Pub Date : 2026-01-17DOI: 10.1016/j.camwa.2026.01.011
Xing Liu , Yumeng Yang
This paper constructs a discretization for the stochastic nonlinear diffusion-wave equation involving the Caputo fractional derivative of order α ∈ (1, 2), driven by fractional Brownian motion with Hurst index 0 < H < 1. For time discretization, we propose a quadrature involving Mittag-Leffler functions . The discretization method combines the integral representation of the solution, the approximation of Mittag-Leffler functions and numerical integration techniques. Two approximation methods for the Mittag-Leffler functions are developed to enhance computational efficiency. The mean square strong convergence order is established by utilizing the confirmed solution regularity. Numerical examples are presented to validate the theoretical results.
本文构造了包含阶α ∈ （1,2）的Caputo分数阶导数的随机非线性扩散波方程的离散化，该方程由Hurst指数为0 <； H <； 1的分数阶布朗运动驱动。对于时间离散，我们提出了一个涉及Mittag-Leffler函数Eβ，η(−t)的正交。离散化方法结合了解的积分表示、Mittag-Leffler函数的逼近和数值积分技术。为了提高计算效率，提出了两种逼近Mittag-Leffler函数的方法。利用确定的解正则性，建立了均方强收敛阶。数值算例验证了理论结果。
{"title":"Strong convergence analysis of time discretization for stochastic nonlinear diffusion-wave equations driven by fractional Brownian motion","authors":"Xing Liu , Yumeng Yang","doi":"10.1016/j.camwa.2026.01.011","DOIUrl":"10.1016/j.camwa.2026.01.011","url":null,"abstract":"<div><div>This paper constructs a discretization for the stochastic nonlinear diffusion-wave equation involving the Caputo fractional derivative of order <em>α</em> ∈ (1, 2), driven by fractional Brownian motion with Hurst index 0 < <em>H</em> < 1. For time discretization, we propose a quadrature involving Mittag-Leffler functions <span><math><mrow><msub><mi>E</mi><mrow><mi>β</mi><mo>,</mo><mi>η</mi></mrow></msub><mrow><mo>(</mo><mo>−</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>. The discretization method combines the integral representation of the solution, the approximation of Mittag-Leffler functions and numerical integration techniques. Two approximation methods for the Mittag-Leffler functions are developed to enhance computational efficiency. The mean square strong convergence order is established by utilizing the confirmed solution regularity. Numerical examples are presented to validate the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"205 ","pages":"Pages 251-267"},"PeriodicalIF":2.5,"publicationDate":"2026-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978224","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 : 2026-01-16DOI: 10.1016/j.camwa.2026.01.008
Hui Yu , Zhaoyang Wang , Ping Lin
In this paper, we propose a variable time-step linear relaxation scheme for time-fractional phase-field equations with a free energy density in general polynomial form. The -CN formula is used to discretize the fractional derivative, and an auxiliary variable is introduced to approximate the nonlinear term by directly solving algebraic equations rather than differential-algebraic equations as in the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV) approaches. The developed semi-discrete scheme is second-order accurate in time, and the inconsistency between the auxiliary and the original variables does not deteriorate over time. Furthermore, we take the time-fractional volume-conserved Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional Swift-Hohenberg equation as examples to demonstrate that the constructed schemes are energy stable and that the discrete energy dissipation law is asymptotically compatible with the classical one when the fractional-order parameter . Several numerical examples demonstrate the effectiveness of the proposed scheme. In particular, numerical results confirm that the auxiliary variable remains well aligned with the original variable, and the error between them does not continue to increase over time before the system reaches steady state.
