and accurate with an increase in the number of basis functions and smaller time steps, while the effect of higher viscosity leads to a decrease in relative errors. It is found that when the number of basis functions is large, the higher-order IMEX RK schemes in conjunction with the Petrov–Galerkin spectral method efficiently discretize time and space and achieve high accuracy at various viscosities due to their well-conditioned band matrices. In addition, the problem of the influence of magnetic fields on the efficiency of different IMEX RK schemes is considered within the framework of a two-dimensional MHD system by studying the evolution of small perturbations in a conducting fluid flow. The analyses show that the higher-order IMEX RK schemes provide enhanced stability and accuracy, while the lower-order schemes offer computational efficiency at the expense of some accuracy.
{"title":"A comparison of implicit–explicit Runge–Kutta time integration schemes in numerical solvers based on the Galerkin and Petrov–Galerkin spectral methods for two-dimensional magneto-hydrodynamic problems","authors":"Anna Piterskaya, Mikael Mortensen","doi":"10.1016/j.rinam.2026.100683","DOIUrl":"10.1016/j.rinam.2026.100683","url":null,"abstract":"<div><div>The research adopts the Galerkin and Petrov–Galerkin spectral methods to analyze the effects of implicit–explicit Runge–Kutta (IMEX RK) time schemes on the stability, accuracy, precision, and efficiency of computations when used in numerical simulations of the diffusion, Burgers’, and magneto-hydrodynamic (MHD) equations. The maximum time steps suitable for the Galerkin and Petrov–Galerkin spectral methods are determined based on an analysis of the diffusion equation. It is shown that for the Petrov–Galerkin spectral method it is possible to use a time step size that is nearly three times as large as that of the Galerkin spectral method. The results of the relative error analysis of Burgers’ equation show that the IMEX RK schemes become more stable and accurate with an increase in the number of basis functions and smaller time steps, while the effect of higher viscosity leads to a decrease in relative errors. It is found that when the number of basis functions is large, the higher-order IMEX RK schemes in conjunction with the Petrov–Galerkin spectral method efficiently discretize time and space and achieve high accuracy at various viscosities due to their well-conditioned band matrices. In addition, the problem of the influence of magnetic fields on the efficiency of different IMEX RK schemes is considered within the framework of a two-dimensional MHD system by studying the evolution of small perturbations in a conducting fluid flow. The analyses show that the higher-order IMEX RK schemes provide enhanced stability and accuracy, while the lower-order schemes offer computational efficiency at the expense of some accuracy.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100683"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2026-02-07DOI: 10.1016/j.rinam.2026.100688
David Wei Ge
Analytical solutions to Maxwell’s equations are essential for understanding the causal and instantaneous behavior of electromagnetic fields, yet they are challenging to obtain in open-space settings with general initial values and source terms. This work uses an infinite-order estimation scheme that yields closed-form analytical solutions to Maxwell’s equations. The scheme is expressed as general function-to-function transformations that directly map initial values and source terms to the fields. Several case studies demonstrate the method, producing exact solutions for different initial and source configurations. The results provide both theoretical insight and practical benchmarks for computational electromagnetics.
