Pub Date : 2025-10-19DOI: 10.1016/j.apnum.2025.10.010
T. Baghban , M.H. Heydari , M. Bayram , M.A. Zaky
This paper provides a numerical strategy for solving nonlinear third-kind fractional integro-differential equations (FIDEs) involving the Caputo-Hadamard derivative. To effectively address the challenges posed by non-local logarithmic kernels, we introduce a novel class of basis functions known as the piecewise logarithmic Jacobi cardinal functions (JCFs). Two corresponding operational matrices, associated with the logarithmic and Hadamard fractional integrals, are developed to transform the original FIDE into a system of nonlinear algebraic equations. Using fixed-point theory, it is verified that a unique solution exists. Moreover, comprehensive error analysis confirms the spectral accuracy and exponential convergence of the proposed method, especially when Chebyshev-type parameters are employed. Numerical experiments support the theoretical findings and reveal substantial accuracy gains with increasing polynomial order.
{"title":"A piecewise logarithmic Jacobi cardinal scheme for nonlinear third-kind fractional integro-differential equations","authors":"T. Baghban , M.H. Heydari , M. Bayram , M.A. Zaky","doi":"10.1016/j.apnum.2025.10.010","DOIUrl":"10.1016/j.apnum.2025.10.010","url":null,"abstract":"<div><div>This paper provides a numerical strategy for solving nonlinear third-kind fractional integro-differential equations (FIDEs) involving the Caputo-Hadamard derivative. To effectively address the challenges posed by non-local logarithmic kernels, we introduce a novel class of basis functions known as the piecewise logarithmic Jacobi cardinal functions (JCFs). Two corresponding operational matrices, associated with the logarithmic and Hadamard fractional integrals, are developed to transform the original FIDE into a system of nonlinear algebraic equations. Using fixed-point theory, it is verified that a unique solution exists. Moreover, comprehensive error analysis confirms the spectral accuracy and exponential convergence of the proposed method, especially when Chebyshev-type parameters are employed. Numerical experiments support the theoretical findings and reveal substantial accuracy gains with increasing polynomial order.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 167-188"},"PeriodicalIF":2.4,"publicationDate":"2025-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145413756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-14DOI: 10.1016/j.apnum.2025.10.008
Xuewei Liu , Zhenyu Wang , Xiaohua Ding , Shao-Liang Zhang
The stochastic two-dimensional KdV equation arises as a mathematical model for shallow water wave dynamics in physical systems. To efficiently handle the equation’s high-order spatial derivatives and stochastic terms, a local discontinuous Galerkin method is proposed. The method is proved to be -stable and to achieve the optimal mean-square convergence rate of order when degree- polynomials are used. For temporal discretization, the implicit midpoint method is applied, and the restarted Generalized Minimum Residual method is employed to solve the resulting linear systems in two-dimensional simulations. Numerical experiments demonstrate optimal convergence rates and confirm both the theoretical analysis and the effectiveness of the method.
{"title":"Optimal error estimates and stability of a local discontinuous Galerkin method for the stochastic two-dimensional KdV equation","authors":"Xuewei Liu , Zhenyu Wang , Xiaohua Ding , Shao-Liang Zhang","doi":"10.1016/j.apnum.2025.10.008","DOIUrl":"10.1016/j.apnum.2025.10.008","url":null,"abstract":"<div><div>The stochastic two-dimensional KdV equation arises as a mathematical model for shallow water wave dynamics in physical systems. To efficiently handle the equation’s high-order spatial derivatives and stochastic terms, a local discontinuous Galerkin method is proposed. The method is proved to be <span><math><msup><mi>L</mi><mn>2</mn></msup></math></span>-stable and to achieve the optimal mean-square convergence rate of order <span><math><mrow><mi>N</mi><mo>+</mo><mn>1</mn></mrow></math></span> when degree-<span><math><mi>N</mi></math></span> polynomials are used. For temporal discretization, the implicit midpoint method is applied, and the restarted Generalized Minimum Residual method is employed to solve the resulting linear systems in two-dimensional simulations. Numerical experiments demonstrate optimal convergence rates and confirm both the theoretical analysis and the effectiveness of the method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 310-328"},"PeriodicalIF":2.4,"publicationDate":"2025-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145526261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-14DOI: 10.1016/j.apnum.2025.10.007
Mengchun Yuan, Qi Li
Numerous efficient numerical algorithms have been developed for phase-field surfactant models, including the convex splitting method, the scalar auxiliary variable (SAV) approach, the invariant energy quadratization (IEQ) approach, and the new Lagrange multiplier method. In this paper, we introduce novel numerical schemes based on the supplementary variable method for solving and simulating the binary fluid phase-field surfactant model of the Cahn-Hilliard type. The key innovation of our method is the introduction of an auxiliary variable, which reformulates the original problem into a constrained optimization framework. This reformulation offers several significant advantages over existing approaches. Firstly, our schemes require solving only a few Poisson-type systems with constant coefficient matrices, significantly reducing computational costs. Secondly, the proposed schemes preserve mass conservation and adhere to the original energy dissipation law at the discrete level, in contrast to the SAV and IEQ approaches, which adhere to modified dissipation laws. Additionally, we rigorously establish the energy stability of the schemes. Extensive 2D and 3D numerical experiments confirm the accuracy and efficiency of the proposed schemes.