{"title":"Linear relaxation schemes with asymptotically compatible energy law for time-fractional phase-field models","authors":"Hui Yu , Zhaoyang Wang , Ping Lin","doi":"10.1016/j.camwa.2026.01.008","DOIUrl":"10.1016/j.camwa.2026.01.008","url":null,"abstract":"<div><div>In this paper, we propose a variable time-step linear relaxation scheme for time-fractional phase-field equations with a free energy density in general polynomial form. The <span><math><mrow><mi>L</mi><msup><mn>1</mn><mo>+</mo></msup></mrow></math></span>-CN formula is used to discretize the fractional derivative, and an auxiliary variable is introduced to approximate the nonlinear term by directly solving algebraic equations rather than differential-algebraic equations as in the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV) approaches. The developed semi-discrete scheme is second-order accurate in time, and the inconsistency between the auxiliary and the original variables does not deteriorate over time. Furthermore, we take the time-fractional volume-conserved Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional Swift-Hohenberg equation as examples to demonstrate that the constructed schemes are energy stable and that the discrete energy dissipation law is asymptotically compatible with the classical one when the fractional-order parameter <span><math><mrow><mi>α</mi><mo>→</mo><msup><mn>1</mn><mo>−</mo></msup></mrow></math></span>. Several numerical examples demonstrate the effectiveness of the proposed scheme. In particular, numerical results confirm that the auxiliary variable remains well aligned with the original variable, and the error between them does not continue to increase over time before the system reaches steady state.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 192-211"},"PeriodicalIF":2.5,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979432","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 : 2026-01-14DOI: 10.1016/j.camwa.2025.12.031
Haihong Zhou , Huan Zhang , Xiaomin Pan
In this paper, we extend the selection of auxiliary variables by proposing a hyperbolic tangent scalar auxiliary variable (tanh-SAV) approach for solving gradient flows. The proposed tanh-SAV schemes introduce an auxiliary variable based on the hyperbolic tangent function, providing a well-defined formulation that enables the construction of decoupled, linear, and efficient numerical schemes. We demonstrate the construction of first-order, second-order, and higher-order unconditionally energy-stable schemes, utilizing either the Crank–Nicolson method or a k-step backward differentiation formula for time discretization. Only one constant-coefficient equation needs to be solved per time step, and we prove that all resulting tanh-SAV schemes are uniquely solvable at each time level. Furthermore, the theoretical analysis demonstrates the discrete energy stability of the proposed numerical schemes and proves the positivity property of the auxiliary variable. For the tanh-SAV/BDFk schemes () combined with a Fourier pseudo-spectral spatial discretization, we further establish fully discrete optimal-order error estimates. In addition, we provide numerical simulations of one- and two-dimensional Cahn–Hilliard, Allen–Cahn, and phase-field crystal models. The results demonstrate that, consistent with the theoretical analysis, the proposed schemes preserve the positivity of the auxiliary variable, maintain excellent stability, and achieve the desired temporal accuracy.
{"title":"Unconditionally energy-stable and accurate schemes based on hyperbolic tangent scalar auxiliary variable approach for gradient flows","authors":"Haihong Zhou , Huan Zhang , Xiaomin Pan","doi":"10.1016/j.camwa.2025.12.031","DOIUrl":"10.1016/j.camwa.2025.12.031","url":null,"abstract":"<div><div>In this paper, we extend the selection of auxiliary variables by proposing a hyperbolic tangent scalar auxiliary variable (tanh-SAV) approach for solving gradient flows. The proposed tanh-SAV schemes introduce an auxiliary variable based on the hyperbolic tangent function, providing a well-defined formulation that enables the construction of decoupled, linear, and efficient numerical schemes. We demonstrate the construction of first-order, second-order, and higher-order unconditionally energy-stable schemes, utilizing either the Crank–Nicolson method or a <em>k</em>-step backward differentiation formula for time discretization. Only one constant-coefficient equation needs to be solved per time step, and we prove that all resulting tanh-SAV schemes are uniquely solvable at each time level. Furthermore, the theoretical analysis demonstrates the discrete energy stability of the proposed numerical schemes and proves the positivity property of the auxiliary variable. For the tanh-SAV/BDF<em>k</em> schemes (<span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></mrow></math></span>) combined with a Fourier pseudo-spectral spatial discretization, we further establish fully discrete optimal-order error estimates. In addition, we provide numerical simulations of one- and two-dimensional Cahn–Hilliard, Allen–Cahn, and phase-field crystal models. The results demonstrate that, consistent with the theoretical analysis, the proposed schemes preserve the positivity of the auxiliary variable, maintain excellent stability, and achieve the desired temporal accuracy.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"204 ","pages":"Pages 263-282"},"PeriodicalIF":2.5,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977942","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 : 2026-01-14DOI: 10.1016/j.camwa.2025.12.030
Wenxin Zheng , Fawang Liu , Shujuan Lü , Ian Turner , Libo Feng
In this work, the magnetohydrodynamic (MHD) flow and heat transfer for a Jeffrey nanofluid flowing over a semi-infinite plate subjected to slip effects is considered. Firstly, the standard constitutive equation of a Jeffrey fluid is generalized to incorporate the tempered fractional derivative to describe the nonlocal but finite relaxation process, which leads to a novel tempered momentum equation. Combined with a tempered energy equation, the coupled momentum and energy system of a Jeffrey nanofluid is formulated. Secondly, the Legendre spectral method is employed to the tempered fractional coupled model, in which the modified shifted Grünwald superconvergence formula is proposed to discretize the tempered fractional derivative. To enhance computational efficiency, a fast algorithm is further developed, for which the stability and convergence analysis are rigorously established. Finally, some numerical examples are performed to confirm the efficiency of the proposed numerical schemes and to investigate the impacts of important model parameters on the variations of fluid movement and thermal transfer.