{"title":"Analytical solutions of 1D Maxwell’s equations via infinite-order expansions","authors":"David Wei Ge","doi":"10.1016/j.rinam.2026.100688","DOIUrl":"10.1016/j.rinam.2026.100688","url":null,"abstract":"<div><div>Analytical solutions to Maxwell’s equations are essential for understanding the causal and instantaneous behavior of electromagnetic fields, yet they are challenging to obtain in open-space settings with general initial values and source terms. This work uses an infinite-order estimation scheme that yields closed-form analytical solutions to Maxwell’s equations. The scheme is expressed as general function-to-function transformations that directly map initial values and source terms to the fields. Several case studies demonstrate the method, producing exact solutions for different initial and source configurations. The results provide both theoretical insight and practical benchmarks for computational electromagnetics.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100688"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2026-02-24DOI: 10.1016/j.rinam.2026.100691
Kankolongo Kadilu Patient , Mpanda Mukendi Marc , Kumwimba Seya Didier , Dorsaf Cherif , Panga Lutanda Grégoire
<div><div>We propose a deterministic and conservative method for pricing European options under parameter ambiguity in the Heston model. The primary inputs <span><math><mrow><mo>(</mo><mi>κ</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>σ</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span> are represented as trapezoidal fuzzy numbers, and uncertainty is propagated via <span><math><mi>α</mi></math></span>-cuts through the characteristic-function representation. Our main contribution is a backward-recursive enclosure pipeline: we first construct conservative <span><math><mi>α</mi></math></span>-cut bands for the volatility and stock-price processes at a prescribed normal-quantile level <span><math><mi>p</mi></math></span>, then derive principal-branch complex-interval (rectangle) enclosures for the Heston discriminant and auxiliary terms <span><math><mrow><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></math></span> and the affine coefficients <span><math><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></math></span>. These bounds yield conservative enclosures for the risk-neutral probabilities <span><math><mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></math></span> through Fourier quadrature with interval-valued integrands, and finally monotone <span><math><mi>α</mi></math></span>-cut price intervals for call and put options. We further report expected intervals and expected values of the resulting fuzzy option prices by integrating <span><math><mi>α</mi></math></span>-cut endpoints, providing concise summaries alongside the full price bands. Numerical experiments show that the exact (generally nonlinear) <span><math><mi>α</mi></math></span>-cut bounds vary smoothly across <span><math><mi>α</mi></math></span>, and that a simple trapezoidal surrogate obtained by linear interpolation between <span><math><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></math></span> approximates the exact bounds with very small relative errors under moderate uncertainty. A core/support sensitivity study highlights that the support width dominates the effect on <span><math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mover><mrow><mi>P</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, consistent with the support governing extrem
{"title":"Heston option pricing with trapezoidal fuzzy parameters","authors":"Kankolongo Kadilu Patient , Mpanda Mukendi Marc , Kumwimba Seya Didier , Dorsaf Cherif , Panga Lutanda Grégoire","doi":"10.1016/j.rinam.2026.100691","DOIUrl":"10.1016/j.rinam.2026.100691","url":null,"abstract":"<div><div>We propose a deterministic and conservative method for pricing European options under parameter ambiguity in the Heston model. The primary inputs <span><math><mrow><mo>(</mo><mi>κ</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>σ</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span> are represented as trapezoidal fuzzy numbers, and uncertainty is propagated via <span><math><mi>α</mi></math></span>-cuts through the characteristic-function representation. Our main contribution is a backward-recursive enclosure pipeline: we first construct conservative <span><math><mi>α</mi></math></span>-cut bands for the volatility and stock-price processes at a prescribed normal-quantile level <span><math><mi>p</mi></math></span>, then derive principal-branch complex-interval (rectangle) enclosures for the Heston discriminant and auxiliary terms <span><math><mrow><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></math></span> and the affine coefficients <span><math><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></math></span>. These bounds yield conservative enclosures for the risk-neutral probabilities <span><math><mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></math></span> through Fourier quadrature with interval-valued integrands, and finally monotone <span><math><mi>α</mi></math></span>-cut price intervals for call and put options. We further report expected intervals and expected values of the resulting fuzzy option prices by integrating <span><math><mi>α</mi></math></span>-cut endpoints, providing concise summaries alongside the full price bands. Numerical experiments show that the exact (generally nonlinear) <span><math><mi>α</mi></math></span>-cut bounds vary smoothly across <span><math><mi>α</mi></math></span>, and that a simple trapezoidal surrogate obtained by linear interpolation between <span><math><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></math></span> approximates the exact bounds with very small relative errors under moderate uncertainty. A core/support sensitivity study highlights that the support width dominates the effect on <span><math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mover><mrow><mi>P</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, consistent with the support governing extrem","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100691"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2026-02-12DOI: 10.1016/j.rinam.2026.100687
Akram Karimi, Mostafa Abbaszadeh
This paper introduces an Adaptive Fidelity Gradient-Enhanced Physics-Informed Neural Network (AF-GPINN) for pricing options under regime-switching models. These models use multiple discrete market states (“regimes”), leading to coupled PDE systems that are computationally challenging. We build upon Physics-Informed Neural Networks (PINNs), which embed PDEs into the loss function. Our method enhances this by incorporating PDE residual gradients (GPINN) for improved accuracy with financial derivatives. The key innovation is a novel adaptive strategy for the total loss function. Training for coupled systems involves balancing many competing terms: PDE residuals and gradients per regime, boundary/initial conditions, and market data. Fixed weights often cause poor performance. Our AF-GPINN framework dynamically adapts these weights during training. Initially, it prioritizes satisfying the physics (PDEs and their gradients) to establish a structurally sound solution. The importance of observational data is then gradually increased. This phased approach refines the solution to match real data without violating physical consistency, generalizing standard PINN/GPINN methods. Numerical experiments on two- and three-regime models validate the framework, demonstrating accurate, efficient option pricing and its potential as a robust tool in financial engineering.
