Pub Date : 2026-05-01Epub Date: 2026-02-14DOI: 10.1016/j.camwa.2026.02.005
Ruo-Xi Yu
This work develops an adaptive isogeometric analysis (IGA) framework based on truncated hierarchical B-spline (THB-spline) for pricing nonlinear multi-asset European options. The framework effectively handles both Black-Scholes and Heston stochastic volatility models by employing Newton linearization method for the nonlinear PDEs and the Crank-Nicolson scheme for temporal discretization. The discrete governing equation is derived via the Galerkin weighted residual method. A least-squares technique is utilized to accurately enforce initial and boundary conditions, while a smoothing-based error estimator drives the adaptive process. The precision and computational efficiency of the proposed framework are validated through comprehensive numerical analysis, establishing it as a robust tool for pricing multi-asset options with nonlinear features.
{"title":"Adaptive isogeometric analysis with truncated hierarchical B-splines for nonlinear option pricing problems","authors":"Ruo-Xi Yu","doi":"10.1016/j.camwa.2026.02.005","DOIUrl":"10.1016/j.camwa.2026.02.005","url":null,"abstract":"<div><div>This work develops an adaptive isogeometric analysis (IGA) framework based on truncated hierarchical B-spline (THB-spline) for pricing nonlinear multi-asset European options. The framework effectively handles both Black-Scholes and Heston stochastic volatility models by employing Newton linearization method for the nonlinear PDEs and the Crank-Nicolson scheme for temporal discretization. The discrete governing equation is derived via the Galerkin weighted residual method. A least-squares technique is utilized to accurately enforce initial and boundary conditions, while a smoothing-based error estimator drives the adaptive process. The precision and computational efficiency of the proposed framework are validated through comprehensive numerical analysis, establishing it as a robust tool for pricing multi-asset options with nonlinear features.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"209 ","pages":"Pages 44-56"},"PeriodicalIF":2.5,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146193031","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 : 2026-05-01Epub Date: 2026-02-12DOI: 10.1016/j.camwa.2026.01.042
Xiaozhuang Ma, Lizhen Chen
In this paper, we propose an efficient, fully discrete numerical scheme for the phase field crystal model, combining second-order accuracy in time with spectral accuracy in space. First, we employ a multi-step strategy for time discretization, obtaining a second-order semi-discrete Crank–Nicolson leap-frog scheme. We rigorously prove that this scheme satisfies total mass conservation, unconditional energy stability, and linear, unique solvability. A detailed error analysis confirms its second-order convergence in time. Next, we discretize the semi-discrete scheme in space using the Fourier pseudo-spectral method, ensuring that the fully discrete scheme retains mass conservation and energy dissipation. Convergence and error estimates are also rigorously derived. Numerical experiments demonstrate the scheme’s accuracy and efficiency, particularly in capturing effective energy decay during long-time coarsening dynamics.
{"title":"Stability and error estimate of the second-order Crank–Nicolson leap-frog scheme for the phase field crystal model","authors":"Xiaozhuang Ma, Lizhen Chen","doi":"10.1016/j.camwa.2026.01.042","DOIUrl":"10.1016/j.camwa.2026.01.042","url":null,"abstract":"<div><div>In this paper, we propose an efficient, fully discrete numerical scheme for the phase field crystal model, combining second-order accuracy in time with spectral accuracy in space. First, we employ a multi-step strategy for time discretization, obtaining a second-order semi-discrete Crank–Nicolson leap-frog scheme. We rigorously prove that this scheme satisfies total mass conservation, unconditional energy stability, and linear, unique solvability. A detailed error analysis confirms its second-order convergence in time. Next, we discretize the semi-discrete scheme in space using the Fourier pseudo-spectral method, ensuring that the fully discrete scheme retains mass conservation and energy dissipation. Convergence and error estimates are also rigorously derived. Numerical experiments demonstrate the scheme’s accuracy and efficiency, particularly in capturing effective energy decay during long-time coarsening dynamics.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"209 ","pages":"Pages 1-15"},"PeriodicalIF":2.5,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146162101","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 : 2026-05-01Epub Date: 2026-02-13DOI: 10.1016/j.camwa.2026.01.016
T. Chaumont-Frelet , D. Paredes , F. Valentin
The flux variable determines the approximation quality of hybridization-based numerical methods. This work proves that approximating flux variables in discontinuous polynomial spaces from the L2 orthogonal projection is super-convergent on meshes that are not necessarily aligned with jumping coefficient interfaces. The results assume only the local regularity of exact solutions in physical partitions. Based on the proposed flux approximation, we demonstrate that the mixed hybrid multiscale (MHM) finite element method is superconvergent on unfitted meshes, supporting the numerics presented in MHM seminal works.
