Pub Date : 2024-09-26DOI: 10.1016/j.matcom.2024.09.018
Combining kernel-based collocation methods with time-stepping methods to solve parabolic partial differential equations can potentially introduce challenges in balancing temporal and spatial discretization errors. Typically, using kernels with high orders of smoothness on some sufficiently dense set of trial centers provides high spatial approximation accuracy that can exceed the accuracy of finite difference methods in time. The paper proposes a greedy approach for selecting trial subspaces in the kernel-based collocation method applied to time-stepping to balance errors in both well-conditioned and ill-conditioned scenarios. The approach involves selecting trial centers using a fast block-greedy algorithm with new stopping criteria that aim to balance temporal and spatial errors. Numerical simulations of coupled bulk-surface pattern formations, a system involving two functions in the domain and two on the boundary, illustrate the effectiveness of the proposed method in reducing trial space dimensions while maintaining accuracy.
{"title":"Greedy trial subspace selection in meshfree time-stepping scheme with applications in coupled bulk-surface pattern formations","authors":"","doi":"10.1016/j.matcom.2024.09.018","DOIUrl":"10.1016/j.matcom.2024.09.018","url":null,"abstract":"<div><div>Combining kernel-based collocation methods with time-stepping methods to solve parabolic partial differential equations can potentially introduce challenges in balancing temporal and spatial discretization errors. Typically, using kernels with high orders of smoothness on some sufficiently dense set of trial centers provides high spatial approximation accuracy that can exceed the accuracy of finite difference methods in time. The paper proposes a greedy approach for selecting trial subspaces in the kernel-based collocation method applied to time-stepping to balance errors in both well-conditioned and ill-conditioned scenarios. The approach involves selecting trial centers using a fast block-greedy algorithm with new stopping criteria that aim to balance temporal and spatial errors. Numerical simulations of coupled bulk-surface pattern formations, a system involving two functions in the domain and two on the boundary, illustrate the effectiveness of the proposed method in reducing trial space dimensions while maintaining accuracy.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328105","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 : 2024-09-26DOI: 10.1016/j.matcom.2024.09.020
In this paper, we study the global well-posedness and global dynamics of a reaction–diffusion HIV infection model with the chemotactic movement of CTLs (Cytotoxic T lymphocytes). We first show the global existence and uniform boundedness for solutions of the system with general functional incidences. Then, for the model with bilinear incidence rate, we discuss the existence conditions of the three equilibria (infection-free, chemokines-extinct, chemokines-acute equilibria) of the model and obtain the conclusion of the local asymptotic stability of these equilibria by analyzing the linearized system at these equilibria. Moreover, by constructing reasonable Lyapunov functionals, we investigate the global stability and attractivity of the equilibria. Applying the estimate, Young’s inequality, Gagiardo-Nirenberg inequality and the parabolic regularity theorem, we also discuss the convergence rates of the equilibria. Finally, some numerical simulations are conducted to verify the theoretical results.