{"title":"Numerical methods and analysis for magnetohydrodynamics slip flow and heat transfer of Jeffrey nanofluid with tempered fractional constitutive relationship","authors":"Wenxin Zheng , Fawang Liu , Shujuan Lü , Ian Turner , Libo Feng","doi":"10.1016/j.camwa.2025.12.030","DOIUrl":"10.1016/j.camwa.2025.12.030","url":null,"abstract":"<div><div>In this work, the magnetohydrodynamic (MHD) flow and heat transfer for a Jeffrey nanofluid flowing over a semi-infinite plate subjected to slip effects is considered. Firstly, the standard constitutive equation of a Jeffrey fluid is generalized to incorporate the tempered fractional derivative to describe the nonlocal but finite relaxation process, which leads to a novel tempered momentum equation. Combined with a tempered energy equation, the coupled momentum and energy system of a Jeffrey nanofluid is formulated. Secondly, the Legendre spectral method is employed to the tempered fractional coupled model, in which the modified shifted Grünwald superconvergence formula is proposed to discretize the tempered fractional derivative. To enhance computational efficiency, a fast algorithm is further developed, for which the stability and convergence analysis are rigorously established. Finally, some numerical examples are performed to confirm the efficiency of the proposed numerical schemes and to investigate the impacts of important model parameters on the variations of fluid movement and thermal transfer.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"205 ","pages":"Pages 229-250"},"PeriodicalIF":2.5,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978238","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 : 2026-01-12DOI: 10.1016/j.camwa.2026.01.003
Quanxiang Wang , Jianqiang Xie , Yu Zheng , Liqun Wang
In this paper, we propose a FE-MIFVE method for the solution of the three-dimensional elliptic interface problems on structured grids. The method uses the standard finite element discretization on regular elements, and uses the modified immersed finite volume element discretization on interface elements. By designing a suitable function which has the same jumps as the solution, we turn the original elliptic interface problem with nonhomogeneous jump conditions to be an elliptic interface problem with homogeneous jump conditions. Numerical experiments for various problems show the new proposed FE-MIFVE method can serve as an efficient field solver in a simulation on structured grids, such as predicting the electrostatics of solvated biomolecules.
{"title":"A FE-MIFVE method for three-dimensional (3D) elliptic interface problems","authors":"Quanxiang Wang , Jianqiang Xie , Yu Zheng , Liqun Wang","doi":"10.1016/j.camwa.2026.01.003","DOIUrl":"10.1016/j.camwa.2026.01.003","url":null,"abstract":"<div><div>In this paper, we propose a FE-MIFVE method for the solution of the three-dimensional elliptic interface problems on structured grids. The method uses the standard finite element discretization on regular elements, and uses the modified immersed finite volume element discretization on interface elements. By designing a suitable function which has the same jumps as the solution, we turn the original elliptic interface problem with nonhomogeneous jump conditions to be an elliptic interface problem with homogeneous jump conditions. Numerical experiments for various problems show the new proposed FE-MIFVE method can serve as an efficient field solver in a simulation on structured grids, such as predicting the electrostatics of solvated biomolecules.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 176-191"},"PeriodicalIF":2.5,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145957177","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 : 2026-01-12DOI: 10.1016/j.camwa.2026.01.005
Min Xiao , Xihua Xu
The aim of this study is to develop a generic framework to construct three-dimensional nodal solvers for Lagrangian cell-centered hydrodynamics. By utilizing the specific volume as a medium, a new relationship between pressure and velocity is derived to figure out the nodal velocity. The nodal control volume adheres to Newton’s first law, resulting in the compatible equation. This generic framework ensures conservation of total mass, total momentum, and total energy, and satisfies an entropy inequality. Furthermore, it can be reduced to well-known schemes such as GLACE (Godunov-type LAgrangian scheme Conservative for total Energy) and EUCCLHYD (Explicit Unstructured Cell-Centered Lagrangian HYDrodynamics), demonstrating its versatility and ability to generate various multi-point schemes.