{"title":"Flux approximation on unfitted meshes and application to multiscale hybrid-mixed methods","authors":"T. Chaumont-Frelet , D. Paredes , F. Valentin","doi":"10.1016/j.camwa.2026.01.016","DOIUrl":"10.1016/j.camwa.2026.01.016","url":null,"abstract":"<div><div>The flux variable determines the approximation quality of hybridization-based numerical methods. This work proves that approximating flux variables in discontinuous polynomial spaces from the <em>L</em><sup>2</sup> orthogonal projection is super-convergent on meshes that are not necessarily aligned with jumping coefficient interfaces. The results assume only the local regularity of exact solutions in physical partitions. Based on the proposed flux approximation, we demonstrate that the mixed hybrid multiscale (MHM) finite element method is superconvergent on unfitted meshes, supporting the numerics presented in MHM seminal works.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"209 ","pages":"Pages 16-27"},"PeriodicalIF":2.5,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146162102","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 : 2026-04-15Epub Date: 2026-02-07DOI: 10.1016/j.camwa.2026.01.038
Karel Van Bockstal , Khonatbek Khompysh
In this paper, we study the inverse problem for determining an unknown time-dependent source coefficient in a semilinear pseudo-parabolic equation with variable coefficients and Neumann boundary condition. This unknown source term is recovered from the integral measurement over the domain Ω. Based on Rothe’s method, the existence and uniqueness of a weak solution, under suitable assumptions on the data, is established. A numerical time-discrete scheme for the unique weak solution and the unknown source coefficient is designed, and the convergence of the approximations is proven. Numerical experiments are presented to support the theoretical results. Noisy data is handled through polynomial regularisation.
{"title":"A time-dependent inverse source problem for a semilinear pseudo-parabolic equation with Neumann boundary condition","authors":"Karel Van Bockstal , Khonatbek Khompysh","doi":"10.1016/j.camwa.2026.01.038","DOIUrl":"10.1016/j.camwa.2026.01.038","url":null,"abstract":"<div><div>In this paper, we study the inverse problem for determining an unknown time-dependent source coefficient in a semilinear pseudo-parabolic equation with variable coefficients and Neumann boundary condition. This unknown source term is recovered from the integral measurement over the domain Ω. Based on Rothe’s method, the existence and uniqueness of a weak solution, under suitable assumptions on the data, is established. A numerical time-discrete scheme for the unique weak solution and the unknown source coefficient is designed, and the convergence of the approximations is proven. Numerical experiments are presented to support the theoretical results. Noisy data is handled through polynomial regularisation.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"208 ","pages":"Pages 97-112"},"PeriodicalIF":2.5,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146134759","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 : 2026-04-15Epub Date: 2026-02-11DOI: 10.1016/j.camwa.2026.02.003
Yong Chen
This paper is devoted to the variable step-size implicit-explicit (IMEX) difference scheme for a partial differential equation (PDE) in two space dimensions with spatial delay term and mixed derivative term, which arises from the option pricing problem under hard-to-borrow stock model. First, a mesh-dependent Taylor expansion on nonuniform grids is proposed to approximate the spatial delay term. Second, the variable step-size IMEX scheme is constructed on nonuniform grids for both time and space. The consistency errors of the studied scheme are evaluated. Then, the theoretical results including unconditional stability and second-order convergence rates are established rigorously. Finally, some numerical examples support the theoretical analysis and show the efficacy of the proposed scheme.
{"title":"Variable step-size IMEX scheme for a partial differential equation with delays and mixed derivative from option pricing under hard-to-borrow model","authors":"Yong Chen","doi":"10.1016/j.camwa.2026.02.003","DOIUrl":"10.1016/j.camwa.2026.02.003","url":null,"abstract":"<div><div>This paper is devoted to the variable step-size implicit-explicit (IMEX) difference scheme for a partial differential equation (PDE) in two space dimensions with spatial delay term and mixed derivative term, which arises from the option pricing problem under hard-to-borrow stock model. First, a mesh-dependent Taylor expansion on nonuniform grids is proposed to approximate the spatial delay term. Second, the variable step-size IMEX scheme is constructed on nonuniform grids for both time and space. The consistency errors of the studied scheme are evaluated. Then, the theoretical results including unconditional stability and second-order convergence rates are established rigorously. Finally, some numerical examples support the theoretical analysis and show the efficacy of the proposed scheme.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"208 ","pages":"Pages 113-126"},"PeriodicalIF":2.5,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146152823","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 : 2026-04-15Epub Date: 2026-02-06DOI: 10.1016/j.camwa.2026.01.035
Ying Zhang , Chong-Jun Li
An approach for formulating polygonal and polyhedral elements is presented based on the scaled boundary coordinate system, which serves as a core component of the scaled boundary finite element method (SBFEM). Within the scaled boundary coordinate system, an arbitrary bivariate polynomials can be accurately represented as a linear combination of some power functions. Therefore, interpolation basis functions with high-order polynomial completeness over polygonal elements can be constructed using only boundary nodal displacements. To demonstrate the effectiveness of the proposed method, two polygonal elements with the second- and third-order completeness are presented as illustrative examples. Further, a faceted polyhedral element depending only on boundary nodal displacements is also constructed, exhibiting second-order completeness. Different from the classic SBFEM, the shape functions of the 2D and 3D elements are explicitly expressed, and the completeness are independent of the body forces (or without additional bubble functions). Numerical results demonstrate the proposed polygonal elements and faceted polyhedral element have corresponding completeness and convergence.
