Fermín S. V. Bazán, Luciano Bedin, Mansur I. Ismailov, Leonardo S. Borges
In this work, we consider the problem of recovering the heat source term for the heat equation with a nonlocal Wentzell-Neumann boundary condition subject to an integral overdetermination condition. Conditions for the existence and uniqueness of the classical solution of the inverse problem are revisited, and a numerical method for practical source reconstruction is introduced. Unlike all of the source reconstruction methods found in literature, the method introduced in this work computes regularized solutions from a triangular linear system arising from a semi-discretization in the space of the continuous model. Regularization is introduced by applying the generalized singular value decomposition of a proper matrix pair along with truncation. Numerical results illustrate the effectiveness of the method.
{"title":"Inverse time-dependent source problem for the heat equation with a nonlocal Wentzell-Neumann boundary condition","authors":"Fermín S. V. Bazán, Luciano Bedin, Mansur I. Ismailov, Leonardo S. Borges","doi":"10.3934/nhm.2023076","DOIUrl":"https://doi.org/10.3934/nhm.2023076","url":null,"abstract":"<abstract><p>In this work, we consider the problem of recovering the heat source term for the heat equation with a nonlocal Wentzell-Neumann boundary condition subject to an integral overdetermination condition. Conditions for the existence and uniqueness of the classical solution of the inverse problem are revisited, and a numerical method for practical source reconstruction is introduced. Unlike all of the source reconstruction methods found in literature, the method introduced in this work computes regularized solutions from a triangular linear system arising from a semi-discretization in the space of the continuous model. Regularization is introduced by applying the generalized singular value decomposition of a proper matrix pair along with truncation. Numerical results illustrate the effectiveness of the method.</p></abstract>","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, two high-order compact difference schemes with graded meshes are proposed for solving the time-fractional Black-Scholes equation. We first eliminate the convection term in the equivalent form of the considered equation by using exponential transformation, then combine the sixth-order/eighth-order compact difference method with a temporal graded meshes-based trapezoidal formulation for the temporal integral term to obtain the fully discrete high-order compact difference schemes. The stability and convergence analysis of the two proposed schemes are studied by applying Fourier analysis. Finally, the effectiveness of the proposed schemes and the correctness of the theoretical results are verified by two numerical examples.
{"title":"Two high-order compact difference schemes with temporal graded meshes for time-fractional Black-Scholes equation","authors":"Jie Gu, Lijuan Nong, Qian Yi, An Chen","doi":"10.3934/nhm.2023074","DOIUrl":"https://doi.org/10.3934/nhm.2023074","url":null,"abstract":"<abstract><p>In this paper, two high-order compact difference schemes with graded meshes are proposed for solving the time-fractional Black-Scholes equation. We first eliminate the convection term in the equivalent form of the considered equation by using exponential transformation, then combine the sixth-order/eighth-order compact difference method with a temporal graded meshes-based trapezoidal formulation for the temporal integral term to obtain the fully discrete high-order compact difference schemes. The stability and convergence analysis of the two proposed schemes are studied by applying Fourier analysis. Finally, the effectiveness of the proposed schemes and the correctness of the theoretical results are verified by two numerical examples.</p></abstract>","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136303432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Kamal, M. Alsolmi, Nayabuddin, Aned Al Mutairi, Eslam Hussam, M. Mustafa, S. G. Nassr
This paper introduces the generalized exponential-$ U $ family of distributions as a novel methodological approach to enhance the distributional flexibility of existing classical and modified distributions. The new family is derived by combining the T-$ X $ family method with the exponential model. The paper presents the generalized exponential-Weibull model, an updated version of the Weibull model. Estimators and heavy-tailed characteristics of the proposed method are derived. The new model is applied to three healthcare data sets, including COVID-19 patient survival times and mortality rate data set from Mexico and Holland. The proposed model outperforms other models in terms of analyzing healthcare data sets by evaluating the best model selection measures. The findings suggest that the proposed model holds promise for broader utilization in the area of predicting and modeling healthcare phenomena.