{"title":"Structure-preserving numerical methods for phase-field surfactant models based on the supplemental variable method (SVM)","authors":"Mengchun Yuan, Qi Li","doi":"10.1016/j.apnum.2025.10.007","DOIUrl":"10.1016/j.apnum.2025.10.007","url":null,"abstract":"<div><div>Numerous efficient numerical algorithms have been developed for phase-field surfactant models, including the convex splitting method, the scalar auxiliary variable (SAV) approach, the invariant energy quadratization (IEQ) approach, and the new Lagrange multiplier method. In this paper, we introduce novel numerical schemes based on the supplementary variable method for solving and simulating the binary fluid phase-field surfactant model of the Cahn-Hilliard type. The key innovation of our method is the introduction of an auxiliary variable, which reformulates the original problem into a constrained optimization framework. This reformulation offers several significant advantages over existing approaches. Firstly, our schemes require solving only a few Poisson-type systems with constant coefficient matrices, significantly reducing computational costs. Secondly, the proposed schemes preserve mass conservation and adhere to the original energy dissipation law at the discrete level, in contrast to the SAV and IEQ approaches, which adhere to modified dissipation laws. Additionally, we rigorously establish the energy stability of the schemes. Extensive 2D and 3D numerical experiments confirm the accuracy and efficiency of the proposed schemes.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 123-143"},"PeriodicalIF":2.4,"publicationDate":"2025-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-10DOI: 10.1016/j.apnum.2025.10.005
Congpei An , Mou Cai
This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific cases. With the aid of the de la Vallée-Poussin approximation, we derive a uniform error bound and a concrete error bound. These error estimates demonstrate the effectiveness of Tikhonov regularization in the denoising process. A new regularity condition for the selection of regularization parameters is proposed. We investigate three strategies for choosing regularization parameters: Morozov’s discrepancy principle, the L-curve, and generalized cross-validation, by explicitly combining these error bounds of the approximating trigonometric polynomial. We show that Morozov’s discrepancy principle satisfies the proposed regularity condition, while the other two methods do not. Finally, numerical examples are provided to illustrate how the aforementioned methodologies, when applied with well-chosen parameters, can significantly improve the quality of approximation.