在本文中，我们研究了带有 CTL（细胞毒性 T 淋巴细胞）趋化运动的反应扩散型 HIV 感染模型的全局拟合性和全局动力学。我们首先证明了具有一般函数发生率的系统解的全局存在性和均匀有界性。然后，对于具有双线性发病率的模型，我们讨论了模型的三个平衡点（无感染平衡点、趋化因子-灭绝平衡点、趋化因子-急性平衡点）的存在条件，并通过分析这些平衡点处的线性化系统，得出了这些平衡点的局部渐近稳定性结论。此外，通过构建合理的 Lyapunov 函数，我们还研究了均衡点的全局稳定性和吸引力。应用 Lp-Lq 估计、Young 不等式、Gagiardo-Nirenberg 不等式和抛物线正则定理，我们还讨论了均衡点的收敛率。最后，我们进行了一些数值模拟来验证理论结果。
{"title":"Global well-posedness and dynamics of spatial diffusion HIV model with CTLs response and chemotaxis","authors":"","doi":"10.1016/j.matcom.2024.09.020","DOIUrl":"10.1016/j.matcom.2024.09.020","url":null,"abstract":"<div><div>In this paper, we study the global well-posedness and global dynamics of a reaction–diffusion HIV infection model with the chemotactic movement of CTLs (Cytotoxic T lymphocytes). We first show the global existence and uniform boundedness for solutions of the system with general functional incidences. Then, for the model with bilinear incidence rate, we discuss the existence conditions of the three equilibria (infection-free, chemokines-extinct, chemokines-acute equilibria) of the model and obtain the conclusion of the local asymptotic stability of these equilibria by analyzing the linearized system at these equilibria. Moreover, by constructing reasonable Lyapunov functionals, we investigate the global stability and attractivity of the equilibria. Applying the <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>−</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></mrow></math></span> estimate, Young’s inequality, Gagiardo-Nirenberg inequality and the parabolic regularity theorem, we also discuss the convergence rates of the equilibria. Finally, some numerical simulations are conducted to verify the theoretical results.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328274","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 : 2024-09-24DOI: 10.1016/j.matcom.2024.09.019
The evolution of a dynamic system on complex curved 3D surfaces is essential for the understanding of natural phenomena, the development of new materials, and engineering design optimization. In this work, we study the viscous Cahn–Hilliard equation on curved surfaces and develop two linear energy stable finite element schemes based on the lumped mass method. Two stabilizing terms are added to ensure both the unique solvability and unconditional energy stability. We prove rigorously that two schemes are unconditionally energy stable . Numerical experiments are presented to verify theoretical results and to show the robustness and accuracy of the proposed method.
{"title":"Two linear energy stable lumped mass finite element schemes for the viscous Cahn–Hilliard equation on curved surfaces in 3D","authors":"","doi":"10.1016/j.matcom.2024.09.019","DOIUrl":"10.1016/j.matcom.2024.09.019","url":null,"abstract":"<div><div>The evolution of a dynamic system on complex curved 3D surfaces is essential for the understanding of natural phenomena, the development of new materials, and engineering design optimization. In this work, we study the viscous Cahn–Hilliard equation on curved surfaces and develop two linear energy stable finite element schemes based on the lumped mass method. Two stabilizing terms are added to ensure both the unique solvability and unconditional energy stability. We prove rigorously that two schemes are unconditionally energy stable . Numerical experiments are presented to verify theoretical results and to show the robustness and accuracy of the proposed method.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328275","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 : 2024-09-24DOI: 10.1016/j.matcom.2024.09.022
To reflect the harvesting effect, a nonsmooth Filippov Leslie–Gower predator–prey model is proposed. Unlike traditional Filippov models, the time delay and reaction–diffusion under the condition of homogeneous Neumann boundary are considered in our system. The stability of equilibrium and the existence of the spatial Hopf bifurcation of the subsystems at the positive equilibrium are investigated. Furthermore, a comprehensive analysis is conducted on the sliding mode dynamics as well as the regular, virtual, and pseudoequilibria. The findings reveal that our Filippov system exhibits either a globally asymptotically stable regular equilibrium, a globally asymptotically stable time periodic solution, or a globally asymptotically stable pseudoequilibrium, contingent upon the specific values of the time delay and threshold level. A boundary point bifurcation, which transform a stable equilibrium point or periodic solution into a stable pseudoequilibrium, is demonstrated to emphasize the impact of time delay on our Filippov system and the significance of threshold control. Meanwhile, two kinds of global sliding bifurcations are exhibited, which sequentially transform a stable periodic solutions below the threshold into a grazing, sliding switching, and crossing bifurcations, depending on changes in the time delay or threshold level. Our results indicate that bucking bifurcation and crossing bifurcation pose significant challenges to the control of our Filippov system.