本文介绍了广义指数-$ U $分布族,作为一种新的方法来提高现有经典分布和修正分布的分布灵活性。将T-$ X $族方法与指数模型相结合,得到了新的族。本文提出了广义指数威布尔模型,这是威布尔模型的更新版本。推导了该方法的估计量和重尾特性。新模型应用于三个医疗保健数据集,包括墨西哥和荷兰的COVID-19患者生存时间和死亡率数据集。通过评估最佳模型选择度量,所提出的模型在分析医疗数据集方面优于其他模型。研究结果表明,该模型有望在医疗保健现象的预测和建模领域得到更广泛的应用。
{"title":"A new distributional approach: estimation, Monte Carlo simulation and applications to the biomedical data sets","authors":"M. Kamal, M. Alsolmi, Nayabuddin, Aned Al Mutairi, Eslam Hussam, M. Mustafa, S. G. Nassr","doi":"10.3934/nhm.2023069","DOIUrl":"https://doi.org/10.3934/nhm.2023069","url":null,"abstract":"This paper introduces the generalized exponential-$ U $ family of distributions as a novel methodological approach to enhance the distributional flexibility of existing classical and modified distributions. The new family is derived by combining the T-$ X $ family method with the exponential model. The paper presents the generalized exponential-Weibull model, an updated version of the Weibull model. Estimators and heavy-tailed characteristics of the proposed method are derived. The new model is applied to three healthcare data sets, including COVID-19 patient survival times and mortality rate data set from Mexico and Holland. The proposed model outperforms other models in terms of analyzing healthcare data sets by evaluating the best model selection measures. The findings suggest that the proposed model holds promise for broader utilization in the area of predicting and modeling healthcare phenomena.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70228410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the problem of non-preemptively scheduling jobs from two agents on an unbounded serial-batch machine. Agents $ A $ and $ B $ have $ n_A $ and $ n_B $ jobs. The machine can process any number of jobs sequentially as a batch, and the processing time of the batch is equal to the total processing time of the jobs in it. Each batch requires a setup time before it is processed. Compatibility means that the jobs from different agents can be processed in a common batch; Otherwise, the jobs from different agents are incompatible. Both the compatible and incompatible models are considered, under both the batch availability and item availability assumptions. Batch availability means that any job in a batch is not available until all the jobs in this batch are completed. Item availability means that a job in a batch becomes available immediately after it is completed processing. The completion time of a job is defined to be the moment when it is available. The goal is to minimize the makespan of agent $ A $ and the maximum lateness of agent $ B $ simultaneously. For the compatible model with batch availability, an $ O(n_A+n_B^2log n_B) $-time algorithm is presented which improves the existing $ O(n_A+n_B^4log n_B) $-time algorithm. A slight modification of the algorithm solves the incompatible model with batch availability in $ O(n_A+n_B^2log n_B) $ time, which has the same time complexity as the existing algorithm. For the compatible model with item availability, the analysis shows that it is easy and admits an $ O(n_A+n_Blog n_B) $-time algorithm. For the incompatible model with item availability, an $ O(n_A+n_Blog n_B) $-time algorithm is also obtained which improves the existing $ O(n_A+n_B^2) $-time algorithm. The algorithms can generate all Pareto optimal points and find a corresponding Pareto optimal schedule for each Pareto optimal point.