{"title":"Construction of the polygonal and faceted polyhedral elements with high order completeness based on the scaled boundary coordinates","authors":"Ying Zhang , Chong-Jun Li","doi":"10.1016/j.camwa.2026.01.035","DOIUrl":"10.1016/j.camwa.2026.01.035","url":null,"abstract":"<div><div>An approach for formulating polygonal and polyhedral elements is presented based on the scaled boundary coordinate system, which serves as a core component of the scaled boundary finite element method (SBFEM). Within the scaled boundary coordinate system, an arbitrary bivariate polynomials can be accurately represented as a linear combination of some power functions. Therefore, interpolation basis functions with high-order polynomial completeness over polygonal elements can be constructed using only boundary nodal displacements. To demonstrate the effectiveness of the proposed method, two polygonal elements with the second- and third-order completeness are presented as illustrative examples. Further, a faceted polyhedral element depending only on boundary nodal displacements is also constructed, exhibiting second-order completeness. Different from the classic SBFEM, the shape functions of the 2D and 3D elements are explicitly expressed, and the completeness are independent of the body forces (or without additional bubble functions). Numerical results demonstrate the proposed polygonal elements and faceted polyhedral element have corresponding completeness and convergence.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"208 ","pages":"Pages 80-96"},"PeriodicalIF":2.5,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146134767","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 : 2026-04-15Epub Date: 2026-02-04DOI: 10.1016/j.camwa.2026.01.009
Ming Huang , Si Qi Zhang , Xiao Dan Chao , Hong Kai Song , Jin Long Yuan , Suo Suo Yang
In this research, we introduce an innovative composite proximal bundle method aimed at resolving generalized variational inequalities with the integration of inexact oracles. The essence of the problem is distilled into the pursuit of the null space of the superposition of two multivalued operators, all within the realm of a real Hilbert space and underpinned by an optimality criterion. We define the non-differentiable composite convex function and the monotone operator as f and Γ, respectively, both characterized by their roles as subdifferentials of functions that are lower semicontinuous. Central to our methodology is the proximal point approach, which entails the iterative refinement of subproblems through the lens of piecewise linear convex functions, augmented by the incorporation of inexact data. To enhance the computational efficiency and precision, we introduce a novel stopping criterion, designed to evaluate the precision of the current approximation. This innovation is pivotal in streamlining the management of subproblems. Moreover, to safeguard the convergence properties of our algorithm against the potential perturbations induced by inexact data, we have integrated a denoising technique. Subsequent to these enhancements, we demonstrate the convergence of our algorithm under specific operator properties, thereby providing a robust theoretical underpinning for its application. To verify the practical efficacy of our approach, we present the outcomes of our numerical experiments, which affirm the effectiveness of our method in the context of non-smooth optimization.