本文探讨了利用单位圆上的三角多项式将Tikhonov正则化纳入最小二乘近似方案。这种方法包括插值和超插值作为具体案例。利用de la vall - poussin近似，导出了统一的误差界和具体的L2误差界。这些误差估计证明了吉洪诺夫正则化在去噪过程中的有效性。提出了一种新的正则化参数选择的正则性条件。我们研究了三种选择正则化参数的策略：Morozov的差异原理、l曲线和广义交叉验证，通过显式地组合这些近似三角多项式的误差界限。结果表明，Morozov的差异原理满足所提出的正则性条件，而其他两种方法则不满足。最后，提供了数值实例来说明上述方法如何在选择良好的参数时应用，可以显着提高近似的质量。
{"title":"Parameter choice strategies for regularized least squares approximation of noisy continuous functions on the unit circle","authors":"Congpei An , Mou Cai","doi":"10.1016/j.apnum.2025.10.005","DOIUrl":"10.1016/j.apnum.2025.10.005","url":null,"abstract":"<div><div>This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific cases. With the aid of the de la Vallée-Poussin approximation, we derive a uniform error bound and a concrete <span><math><msub><mi>L</mi><mn>2</mn></msub></math></span> error bound. These error estimates demonstrate the effectiveness of Tikhonov regularization in the denoising process. A new regularity condition for the selection of regularization parameters is proposed. We investigate three strategies for choosing regularization parameters: Morozov’s discrepancy principle, the L-curve, and generalized cross-validation, by explicitly combining these error bounds of the approximating trigonometric polynomial. We show that Morozov’s discrepancy principle satisfies the proposed regularity condition, while the other two methods do not. Finally, numerical examples are provided to illustrate how the aforementioned methodologies, when applied with well-chosen parameters, can significantly improve the quality of approximation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 84-103"},"PeriodicalIF":2.4,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-10DOI: 10.1016/j.apnum.2025.10.002
Jon Henshaw , Aviv Gibali , Thomas Humphries
The superiorization methodology (SM) is an optimization heuristic in which an iterative algorithm, which aims to solve a particular problem, is “superiorized” to promote solutions that are improved with respect to some secondary criterion. This superiorization is achieved by perturbing iterates of the algorithm in nonascending directions of a prescribed function that penalizes undesirable characteristics in the solution; the solution produced by the superiorized algorithm should therefore be improved with respect to the value of this function. In this paper, we broaden the SM to allow for the perturbations to be introduced by an arbitrary procedure instead, using a plug-and-play approach. This allows for operations such as image denoisers or deep neural networks, which have applications to a broad class of problems, to be incorporated within the superiorization methodology. As proof of concept, we perform numerical simulations involving low-dose and sparse-view computed tomography image reconstruction, comparing the plug-and-play approach to two conventionally superiorized algorithms, as well as a post-processing approach. The plug-and-play approach provides comparable or better image quality in most cases, while also providing advantages in terms of computing time, and data fidelity of the solutions.
{"title":"Plug-and-play superiorization","authors":"Jon Henshaw , Aviv Gibali , Thomas Humphries","doi":"10.1016/j.apnum.2025.10.002","DOIUrl":"10.1016/j.apnum.2025.10.002","url":null,"abstract":"<div><div>The superiorization methodology (SM) is an optimization heuristic in which an iterative algorithm, which aims to solve a particular problem, is “superiorized” to promote solutions that are improved with respect to some secondary criterion. This superiorization is achieved by perturbing iterates of the algorithm in nonascending directions of a prescribed function that penalizes undesirable characteristics in the solution; the solution produced by the superiorized algorithm should therefore be improved with respect to the value of this function. In this paper, we broaden the SM to allow for the perturbations to be introduced by an arbitrary procedure instead, using a plug-and-play approach. This allows for operations such as image denoisers or deep neural networks, which have applications to a broad class of problems, to be incorporated within the superiorization methodology. As proof of concept, we perform numerical simulations involving low-dose and sparse-view computed tomography image reconstruction, comparing the plug-and-play approach to two conventionally superiorized algorithms, as well as a post-processing approach. The plug-and-play approach provides comparable or better image quality in most cases, while also providing advantages in terms of computing time, and data fidelity of the solutions.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 29-43"},"PeriodicalIF":2.4,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-10DOI: 10.1016/j.apnum.2025.10.004
Behzad Kafash
In this study, a modified Rayleigh-Ritz method is presented for the solution of optimal control problems governed by time-delayed dynamical systems, considering both constrained and unconstrained control and state variables. In this approach, the control or state variables are approximated using shifted Chebyshev polynomials with unknown coefficients. The proposed modified Rayleigh-Ritz method transforms the constrained optimal control problems governed by time-delayed dynamical systems into an optimization problem with constraints. Furthermore, a computational algorithm is developed for implementing the proposed method, and its convergence is proven analytically. To evaluate the efficiency and accuracy of the proposed algorithm, several numerical examples are presented. These include the single-input/single-output system as a case study with control and final state constraints, and an optimal control problem of the harmonic oscillator under different scenarios, which involve constraints on state and control variables.