{"title":"A generic framework for solving three-dimensional gas dynamics equations","authors":"Min Xiao , Xihua Xu","doi":"10.1016/j.camwa.2026.01.005","DOIUrl":"10.1016/j.camwa.2026.01.005","url":null,"abstract":"<div><div>The aim of this study is to develop a generic framework to construct three-dimensional nodal solvers for Lagrangian cell-centered hydrodynamics. By utilizing the specific volume as a medium, a new relationship between pressure and velocity is derived to figure out the nodal velocity. The nodal control volume adheres to Newton’s first law, resulting in the compatible equation. This generic framework ensures conservation of total mass, total momentum, and total energy, and satisfies an entropy inequality. Furthermore, it can be reduced to well-known schemes such as GLACE (Godunov-type LAgrangian scheme Conservative for total Energy) and EUCCLHYD (Explicit Unstructured Cell-Centered Lagrangian HYDrodynamics), demonstrating its versatility and ability to generate various multi-point schemes.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 152-175"},"PeriodicalIF":2.5,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145957176","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 : 2026-01-10DOI: 10.1016/j.camwa.2026.01.004
Yun Zhang , Xiaoli Feng , Xiongbin Yan
This paper is dedicated to addressing the simultaneous inversion problem involving the initial value and space-dependent source term in a time-fractional diffusion-wave equation. Firstly, we establish the uniqueness of the inverse problem by leveraging the asymptotic expansion of Mittag-Leffler functions. Subsequently, we decompose the inverse problem into two subproblems and introduce an alternating iteration reconstruction method, complemented by a regularization strategy. Additionally, a comprehensive convergence analysis for this method is provided. To solve the inverse problem numerically, we introduce two semidiscrete schemes based on standard Galerkin method and lumped mass method, respectively. Furthermore, we establish error estimates that are associated with the noise level, iteration step, regularization parameter, and spatial discretization parameter. Finally, we present several numerical experiments in both one-dimensional and two-dimensional cases to validate the theoretical results and demonstrate the effectiveness of our proposed method.
{"title":"An alternating approach for reconstructing the initial value and source term in a time-fractional diffusion-wave equation","authors":"Yun Zhang , Xiaoli Feng , Xiongbin Yan","doi":"10.1016/j.camwa.2026.01.004","DOIUrl":"10.1016/j.camwa.2026.01.004","url":null,"abstract":"<div><div>This paper is dedicated to addressing the simultaneous inversion problem involving the initial value and space-dependent source term in a time-fractional diffusion-wave equation. Firstly, we establish the uniqueness of the inverse problem by leveraging the asymptotic expansion of Mittag-Leffler functions. Subsequently, we decompose the inverse problem into two subproblems and introduce an alternating iteration reconstruction method, complemented by a regularization strategy. Additionally, a comprehensive convergence analysis for this method is provided. To solve the inverse problem numerically, we introduce two semidiscrete schemes based on standard Galerkin method and lumped mass method, respectively. Furthermore, we establish error estimates that are associated with the noise level, iteration step, regularization parameter, and spatial discretization parameter. Finally, we present several numerical experiments in both one-dimensional and two-dimensional cases to validate the theoretical results and demonstrate the effectiveness of our proposed method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"204 ","pages":"Pages 231-262"},"PeriodicalIF":2.5,"publicationDate":"2026-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977941","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 : 2026-01-10DOI: 10.1016/j.camwa.2025.12.029
Zhi-Feng Pang , Xiaojie Peng , Bing Li , Hong Ge
Selective segmentation represents a pivotal image processing technique within the domain of computer vision. Its objective is to facilitate the precise identification and extraction of a region of interest (ROI) within an image, while excluding background regions that are irrelevant to the task at hand. Nevertheless, the challenge of segmentation for images with noise, particularly those with grayscale inhomogeneity, still exists in many state-of-the-art segmentation models. In order to address this challenge, this paper proposes a novel two-step selective segmentation method based on an exponential weighted geodesic distance-driven scheme and a thresholding method inspired by the K-means method. In particular, the exponential weighted geodesic distance is first leveraged to enhance the contrast between the ROI and the background. Subsequently, the segmentation of the ROI is obtained through thresholding on the enhanced image generated in the initial stage. The experimental results demonstrate the efficacy of the proposed method in suppressing the influence of noise and grayscale inhomogeneity to a certain extent. Furthermore, the results show that the proposed method yields a higher segmentation accuracy than several existing state-of-the-art variation-based selective segmentation models and two classical deep learning segmentation models.