{"title":"Algorithms for two-agent unbounded serial-batch scheduling with makespan and maximum lateness objectives","authors":"Shuguang Li, Mingsong Li, Muhammad Ijaz Khan","doi":"10.3934/nhm.2023073","DOIUrl":"https://doi.org/10.3934/nhm.2023073","url":null,"abstract":"<abstract><p>We study the problem of non-preemptively scheduling jobs from two agents on an unbounded serial-batch machine. Agents $ A $ and $ B $ have $ n_A $ and $ n_B $ jobs. The machine can process any number of jobs sequentially as a batch, and the processing time of the batch is equal to the total processing time of the jobs in it. Each batch requires a setup time before it is processed. Compatibility means that the jobs from different agents can be processed in a common batch; Otherwise, the jobs from different agents are incompatible. Both the compatible and incompatible models are considered, under both the batch availability and item availability assumptions. Batch availability means that any job in a batch is not available until all the jobs in this batch are completed. Item availability means that a job in a batch becomes available immediately after it is completed processing. The completion time of a job is defined to be the moment when it is available. The goal is to minimize the makespan of agent $ A $ and the maximum lateness of agent $ B $ simultaneously. For the compatible model with batch availability, an $ O(n_A+n_B^2log n_B) $-time algorithm is presented which improves the existing $ O(n_A+n_B^4log n_B) $-time algorithm. A slight modification of the algorithm solves the incompatible model with batch availability in $ O(n_A+n_B^2log n_B) $ time, which has the same time complexity as the existing algorithm. For the compatible model with item availability, the analysis shows that it is easy and admits an $ O(n_A+n_Blog n_B) $-time algorithm. For the incompatible model with item availability, an $ O(n_A+n_Blog n_B) $-time algorithm is also obtained which improves the existing $ O(n_A+n_B^2) $-time algorithm. The algorithms can generate all Pareto optimal points and find a corresponding Pareto optimal schedule for each Pareto optimal point.</p></abstract>","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135799803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Pilyugin, M. Tarasova, Aleksandr S. Tarasov, G. Monakov
In this paper, we study a model of opinion dynamics based on the so-called "bounded confidence" principle introduced by Hegselmann and Krause. Following this principle, voters participating in an electoral decision with two options are influenced by individuals sharing an opinion similar to their own.We consider a modification of this model where the operator generating the dynamical system which describes the process of formation the final distribution of opinions in the society is defined in two steps. First, to the opinion of an agent, a value proportional to opinions in his/her "influence group" is added, and then the elements of the resulting array are divided by the maximal absolute value of elements to keep the opinions in the prescribed interval. We show that under appropriate conditions, any trajectory tends to a fixed point, and all the remaining fixed points are Lyapunov stable.
{"title":"A model of voting dynamics under bounded confidence with nonstandard norming","authors":"S. Pilyugin, M. Tarasova, Aleksandr S. Tarasov, G. Monakov","doi":"10.3934/nhm.2022032","DOIUrl":"https://doi.org/10.3934/nhm.2022032","url":null,"abstract":"In this paper, we study a model of opinion dynamics based on the so-called \"bounded confidence\" principle introduced by Hegselmann and Krause. Following this principle, voters participating in an electoral decision with two options are influenced by individuals sharing an opinion similar to their own.We consider a modification of this model where the operator generating the dynamical system which describes the process of formation the final distribution of opinions in the society is defined in two steps. First, to the opinion of an agent, a value proportional to opinions in his/her \"influence group\" is added, and then the elements of the resulting array are divided by the maximal absolute value of elements to keep the opinions in the prescribed interval. We show that under appropriate conditions, any trajectory tends to a fixed point, and all the remaining fixed points are Lyapunov stable.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70227935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the following work, we consider the Boltzmann equation that models a diatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section begin{document}$ mathcal{B} $end{document}, we prove that the linearized Boltzmann operator begin{document}$ mathcal{L} $end{document} of this model is a Fredholm operator. For this, we write begin{document}$ mathcal{L} $end{document} as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator begin{document}$ mathcal{K} $end{document} is compact. The result is established after inspecting the kernel form of begin{document}$ mathcal{K} $end{document} and proving it to be begin{document}$ L^2 $end{document} integrable over its domain using elementary arguments.This implies that begin{document}$ mathcal{K} $end{document} is a Hilbert-Schmidt operator.
In the following work, we consider the Boltzmann equation that models a diatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section begin{document}$ mathcal{B} $end{document}, we prove that the linearized Boltzmann operator begin{document}$ mathcal{L} $end{document} of this model is a Fredholm operator. For this, we write begin{document}$ mathcal{L} $end{document} as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator begin{document}$ mathcal{K} $end{document} is compact. The result is established after inspecting the kernel form of begin{document}$ mathcal{K} $end{document} and proving it to be begin{document}$ L^2 $end{document} integrable over its domain using elementary arguments.This implies that begin{document}$ mathcal{K} $end{document} is a Hilbert-Schmidt operator.