{"title":"Numerical approximation of constrained optimal control problems in delayed systems using an enhanced Rayleigh-Ritz algorithm","authors":"Behzad Kafash","doi":"10.1016/j.apnum.2025.10.004","DOIUrl":"10.1016/j.apnum.2025.10.004","url":null,"abstract":"<div><div>In this study, a modified Rayleigh-Ritz method is presented for the solution of optimal control problems governed by time-delayed dynamical systems, considering both constrained and unconstrained control and state variables. In this approach, the control or state variables are approximated using shifted Chebyshev polynomials with unknown coefficients. The proposed modified Rayleigh-Ritz method transforms the constrained optimal control problems governed by time-delayed dynamical systems into an optimization problem with constraints. Furthermore, a computational algorithm is developed for implementing the proposed method, and its convergence is proven analytically. To evaluate the efficiency and accuracy of the proposed algorithm, several numerical examples are presented. These include the single-input/single-output system as a case study with control and final state constraints, and an optimal control problem of the harmonic oscillator under different scenarios, which involve constraints on state and control variables.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 104-122"},"PeriodicalIF":2.4,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-10DOI: 10.1016/j.apnum.2025.10.003
Xiaoni Chi , Lin Gan , Zhuoran Gao , Jein-Shan Chen
This paper investigates feasible interior-point method (IPM) with full-Newton step for -weighted linear complementarity problem (WLCP). In particular, by applying the algebraically equivalent transformation (AET) for linear optimization, we obtain the new search directions by solving the perturbed Newton system. The AET of the Newton system is based on the kernel function , which is used for solving WLCP for the first time. At each iteration, our algorithm takes only full-Newton steps. Therefore, no line-searches are needed to update the iterates. We show the strict feasibility of the full-Newton step and the polynomial iteration complexity of our algorithm under suitable assumptions. Some numerical experiments demonstrate the effectiveness of the proposed algorithm.
{"title":"A feasible interior-point method with full-Newton step for P*(κ)-weighted linear complementarity problem via the algebraically equivalent transformation","authors":"Xiaoni Chi , Lin Gan , Zhuoran Gao , Jein-Shan Chen","doi":"10.1016/j.apnum.2025.10.003","DOIUrl":"10.1016/j.apnum.2025.10.003","url":null,"abstract":"<div><div>This paper investigates feasible interior-point method (IPM) with full-Newton step for <span><math><mrow><msub><mi>P</mi><mo>*</mo></msub><mrow><mo>(</mo><mi>κ</mi><mo>)</mo></mrow></mrow></math></span>-weighted linear complementarity problem (WLCP). In particular, by applying the algebraically equivalent transformation (AET) for linear optimization, we obtain the new search directions by solving the perturbed Newton system. The AET of the Newton system is based on the kernel function <span><math><mrow><mi>φ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>t</mi><mo>−</mo><msqrt><mi>t</mi></msqrt></mrow></math></span>, which is used for solving WLCP for the first time. At each iteration, our algorithm takes only full-Newton steps. Therefore, no line-searches are needed to update the iterates. We show the strict feasibility of the full-Newton step and the polynomial iteration complexity of our algorithm under suitable assumptions. Some numerical experiments demonstrate the effectiveness of the proposed algorithm.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 144-166"},"PeriodicalIF":2.4,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-09DOI: 10.1016/j.apnum.2025.10.006
Feng Shao , Hu Shao , Bin Wu , Haijun Wang , Pengjie Liu , Meixing Liu
In this paper, we aim to develop a general conjugate gradient (CG) algorithmic framework for solving unconstrained optimization problems. Additionally, we employ different hybrid techniques to derive two hybrid conjugate parameters, which are then integrated into the algorithmic framework to develop two effective hybrid CG methods. Under common assumptions, we establish the global convergence of the proposed framework without any convexity assumption. Furthermore, we reveal the convergence rate of the framework under the uniformly convex condition. Preliminary numerical experiment results, including applications to unconstrained optimization and image restoration problems, are presented to explicitly illustrate the performance of the proposed methods in comparison with several existing methods.