{"title":"Two-stage selective segmentation method based on exponential weighted geodesic distance driven model and thresholding method","authors":"Zhi-Feng Pang , Xiaojie Peng , Bing Li , Hong Ge","doi":"10.1016/j.camwa.2025.12.029","DOIUrl":"10.1016/j.camwa.2025.12.029","url":null,"abstract":"<div><div>Selective segmentation represents a pivotal image processing technique within the domain of computer vision. Its objective is to facilitate the precise identification and extraction of a region of interest (ROI) within an image, while excluding background regions that are irrelevant to the task at hand. Nevertheless, the challenge of segmentation for images with noise, particularly those with grayscale inhomogeneity, still exists in many state-of-the-art segmentation models. In order to address this challenge, this paper proposes a novel two-step selective segmentation method based on an exponential weighted geodesic distance-driven scheme and a thresholding method inspired by the K-means method. In particular, the exponential weighted geodesic distance is first leveraged to enhance the contrast between the ROI and the background. Subsequently, the segmentation of the ROI is obtained through thresholding on the enhanced image generated in the initial stage. The experimental results demonstrate the efficacy of the proposed method in suppressing the influence of noise and grayscale inhomogeneity to a certain extent. Furthermore, the results show that the proposed method yields a higher segmentation accuracy than several existing state-of-the-art variation-based selective segmentation models and two classical deep learning segmentation models.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"206 ","pages":"Pages 133-151"},"PeriodicalIF":2.5,"publicationDate":"2026-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940427","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 : 2026-01-10DOI: 10.1016/j.camwa.2026.01.001
Elham Gholamipour , Ahmad Shirzadi , Hossein Hosseinzadeh , Vladimir Sladek , Jan Sladek
While various Meshless Local Petrov-Galerkin (MLPG) method variants exist, primarily distinguished by their choice of test functions in local subdomains, the mathematical relationships among these approaches remain unexplored. This paper establishes rigorous connections between existing MLPG formulations and proposes novel extensions based on an analysis of test function smoothness. It is shown that MLPG5, which employs Heaviside step test functions, corresponds to the mean value of MLPG2 (the collocation method), while MLPG4, using logarithmic test functions, is proven equivalent to the mean value of MLPG5 over the radius of local subdomains. Building on these insights, a systematic framework for generating new MLPG variants is introduced by leveraging the smoothness properties of test functions. All existing and newly proposed MLPG variants are demonstrated to have local weak forms equivalent to the original strong-form equations. This equivalence establishes the unique solvability of the methods, addressing long-standing questions regarding their consistency. Comprehensive numerical experiments validate the theoretical findings, confirming both the inter-variant relationships and the robustness of the newly extended MLPG variants.