{"title":"Rich dynamics of a reaction–diffusion Filippov Leslie–Gower predator–prey model with time delay and discontinuous harvesting","authors":"","doi":"10.1016/j.matcom.2024.09.022","DOIUrl":"10.1016/j.matcom.2024.09.022","url":null,"abstract":"<div><div>To reflect the harvesting effect, a nonsmooth Filippov Leslie–Gower predator–prey model is proposed. Unlike traditional Filippov models, the time delay and reaction–diffusion under the condition of homogeneous Neumann boundary are considered in our system. The stability of equilibrium and the existence of the spatial Hopf bifurcation of the subsystems at the positive equilibrium are investigated. Furthermore, a comprehensive analysis is conducted on the sliding mode dynamics as well as the regular, virtual, and pseudoequilibria. The findings reveal that our Filippov system exhibits either a globally asymptotically stable regular equilibrium, a globally asymptotically stable time periodic solution, or a globally asymptotically stable pseudoequilibrium, contingent upon the specific values of the time delay and threshold level. A boundary point bifurcation, which transform a stable equilibrium point or periodic solution into a stable pseudoequilibrium, is demonstrated to emphasize the impact of time delay on our Filippov system and the significance of threshold control. Meanwhile, two kinds of global sliding bifurcations are exhibited, which sequentially transform a stable periodic solutions below the threshold into a grazing, sliding switching, and crossing bifurcations, depending on changes in the time delay or threshold level. Our results indicate that bucking bifurcation and crossing bifurcation pose significant challenges to the control of our Filippov system.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323339","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 : 2024-09-20DOI: 10.1016/j.matcom.2024.09.015
Seeking shelter from threats is a widespread instinct across species, specially employed by prey to avoid direct confrontations with predators. The present investigation centers on a three-species food chain model wherein the basal prey, characterized by logistic growth, seeks refuge to evade the intermediate predator, while the intermediate predator, in turn, seeks refuge to avoid encounters with the top predator. Additionally, our model assumes that the presence of the top predator induces a mate-finding Allee effect among the intermediate predator. We investigate how varying levels of refuge, along with other critical parameters such as the reproduction rate of the basal prey and the natural mortality rate of the top predator, influence system’s dynamics within the biparametric planes. Our model displays multistability and undergoes transcritical, saddle–node, Bogdanov–Takens and cusp bifurcations across different parameters. Moreover, the external environmental noise can induce interesting dynamics in the predator–prey system, resulting in noise-induced frequent transitions between distinct interior attractors or from interior to axial attractors. This phenomenon is particularly notable in scenarios where the deterministic model exhibits tristability. In summary, our findings offer potential new avenues for developing control strategies within the realm of community ecology in constant as well as fluctuating environments.
{"title":"Bifurcation analysis and exploration of noise-induced transitions of a food chain model with Allee effect","authors":"","doi":"10.1016/j.matcom.2024.09.015","DOIUrl":"10.1016/j.matcom.2024.09.015","url":null,"abstract":"<div><div>Seeking shelter from threats is a widespread instinct across species, specially employed by prey to avoid direct confrontations with predators. The present investigation centers on a three-species food chain model wherein the basal prey, characterized by logistic growth, seeks refuge to evade the intermediate predator, while the intermediate predator, in turn, seeks refuge to avoid encounters with the top predator. Additionally, our model assumes that the presence of the top predator induces a mate-finding Allee effect among the intermediate predator. We investigate how varying levels of refuge, along with other critical parameters such as the reproduction rate of the basal prey and the natural mortality rate of the top predator, influence system’s dynamics within the biparametric planes. Our model displays multistability and undergoes transcritical, saddle–node, Bogdanov–Takens and cusp bifurcations across different parameters. Moreover, the external environmental noise can induce interesting dynamics in the predator–prey system, resulting in noise-induced frequent transitions between distinct interior attractors or from interior to axial attractors. This phenomenon is particularly notable in scenarios where the deterministic model exhibits tristability. In summary, our findings offer potential new avenues for developing control strategies within the realm of community ecology in constant as well as fluctuating environments.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323338","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 : 2024-09-20DOI: 10.1016/j.matcom.2024.09.009
In this paper, we numerically investigate vector solitons with oscillatory phase backgrounds in the integrable coupled nonlinear Schrödinger equations, which are widely applied to varieties of physical contexts such as the simultaneous propagation of nonlinear optical pulses and the dynamics of two-components Bose–Einstein condensates. We develop the time-splitting Chebyshev–Galerkin method based on a transformation to accurately compute the vector soliton solutions. Compared to the finite difference method, numerical experiments show that the method with spectral accuracy and high efficiency is necessary for simulating the dynamics evolution of vector solitons. Combined with modulation instability conditions, linear stability analysis and direct numerical simulation, we reveal that the bright-dark and dark-dark solitons with various combinations of parameters under perturbations have qualitative differences. Particularly, vector solitons in unstable background with different wave numbers present distinct dynamics evolutions. The results help us to understand soliton dynamics with oscillatory phase backgrounds and the superposition between nonlinear waves.