{"title":"An adaptive-fidelity gradient-enhanced PINN framework for option pricing under regime-switching models","authors":"Akram Karimi, Mostafa Abbaszadeh","doi":"10.1016/j.rinam.2026.100687","DOIUrl":"10.1016/j.rinam.2026.100687","url":null,"abstract":"<div><div>This paper introduces an Adaptive Fidelity Gradient-Enhanced Physics-Informed Neural Network (AF-GPINN) for pricing options under regime-switching models. These models use multiple discrete market states (“regimes”), leading to coupled PDE systems that are computationally challenging. We build upon Physics-Informed Neural Networks (PINNs), which embed PDEs into the loss function. Our method enhances this by incorporating PDE residual gradients (GPINN) for improved accuracy with financial derivatives. The key innovation is a novel adaptive strategy for the total loss function. Training for coupled systems involves balancing many competing terms: PDE residuals and gradients per regime, boundary/initial conditions, and market data. Fixed weights often cause poor performance. Our AF-GPINN framework dynamically adapts these weights during training. Initially, it prioritizes satisfying the physics (PDEs and their gradients) to establish a structurally sound solution. The importance of observational data is then gradually increased. This phased approach refines the solution to match real data without violating physical consistency, generalizing standard PINN/GPINN methods. Numerical experiments on two- and three-regime models validate the framework, demonstrating accurate, efficient option pricing and its potential as a robust tool in financial engineering.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100687"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2026-01-06DOI: 10.1016/j.rinam.2025.100679
Zixuan Shen , Deyi Ma
In this paper, we establish a Liouville-type theorem for smooth solutions of the stationary Navier–Stokes equations under a growth condition on the Lebesgue norms. Based on this condition, we prove a lemma analogous to the Poincaré-type inequality in the curl sense, which serves as a key tool in proving our main result.
{"title":"A Liouville-type theorem for 3D stationary Navier–Stokes equations","authors":"Zixuan Shen , Deyi Ma","doi":"10.1016/j.rinam.2025.100679","DOIUrl":"10.1016/j.rinam.2025.100679","url":null,"abstract":"<div><div>In this paper, we establish a Liouville-type theorem for smooth solutions of the stationary Navier–Stokes equations under a growth condition on the Lebesgue norms. Based on this condition, we prove a lemma analogous to the Poincaré-type inequality in the curl sense, which serves as a key tool in proving our main result.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100679"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2026-01-06DOI: 10.1016/j.rinam.2025.100681
Mohammad Meysami , Ali Lotfi
The screening effect is a central idea in spatial prediction: once nearby observations are used, distant ones add little. While Stein’s classical results explain this effect under strong spectral conditions, many models used in practice fall outside those assumptions. In this paper, we extend Stein’s theory to a broader class of Gaussian random fields. We replace strict regular variation by more flexible O-regular variation, resolve the critical borderline case where the spectral exponent equals the dimension, and derive explicit convergence rates for prediction error. Our analysis also shows that screening is robust to mild nonstationarity and anisotropy, and it applies to irregular sampling designs such as Delone sets. To illustrate these results, we provide examples with Matérn, generalized Cauchy, and anisotropic models, along with numerical experiments confirming the theory. These findings clarify when local kriging can safely replace global prediction, and they provide a solid foundation for scalable methods such as covariance tapering and Vecchia approximations.