{"title":"Mathematical relationships and novel extensions of MLPG variants","authors":"Elham Gholamipour , Ahmad Shirzadi , Hossein Hosseinzadeh , Vladimir Sladek , Jan Sladek","doi":"10.1016/j.camwa.2026.01.001","DOIUrl":"10.1016/j.camwa.2026.01.001","url":null,"abstract":"<div><div>While various Meshless Local Petrov-Galerkin (MLPG) method variants exist, primarily distinguished by their choice of test functions in local subdomains, the mathematical relationships among these approaches remain unexplored. This paper establishes rigorous connections between existing MLPG formulations and proposes novel extensions based on an analysis of test function smoothness. It is shown that MLPG5, which employs Heaviside step test functions, corresponds to the mean value of MLPG2 (the collocation method), while MLPG4, using logarithmic test functions, is proven equivalent to the mean value of MLPG5 over the radius of local subdomains. Building on these insights, a systematic framework for generating new MLPG variants is introduced by leveraging the smoothness properties of test functions. All existing and newly proposed MLPG variants are demonstrated to have local weak forms equivalent to the original strong-form equations. This equivalence establishes the unique solvability of the methods, addressing long-standing questions regarding their consistency. Comprehensive numerical experiments validate the theoretical findings, confirming both the inter-variant relationships and the robustness of the newly extended MLPG variants.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"205 ","pages":"Pages 212-228"},"PeriodicalIF":2.5,"publicationDate":"2026-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927228","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}
We propose a two-compartment mathematical model of myocardial perfusion, representing myocardial tissue at the arteriolar level. The model comprises a simplified two-dimensional geometrical analog of the complex three-dimensional myocardial vasculature. In the advective compartment, consisting of a 2D vasculature analog, fluid flow and contrast agent transport are governed by Navier-Stokes and system of advection-diffusion equations, respectively. The surrounding myocardium, included in porous capillary compartment and modeled as a porous medium, assumes purely diffusive transport without fluid flow. Contrast agent exchange occurs through the interface between the two compartments. The model is numerically solved using the lattice Boltzmann method, with GPU implementation enabling massive parallelization. Sample contrast agent profiles are analyzed for both healthy and defective tissues. The model’s capability to interpret actual MRI perfusion curves is evaluated using mathematical optimization techniques. Furthermore, the model is employed for a binary classification test to evaluate its agreement with the expert opinion of a qualified clinician. Myocardial blood flow approximations from the proposed model compare favorably to results from established medical software utilizing signal-deconvolution methods. Despite its simplifications, the 2D model accurately represents essential perfusion dynamics, matching or exceeding clinical software in agreement with expert evaluations. Although tested on a small number of patients, this proof of concept shows potential for direct application during perfusion exams or generating synthetic data for machine learning.
{"title":"Mathematical modeling of myocardial perfusion using lattice Boltzmann method","authors":"Jan Kovář , Radek Fučík , Tarique Hussain , Munes Fares , Radomír Chabiniok","doi":"10.1016/j.camwa.2025.12.005","DOIUrl":"10.1016/j.camwa.2025.12.005","url":null,"abstract":"<div><div>We propose a two-compartment mathematical model of myocardial perfusion, representing myocardial tissue at the arteriolar level. The model comprises a simplified two-dimensional geometrical analog of the complex three-dimensional myocardial vasculature. In the advective compartment, consisting of a 2D vasculature analog, fluid flow and contrast agent transport are governed by Navier-Stokes and system of advection-diffusion equations, respectively. The surrounding myocardium, included in porous capillary compartment and modeled as a porous medium, assumes purely diffusive transport without fluid flow. Contrast agent exchange occurs through the interface between the two compartments. The model is numerically solved using the lattice Boltzmann method, with GPU implementation enabling massive parallelization. Sample contrast agent profiles are analyzed for both healthy and defective tissues. The model’s capability to interpret actual MRI perfusion curves is evaluated using mathematical optimization techniques. Furthermore, the model is employed for a binary classification test to evaluate its agreement with the expert opinion of a qualified clinician. Myocardial blood flow approximations from the proposed model compare favorably to results from established medical software utilizing signal-deconvolution methods. Despite its simplifications, the 2D model accurately represents essential perfusion dynamics, matching or exceeding clinical software in agreement with expert evaluations. Although tested on a small number of patients, this proof of concept shows potential for direct application during perfusion exams or generating synthetic data for machine learning.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"203 ","pages":"Pages 230-253"},"PeriodicalIF":2.5,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145925930","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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