{"title":"Numerical study of vector solitons with the oscillatory phase backgrounds in the integrable coupled nonlinear Schrödinger equations","authors":"","doi":"10.1016/j.matcom.2024.09.009","DOIUrl":"10.1016/j.matcom.2024.09.009","url":null,"abstract":"<div><div>In this paper, we numerically investigate vector solitons with oscillatory phase backgrounds in the integrable coupled nonlinear Schrödinger equations, which are widely applied to varieties of physical contexts such as the simultaneous propagation of nonlinear optical pulses and the dynamics of two-components Bose–Einstein condensates. We develop the time-splitting Chebyshev–Galerkin method based on a transformation to accurately compute the vector soliton solutions. Compared to the finite difference method, numerical experiments show that the method with spectral accuracy and high efficiency is necessary for simulating the dynamics evolution of vector solitons. Combined with modulation instability conditions, linear stability analysis and direct numerical simulation, we reveal that the bright-dark and dark-dark solitons with various combinations of parameters under perturbations have qualitative differences. Particularly, vector solitons in unstable background with different wave numbers present distinct dynamics evolutions. The results help us to understand soliton dynamics with oscillatory phase backgrounds and the superposition between nonlinear waves.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328103","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 : 2024-09-19DOI: 10.1016/j.matcom.2024.09.006
In this paper, we derive a kinetic description of swarming particle dynamics in an interacting multi-agent system featuring emerging leaders and followers. Agents are classically characterized by their position and velocity plus a continuous parameter quantifying their degree of leadership. The microscopic processes ruling the change of velocity and degree of leadership are independent, non-conservative and non-local in the physical space, so as to account for long-range interactions. Out of the kinetic description, we obtain then a macroscopic model under a hydrodynamic limit reminiscent of that used to tackle the hydrodynamics of weakly dissipative granular gases, thus relying in particular on a regime of small non-conservative and short-range interactions. Numerical simulations in one- and two-dimensional domains show that the limiting macroscopic model is consistent with the original particle dynamics and furthermore can reproduce classical emerging patterns typically observed in swarms.
{"title":"Kinetic description and macroscopic limit of swarming dynamics with continuous leader–follower transitions","authors":"","doi":"10.1016/j.matcom.2024.09.006","DOIUrl":"10.1016/j.matcom.2024.09.006","url":null,"abstract":"<div><div>In this paper, we derive a kinetic description of swarming particle dynamics in an interacting multi-agent system featuring emerging leaders and followers. Agents are classically characterized by their position and velocity plus a continuous parameter quantifying their degree of leadership. The microscopic processes ruling the change of velocity and degree of leadership are independent, non-conservative and non-local in the physical space, so as to account for long-range interactions. Out of the kinetic description, we obtain then a macroscopic model under a hydrodynamic limit reminiscent of that used to tackle the hydrodynamics of weakly dissipative granular gases, thus relying in particular on a regime of small non-conservative and short-range interactions. Numerical simulations in one- and two-dimensional domains show that the limiting macroscopic model is consistent with the original particle dynamics and furthermore can reproduce classical emerging patterns typically observed in swarms.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-19DOI: 10.1016/j.matcom.2024.09.008
This study introduces a novel stochastic SICR (susceptible, infected, cervical cancer and recovered) model to illustrate HPV (Human papillomavirus) infection dynamics and its impact on cervical cancer in the female population of India. We prove the existence of a unique positive global solution that ensures stochastic boundedness and permanence. Moreover, sufficient conditions for HPV extinction are established through the stochastic extinction parameter , indicating that the infection will die out if . Conversely, the persistence of HPV is established by the existence and uniqueness of an ergodic stationary distribution of the solution when the stochastic threshold , using the suitable selection of Lyapunov functions. Furthermore, data on cervical cancer cases in India from 2016 to 2020 is fitted to the model, providing parameter values suitable for the region. The theoretical findings are validated using the Positive-Preserving Truncated Euler–Maruyama method. Additionally, effective control strategies for India are suggested based on model predictions and sensitivity of key parameters.