{"title":"Screening effects in Gaussian random fields under generalized spectral conditions","authors":"Mohammad Meysami , Ali Lotfi","doi":"10.1016/j.rinam.2025.100681","DOIUrl":"10.1016/j.rinam.2025.100681","url":null,"abstract":"<div><div>The screening effect is a central idea in spatial prediction: once nearby observations are used, distant ones add little. While Stein’s classical results explain this effect under strong spectral conditions, many models used in practice fall outside those assumptions. In this paper, we extend Stein’s theory to a broader class of Gaussian random fields. We replace strict regular variation by more flexible O-regular variation, resolve the critical borderline case where the spectral exponent equals the dimension, and derive explicit convergence rates for prediction error. Our analysis also shows that screening is robust to mild nonstationarity and anisotropy, and it applies to irregular sampling designs such as Delone sets. To illustrate these results, we provide examples with Matérn, generalized Cauchy, and anisotropic models, along with numerical experiments confirming the theory. These findings clarify when local kriging can safely replace global prediction, and they provide a solid foundation for scalable methods such as covariance tapering and Vecchia approximations.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100681"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2026-01-06DOI: 10.1016/j.rinam.2025.100682
S. Mansoori Aref , M.H. Heydari , M. Bayram
In this study, we develop an operational matrix technique to address a set of fractional nonlinear integro-differential equations with the Caputo–Hadamard derivative. We utilize a family of the piecewise Chebyshev cardinal functions as basis functions in this regard. Some formulas are introduced for calculating the classical and Hadamard fractional integrals of these functions. In the established strategy, the fractional expression is approximated utilizing these piecewise functions. By employing the aforementioned operational matrices and leveraging the cardinal property of the basis functions, we solve an algebraic system to obtain the solution. Convergence is both analytically demonstrated and confirmed through numerical experiments. Additionally, we compare the results obtained using this method with those derived from the hat functions method.
{"title":"A numerical framework based on piecewise Chebyshev cardinal functions for fractional integro-differential equations","authors":"S. Mansoori Aref , M.H. Heydari , M. Bayram","doi":"10.1016/j.rinam.2025.100682","DOIUrl":"10.1016/j.rinam.2025.100682","url":null,"abstract":"<div><div>In this study, we develop an operational matrix technique to address a set of fractional nonlinear integro-differential equations with the Caputo–Hadamard derivative. We utilize a family of the piecewise Chebyshev cardinal functions as basis functions in this regard. Some formulas are introduced for calculating the classical and Hadamard fractional integrals of these functions. In the established strategy, the fractional expression is approximated utilizing these piecewise functions. By employing the aforementioned operational matrices and leveraging the cardinal property of the basis functions, we solve an algebraic system to obtain the solution. Convergence is both analytically demonstrated and confirmed through numerical experiments. Additionally, we compare the results obtained using this method with those derived from the hat functions method.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100682"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2026-01-28DOI: 10.1016/j.rinam.2026.100686
S. Asghar , Q. Peng , F.J. Vermolen , C. Vuik
The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previously known expressions of the inverse discretized Laplacian in one spatial dimension (Vermolen et al., 2022). The formalism is further extended to obtain closed form expressions for time-dependent problems.
矩阵多项式的有效反演是计算数学中的一个重大挑战。设计了一个求多维拉普拉斯矩阵的多项式逆的程序。该方法基于特征向量和特征值展开。该方法与先前已知的一维空间离散拉普拉斯逆表达式一致（Vermolen et al., 2022）。进一步推广了该形式，得到了时变问题的封闭形式表达式。
{"title":"On the inversion of polynomials of discrete Laplace matrices","authors":"S. Asghar , Q. Peng , F.J. Vermolen , C. Vuik","doi":"10.1016/j.rinam.2026.100686","DOIUrl":"10.1016/j.rinam.2026.100686","url":null,"abstract":"<div><div>The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previously known expressions of the inverse discretized Laplacian in one spatial dimension (Vermolen et al., 2022). The formalism is further extended to obtain closed form expressions for time-dependent problems.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100686"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2026-02-27DOI: 10.1016/j.rinam.2026.100692
M. Soluki , S.H. Rasouli , G.A. Afrouzi
This paper is concerned with the existence and multiplicity of solutions to the following Schrödinger–Kirchhoff–Poisson system where and and is a bounded smooth domain of . Under certain assumptions of nonnegative density charge and superlinear term , we establish the existence of infinitely many nontrivial solutions by means of the symmetric mountain pass theorem.