{"title":"A conjugate gradient algorithmic framework for unconstrained optimization with applications: Convergence and rate analyses","authors":"Feng Shao , Hu Shao , Bin Wu , Haijun Wang , Pengjie Liu , Meixing Liu","doi":"10.1016/j.apnum.2025.10.006","DOIUrl":"10.1016/j.apnum.2025.10.006","url":null,"abstract":"<div><div>In this paper, we aim to develop a general conjugate gradient (CG) algorithmic framework for solving unconstrained optimization problems. Additionally, we employ different hybrid techniques to derive two hybrid conjugate parameters, which are then integrated into the algorithmic framework to develop two effective hybrid CG methods. Under common assumptions, we establish the global convergence of the proposed framework without any convexity assumption. Furthermore, we reveal the convergence rate of the framework under the uniformly convex condition. Preliminary numerical experiment results, including applications to unconstrained optimization and image restoration problems, are presented to explicitly illustrate the performance of the proposed methods in comparison with several existing methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 13-28"},"PeriodicalIF":2.4,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce a Scalar Auxiliary Variable (SAV) approach combined with the Runge-Kutta method for solving the Cahn-Hilliard equation. Within the framework of the SAV method, this Runge-Kutta method is particularly well-suited for obtaining numerical solutions that preserve key structural properties of the system, including energy conservation and momentum conservation. In the SAV-Runge-Kutta method, through a comparison of approximate solutions derived at various time points, the error for each step can be rigorously estimated. This approach thereby guarantees the stability and accuracy of the entire solution process. Finally, to illustrate the effectiveness and precision of our proposed method, we present several numerical examples. These examples demonstrate the capability of the SAV-Runge-Kutta method to accurately capture the intricate dynamics of the Cahn-Hilliard equation while maintaining energy conservation and momentum conservation.
{"title":"Error estimate for the Cahn-Hilliard equation by the SAV-Runge-Kutta scheme","authors":"Lizhen Chen , Xiaozhuang Ma , Guohui Zhang , Jing Zhang","doi":"10.1016/j.apnum.2025.10.001","DOIUrl":"10.1016/j.apnum.2025.10.001","url":null,"abstract":"<div><div>In this paper, we introduce a Scalar Auxiliary Variable (SAV) approach combined with the Runge-Kutta method for solving the Cahn-Hilliard equation. Within the framework of the SAV method, this Runge-Kutta method is particularly well-suited for obtaining numerical solutions that preserve key structural properties of the system, including energy conservation and momentum conservation. In the SAV-Runge-Kutta method, through a comparison of approximate solutions derived at various time points, the error for each step can be rigorously estimated. This approach thereby guarantees the stability and accuracy of the entire solution process. Finally, to illustrate the effectiveness and precision of our proposed method, we present several numerical examples. These examples demonstrate the capability of the SAV-Runge-Kutta method to accurately capture the intricate dynamics of the Cahn-Hilliard equation while maintaining energy conservation and momentum conservation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 66-83"},"PeriodicalIF":2.4,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-19DOI: 10.1016/j.apnum.2025.09.008
Mahmoud A. Zaky
In this paper, we construct and analyze a linearized L1–Galerkin Legendre spectral method for solving two-dimensional time-fractional diffusion equations with delay. The approach combines the L1 temporal discretization of the Caputo derivative with a Legendre spectral approximation in space, while nonlinear source terms are efficiently handled through a linearization strategy. To further enhance computational performance, we employ matrix diagonalization approach to solve the resulting algebraic systems in numerical implementation. Rigorous stability and convergence analyses are carried out using discrete fractional Grönwall and fractional Halanay inequalities, establishing unconditional stability and spectral error estimates. Numerical experiments confirm the theoretical predictions, demonstrating spectral accuracy in space and -order accuracy in time, as well as validating the robustness and efficiency of the proposed method across different fractional orders and delay parameters.
{"title":"A linearized two-dimensional Galerkin-L1 spectral method with diagonalization for time-fractional diffusion equations with delay","authors":"Mahmoud A. Zaky","doi":"10.1016/j.apnum.2025.09.008","DOIUrl":"10.1016/j.apnum.2025.09.008","url":null,"abstract":"<div><div>In this paper, we construct and analyze a linearized L1–Galerkin Legendre spectral method for solving two-dimensional time-fractional diffusion equations with delay. The approach combines the L1 temporal discretization of the Caputo derivative with a Legendre spectral approximation in space, while nonlinear source terms are efficiently handled through a linearization strategy. To further enhance computational performance, we employ matrix diagonalization approach to solve the resulting algebraic systems in numerical implementation. Rigorous stability and convergence analyses are carried out using discrete fractional Grönwall and fractional Halanay inequalities, establishing unconditional stability and spectral error estimates. Numerical experiments confirm the theoretical predictions, demonstrating spectral accuracy in space and <span><math><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>ϑ</mi><mo>)</mo></mrow></math></span>-order accuracy in time, as well as validating the robustness and efficiency of the proposed method across different fractional orders and delay parameters.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 1-12"},"PeriodicalIF":2.4,"publicationDate":"2025-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265082","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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