{"title":"A study of stochastically perturbed epidemic model of HPV infection and cervical cancer in Indian female population","authors":"","doi":"10.1016/j.matcom.2024.09.008","DOIUrl":"10.1016/j.matcom.2024.09.008","url":null,"abstract":"<div><div>This study introduces a novel stochastic SICR (susceptible, infected, cervical cancer and recovered) model to illustrate HPV (Human papillomavirus) infection dynamics and its impact on cervical cancer in the female population of India. We prove the existence of a unique positive global solution that ensures stochastic boundedness and permanence. Moreover, sufficient conditions for HPV extinction are established through the stochastic extinction parameter <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>e</mi></mrow></msubsup></math></span>, indicating that the infection will die out if <span><math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>e</mi></mrow></msubsup><mo><</mo><mn>1</mn></mrow></math></span>. Conversely, the persistence of HPV is established by the existence and uniqueness of an ergodic stationary distribution of the solution when the stochastic threshold <span><math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>></mo><mn>1</mn></mrow></math></span>, using the suitable selection of Lyapunov functions. Furthermore, data on cervical cancer cases in India from 2016 to 2020 is fitted to the model, providing parameter values suitable for the region. The theoretical findings are validated using the Positive-Preserving Truncated Euler–Maruyama method. Additionally, effective control strategies for India are suggested based on model predictions and sensitivity of key parameters.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328102","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 : 2024-09-19DOI: 10.1016/j.matcom.2024.09.012
In this paper, the space-time generalized finite difference scheme is applied to solve the nonlinear high-order Korteweg-de Vries equations in multiple dimensions. The proposed numerical scheme combines the space-time generalized finite difference method, the Levenberg-Marquardt algorithm, and a time-marching approach. The space-time generalized finite difference method treats the temporal axis as a spatial axis, enabling the proposed scheme to discretize all derivatives in the governing equation. This is accomplished through Taylor series expansion and the moving least squares method. Due to the expandability of the Taylor series to any order, the proposed numerical scheme excels in efficiently handling mixed and higher-order derivatives. These capabilities are distinct advantages of the proposed scheme. The resulting system of algebraic equations is sparse but overdetermined. Therefore, the Levenberg-Marquardt algorithm is directly applied to solve this nonlinear algebraic system. During the calculation process, the time-marching approach reduces computational effort and improves efficiency by dividing the space-time domain.