{"title":"On the existence and multiplicity of solutions to a Schrödinger–Kirchhoff–Poisson system with superlinear terms","authors":"M. Soluki , S.H. Rasouli , G.A. Afrouzi","doi":"10.1016/j.rinam.2026.100692","DOIUrl":"10.1016/j.rinam.2026.100692","url":null,"abstract":"<div><div>This paper is concerned with the existence and multiplicity of solutions to the following Schrödinger–Kirchhoff–Poisson system <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><mo>−</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mo>|</mo><mo>∇</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>K</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>ϕ</mi><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>ϕ</mi><mo>=</mo><mi>K</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mi>ϕ</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mi>∂</mi><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mrow><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span><math><mi>Ω</mi></math></span> is a bounded smooth domain of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Under certain assumptions of nonnegative density charge <span><math><mrow><mi>K</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> and superlinear term <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span>, we establish the existence of infinitely many nontrivial solutions by means of the symmetric mountain pass theorem.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100692"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2026-01-27DOI: 10.1016/j.rinam.2026.100684
Vitalii Stelmashchuk
A functionally graded one-dimensional piezoelectric rod excited by a moving heat source is considered. Its dynamic multifield response under this type of thermal load is investigated in the context of coupled Lord-Shulman thermopiezoelectricity theory.
In the literature, the transient response of piezoelectric materials to a moving heat source is extensively studied, but in most cases the solution is obtained using Laplace transform. In this paper we propose a completely numerical approach to solve the problem.
The original initial boundary value problem is transformed into a variational one. The variational problem is then rewritten in a non-dimensional form using appropriate variable substitutions. For space discretization the finite element method is used and for time discretization we utilize a hybrid time integration scheme based on Newmark scheme for hyperbolic equations and generalized trapezoidal rule for parabolic equations. The approximation of Dirac delta function by means of Gaussian probability density function is used to simulate the moving heat source.
The developed numerical scheme is validated against the benchmark solutions available in the literature and our results show great agreement with them. Besides, we empirically analyze the energy conservation properties of proposed time integration scheme and verify spatial and temporal convergence of the obtained solutions. Finally, we study the influence of non-homogeneity index, thermal relaxation time and moving heat source velocity on the solutions of the problem.
{"title":"Non-dimensional FEM analysis of a functionally graded thermopiezoelectric rod subjected to a moving heat source","authors":"Vitalii Stelmashchuk","doi":"10.1016/j.rinam.2026.100684","DOIUrl":"10.1016/j.rinam.2026.100684","url":null,"abstract":"<div><div>A functionally graded one-dimensional piezoelectric rod excited by a moving heat source is considered. Its dynamic multifield response under this type of thermal load is investigated in the context of coupled Lord-Shulman thermopiezoelectricity theory.</div><div>In the literature, the transient response of piezoelectric materials to a moving heat source is extensively studied, but in most cases the solution is obtained using Laplace transform. In this paper we propose a completely numerical approach to solve the problem.</div><div>The original initial boundary value problem is transformed into a variational one. The variational problem is then rewritten in a non-dimensional form using appropriate variable substitutions. For space discretization the finite element method is used and for time discretization we utilize a hybrid time integration scheme based on Newmark scheme for hyperbolic equations and generalized trapezoidal rule for parabolic equations. The approximation of Dirac delta function by means of Gaussian probability density function is used to simulate the moving heat source.</div><div>The developed numerical scheme is validated against the benchmark solutions available in the literature and our results show great agreement with them. Besides, we empirically analyze the energy conservation properties of proposed time integration scheme and verify spatial and temporal convergence of the obtained solutions. Finally, we study the influence of non-homogeneity index, thermal relaxation time and moving heat source velocity on the solutions of the problem.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"29 ","pages":"Article 100684"},"PeriodicalIF":1.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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