{"title":"The new inexact bundle-type technique to solve variational inequalities of composite structures","authors":"Ming Huang , Si Qi Zhang , Xiao Dan Chao , Hong Kai Song , Jin Long Yuan , Suo Suo Yang","doi":"10.1016/j.camwa.2026.01.009","DOIUrl":"10.1016/j.camwa.2026.01.009","url":null,"abstract":"<div><div>In this research, we introduce an innovative composite proximal bundle method aimed at resolving generalized variational inequalities with the integration of inexact oracles. The essence of the problem is distilled into the pursuit of the null space of the superposition of two multivalued operators, all within the realm of a real Hilbert space and underpinned by an optimality criterion. We define the non-differentiable composite convex function and the monotone operator as <em>f</em> and Γ, respectively, both characterized by their roles as subdifferentials of functions that are lower semicontinuous. Central to our methodology is the proximal point approach, which entails the iterative refinement of subproblems through the lens of piecewise linear convex functions, augmented by the incorporation of inexact data. To enhance the computational efficiency and precision, we introduce a novel stopping criterion, designed to evaluate the precision of the current approximation. This innovation is pivotal in streamlining the management of subproblems. Moreover, to safeguard the convergence properties of our algorithm against the potential perturbations induced by inexact data, we have integrated a denoising technique. Subsequent to these enhancements, we demonstrate the convergence of our algorithm under specific operator properties, thereby providing a robust theoretical underpinning for its application. To verify the practical efficacy of our approach, we present the outcomes of our numerical experiments, which affirm the effectiveness of our method in the context of non-smooth optimization.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"208 ","pages":"Pages 55-79"},"PeriodicalIF":2.5,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146116550","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 : 2026-04-15Epub Date: 2026-02-04DOI: 10.1016/j.camwa.2026.01.032
R. Shivananda Rao, M. Ramakrishna
In this paper, we adapt a troubled-cell indicator proposed by Fu and Shu [1] for discontinuous Galerkin (DG) methods to finite volume methods (FVM) employing third-order MUSCL reconstruction. We show that limiting only in troubled-cells is advantageous in terms of convergence but at the expense of the overall solution quality. We investigate the optimal number of troubled-cells required in the vicinity of an oblique shock to obtain a solution with minimal oscillations and enhanced convergence using a novel monotonicity parameter. An oblique shock is characterized primarily by the upstream Mach number, the shock angle β, and the deflection angle θ. We study these factors and their combinations by employing two dimensional compressible Euler equations and find that the degree of the shock misalignment with the grid determines the optimal number of troubled-cells. We show that, on each side of the shock, the optimal set consists of three troubled-cells for aligned shocks, and the troubled-cells identified by tracing the shock and four lines parallel to it, separated by the grid spacing, for nonaligned shocks. We show that, for a threshold constant , the adapted troubled-cell indicator identifies a set of cells that is close to and contains the optimal set of cells, and consequently, produces a solution close to that obtained by limiting everywhere, but with improved convergence. We demonstrate the effectiveness of the adapted troubled-cell indicator for unsteady problems using the double Mach reflection test case.
{"title":"A comprehensive numerical investigation of the application of troubled-cells to finite volume methods using a novel monotonicity parameter","authors":"R. Shivananda Rao, M. Ramakrishna","doi":"10.1016/j.camwa.2026.01.032","DOIUrl":"10.1016/j.camwa.2026.01.032","url":null,"abstract":"<div><div>In this paper, we adapt a troubled-cell indicator proposed by Fu and Shu [1] for discontinuous Galerkin (DG) methods to finite volume methods (FVM) employing third-order MUSCL reconstruction. We show that limiting only in troubled-cells is advantageous in terms of convergence but at the expense of the overall solution quality. We investigate the optimal number of troubled-cells required in the vicinity of an oblique shock to obtain a solution with minimal oscillations and enhanced convergence using a novel monotonicity parameter. An oblique shock is characterized primarily by the upstream Mach number, the shock angle <em>β</em>, and the deflection angle <em>θ</em>. We study these factors and their combinations by employing two dimensional compressible Euler equations and find that the degree of the shock misalignment with the grid determines the optimal number of troubled-cells. We show that, on each side of the shock, the optimal set consists of three troubled-cells for aligned shocks, and the troubled-cells identified by tracing the shock and four lines parallel to it, separated by the grid spacing, for nonaligned shocks. We show that, for a threshold constant <span><math><mrow><mi>K</mi><mo>=</mo><mn>0.05</mn></mrow></math></span>, the adapted troubled-cell indicator identifies a set of cells that is close to and contains the optimal set of cells, and consequently, produces a solution close to that obtained by limiting everywhere, but with improved convergence. We demonstrate the effectiveness of the adapted troubled-cell indicator for unsteady problems using the double Mach reflection test case.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"208 ","pages":"Pages 1-32"},"PeriodicalIF":2.5,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146116551","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 : 2026-04-15Epub Date: 2026-02-04DOI: 10.1016/j.camwa.2026.01.036
Sooncheol Hwang , Patrick J. Lynett , Sangyoung Son
In this paper, we develop a two-dimensional, second-order central-upwind scheme based on the finite volume method for the shallow water equations in the presence of wet/dry fronts. The proposed scheme is positivity-preserving and well-balanced, and remains robust for shallow water flows over complex bathymetry and wet-dry interfaces. We extend existing one-dimensional positivity-preserving and well-balanced schemes to two-dimensional spaces, addressing the challenges posed by partially flooded cells with varying bottom gradients in each direction. A novel draining time step for two-dimensional spaces is introduced to ensure the non-negativity of computed water depths across the entire computational domain. The performance of the proposed scheme is validated through several numerical experiments using both analytical solutions and experimental data, demonstrating its accuracy in capturing wet/dry fronts. The results for the lake-at-rest steady-state case confirm that numerical oscillations near the wet/dry fronts are successfully minimized, maintaining the initial state. Moreover, other examples show reduced numerical oscillations during wave run-up and rundown processes, further confirming the performance of the proposed scheme.