{"title":"A space-time generalized finite difference scheme for long wave propagation based on high-order Korteweg-de Vries type equations","authors":"","doi":"10.1016/j.matcom.2024.09.012","DOIUrl":"10.1016/j.matcom.2024.09.012","url":null,"abstract":"<div><div>In this paper, the space-time generalized finite difference scheme is applied to solve the nonlinear high-order Korteweg-de Vries equations in multiple dimensions. The proposed numerical scheme combines the space-time generalized finite difference method, the Levenberg-Marquardt algorithm, and a time-marching approach. The space-time generalized finite difference method treats the temporal axis as a spatial axis, enabling the proposed scheme to discretize all derivatives in the governing equation. This is accomplished through Taylor series expansion and the moving least squares method. Due to the expandability of the Taylor series to any order, the proposed numerical scheme excels in efficiently handling mixed and higher-order derivatives. These capabilities are distinct advantages of the proposed scheme. The resulting system of algebraic equations is sparse but overdetermined. Therefore, the Levenberg-Marquardt algorithm is directly applied to solve this nonlinear algebraic system. During the calculation process, the time-marching approach reduces computational effort and improves efficiency by dividing the space-time domain.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312014","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 : 2024-09-18DOI: 10.1016/j.matcom.2024.09.010
In this work, we propose, analyze, and solve a generalization of the -dominating set problem in a graph, when we consider a weighted graph. Given a graph with weights in its edges, a set of vertices is a -weighted dominating set if for every vertex outside the set, the sum of the weights from it to its adjacent vertices in the set is bigger than or equal to . The -weighted domination number is the minimum cardinality among all -weighted dominating sets. Since the problem of finding the -weighted domination number is -hard, we have proposed several problem-adapted construction and reconstruction techniques and embedded them in an Iterated Greedy algorithm. The resulting sixteen variants of the Iterated Greedy algorithm have been compared with an exact algorithm. Computational results show that the proposal is able to find optimal or near-optimal solutions within a short computational time. To the best of our knowledge, the -weighted dominating set problem has never been studied before in the literature and, therefore, there is no other state-of-the-art algorithm to solve it. We have also included a comparison with a particular case of our problem, the minimum dominating set problem and, on average, we achieve same quality results within around 50% of computation time.
在这项研究中，我们提出、分析并解决了图中 k 主集问题的一般化，即考虑加权图。给定一个边上有权重的图，如果该图外的每个顶点到图中相邻顶点的权重之和大于或等于 k，则该顶点集是一个 k 加权支配集。由于求 k 加权支配数的问题是 NP 难问题，我们提出了几种与问题相适应的构造和重构技术，并将它们嵌入到迭代贪婪算法中。我们将迭代贪心算法的 16 个变体与精确算法进行了比较。计算结果表明，该建议能够在较短的计算时间内找到最优或接近最优的解决方案。据我们所知，文献中从未研究过 k 加权支配集问题，因此也没有其他最先进的算法来解决这个问题。我们还将其与我们问题的一种特殊情况--最小支配集问题--进行了比较，平均而言，我们只用了大约 50% 的计算时间就获得了相同质量的结果。
{"title":"Finding the minimum k-weighted dominating sets using heuristic algorithms","authors":"","doi":"10.1016/j.matcom.2024.09.010","DOIUrl":"10.1016/j.matcom.2024.09.010","url":null,"abstract":"<div><div>In this work, we propose, analyze, and solve a generalization of the <span><math><mi>k</mi></math></span>-dominating set problem in a graph, when we consider a weighted graph. Given a graph with weights in its edges, a set of vertices is a <span><math><mi>k</mi></math></span>-weighted dominating set if for every vertex outside the set, the sum of the weights from it to its adjacent vertices in the set is bigger than or equal to <span><math><mi>k</mi></math></span>. The <span><math><mi>k</mi></math></span>-weighted domination number is the minimum cardinality among all <span><math><mi>k</mi></math></span>-weighted dominating sets. Since the problem of finding the <span><math><mi>k</mi></math></span>-weighted domination number is <span><math><mi>NP</mi></math></span>-hard, we have proposed several problem-adapted construction and reconstruction techniques and embedded them in an Iterated Greedy algorithm. The resulting sixteen variants of the Iterated Greedy algorithm have been compared with an exact algorithm. Computational results show that the proposal is able to find optimal or near-optimal solutions within a short computational time. To the best of our knowledge, the <span><math><mi>k</mi></math></span>-weighted dominating set problem has never been studied before in the literature and, therefore, there is no other state-of-the-art algorithm to solve it. We have also included a comparison with a particular case of our problem, the minimum dominating set problem and, on average, we achieve same quality results within around 50% of computation time.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":null,"pages":null},"PeriodicalIF":4.4,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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