{"title":"Compactness property of the linearized Boltzmann operator for a diatomic single gas model","authors":"S. Brull, Marwa Shahine, P. Thieullen","doi":"10.3934/nhm.2022029","DOIUrl":"https://doi.org/10.3934/nhm.2022029","url":null,"abstract":"<p style='text-indent:20px;'>In the following work, we consider the Boltzmann equation that models a diatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section <inline-formula><tex-math id=\"M1\">begin{document}$ mathcal{B} $end{document}</tex-math></inline-formula>, we prove that the linearized Boltzmann operator <inline-formula><tex-math id=\"M2\">begin{document}$ mathcal{L} $end{document}</tex-math></inline-formula> of this model is a Fredholm operator. For this, we write <inline-formula><tex-math id=\"M3\">begin{document}$ mathcal{L} $end{document}</tex-math></inline-formula> as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator <inline-formula><tex-math id=\"M4\">begin{document}$ mathcal{K} $end{document}</tex-math></inline-formula> is compact. The result is established after inspecting the kernel form of <inline-formula><tex-math id=\"M5\">begin{document}$ mathcal{K} $end{document}</tex-math></inline-formula> and proving it to be <inline-formula><tex-math id=\"M6\">begin{document}$ L^2 $end{document}</tex-math></inline-formula> integrable over its domain using elementary arguments.This implies that <inline-formula><tex-math id=\"M7\">begin{document}$ mathcal{K} $end{document}</tex-math></inline-formula> is a Hilbert-Schmidt operator.</p>","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70228121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a conical body facing a supersonic stream of air at a uniform velocity. When the opening angle of the obstacle cone is small, the conical shock wave is attached to the vertex. Under the assumption of self-similarity for irrotational motions, the Euler system is transformed into the nonlinear ODE system. We reformulate the problem in a non-dimensional form and analyze the corresponding ODE system. The initial data is given on the obstacle cone and the solution is integrated until the Rankine-Hugoniot condition is satisfied on the shock cone. By applying the fundamental theory of ODE systems and technical estimates, we construct supersonic solutions and also show that no matter how small the opening angle is, a smooth transonic solution always exists as long as the speed of the incoming flow is suitably chosen for this given angle.
{"title":"Smooth Transonic Flows Around Cones","authors":"W. Lien, Yu-Yu Liu, Chen-Chang Peng","doi":"10.3934/nhm.2022028","DOIUrl":"https://doi.org/10.3934/nhm.2022028","url":null,"abstract":"We consider a conical body facing a supersonic stream of air at a uniform velocity. When the opening angle of the obstacle cone is small, the conical shock wave is attached to the vertex. Under the assumption of self-similarity for irrotational motions, the Euler system is transformed into the nonlinear ODE system. We reformulate the problem in a non-dimensional form and analyze the corresponding ODE system. The initial data is given on the obstacle cone and the solution is integrated until the Rankine-Hugoniot condition is satisfied on the shock cone. By applying the fundamental theory of ODE systems and technical estimates, we construct supersonic solutions and also show that no matter how small the opening angle is, a smooth transonic solution always exists as long as the speed of the incoming flow is suitably chosen for this given angle.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70228071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-05DOI: 10.1016/b978-0-12-819880-3.00007-x
{"title":"Notations","authors":"","doi":"10.1016/b978-0-12-819880-3.00007-x","DOIUrl":"https://doi.org/10.1016/b978-0-12-819880-3.00007-x","url":null,"abstract":"","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81230984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1016/B978-0-12-819880-3.00013-5
S. Kanaun
{"title":"Cracks in heterogeneous media","authors":"S. Kanaun","doi":"10.1016/B978-0-12-819880-3.00013-5","DOIUrl":"https://doi.org/10.1016/B978-0-12-819880-3.00013-5","url":null,"abstract":"","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81564655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1016/b978-0-12-819880-3.00017-2
{"title":"Index","authors":"","doi":"10.1016/b978-0-12-819880-3.00017-2","DOIUrl":"https://doi.org/10.1016/b978-0-12-819880-3.00017-2","url":null,"abstract":"","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86045878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}