{"title":"A second-order well-balanced reconstruction for the shallow flows with wet/dry fronts","authors":"Sooncheol Hwang , Patrick J. Lynett , Sangyoung Son","doi":"10.1016/j.camwa.2026.01.036","DOIUrl":"10.1016/j.camwa.2026.01.036","url":null,"abstract":"<div><div>In this paper, we develop a two-dimensional, second-order central-upwind scheme based on the finite volume method for the shallow water equations in the presence of wet/dry fronts. The proposed scheme is positivity-preserving and well-balanced, and remains robust for shallow water flows over complex bathymetry and wet-dry interfaces. We extend existing one-dimensional positivity-preserving and well-balanced schemes to two-dimensional spaces, addressing the challenges posed by partially flooded cells with varying bottom gradients in each direction. A novel draining time step for two-dimensional spaces is introduced to ensure the non-negativity of computed water depths across the entire computational domain. The performance of the proposed scheme is validated through several numerical experiments using both analytical solutions and experimental data, demonstrating its accuracy in capturing wet/dry fronts. The results for the lake-at-rest steady-state case confirm that numerical oscillations near the wet/dry fronts are successfully minimized, maintaining the initial state. Moreover, other examples show reduced numerical oscillations during wave run-up and rundown processes, further confirming the performance of the proposed scheme.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"208 ","pages":"Pages 33-54"},"PeriodicalIF":2.5,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146116549","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 presents an innovative spatio-temporal parabolic framework based on fractional calculus, employing both Caputo and Riemann–Liouville derivatives. The primary objective is to enhance traditional super-resolution methods, with a specific focus on multi-frame image reconstruction. The proposed model incorporates a spatially regularized fractional tensor diffusion mechanism that modulates both the magnitude and orientation of diffusion locally across the image domain. Theoretical analysis begins by addressing the model’s well-posedness. Using the Faedo-Galerkin scheme, we first establish the uniqueness and existence of weak solution to an auxiliary problem involving a time-fractional Caputo derivative. Then, leveraging Schauder fixed point theorem, we show the existence of a unique weak solution for our full model. Numerical experiments illustrate the practical capabilities of the approach, showcasing the advantages of fractional order methods in the context of image denoising and super-resolution. Furthermore, tests on real video sequences confirm the model’s robustness and performance in blind reconstruction scenarios. Comparative evaluations with state-of-the-art techniques underline the efficiency of our fractional model in terms of visual quality and detail preservation.
{"title":"Fractional spatio-temporal modeling for enhanced MRI super-resolution from multi-frame data","authors":"Anouar Ben-Loghfyry , Abderrahim Charkaoui , Shengda Zeng","doi":"10.1016/j.camwa.2026.01.037","DOIUrl":"10.1016/j.camwa.2026.01.037","url":null,"abstract":"<div><div>This work presents an innovative spatio-temporal parabolic framework based on fractional calculus, employing both Caputo and Riemann–Liouville derivatives. The primary objective is to enhance traditional super-resolution methods, with a specific focus on multi-frame image reconstruction. The proposed model incorporates a spatially regularized fractional tensor diffusion mechanism that modulates both the magnitude and orientation of diffusion locally across the image domain. Theoretical analysis begins by addressing the model’s well-posedness. Using the Faedo-Galerkin scheme, we first establish the uniqueness and existence of weak solution to an auxiliary problem involving a time-fractional Caputo derivative. Then, leveraging Schauder fixed point theorem, we show the existence of a unique weak solution for our full model. Numerical experiments illustrate the practical capabilities of the approach, showcasing the advantages of fractional order methods in the context of image denoising and super-resolution. Furthermore, tests on real video sequences confirm the model’s robustness and performance in blind reconstruction scenarios. Comparative evaluations with state-of-the-art techniques underline the efficiency of our fractional model in terms of visual quality and detail preservation.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"208 ","pages":"Pages 147-185"},"PeriodicalIF":2.5,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146160706","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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