Pub Date : 2025-08-01Epub Date: 2025-06-27DOI: 10.1016/j.rinam.2025.100603
Yutaka Sakuma, Yan Linn Aung
In this paper, we consider an queue, where arriving customers decide whether to join the queue or not join based on the queue length at arrival instants. Kerner (2008, Stochastic Models) studies the queue, and derives a recursive formula for the Laplace-Stieltjes transform (LST, for short) of the conditional distribution of the server’s residual service time, given the queue length at arrival instants. This paper aims to analyze the queue in a much simpler way than the previous studies, and to show that our LST of the conditional distribution of the server’s residual service time is given in a more numerically stable form than that of the previous studies, specifically by avoiding the indeterminate form such as . We then use the formula to compute the customers joining probabilities in Nash equilibrium.
{"title":"A numerically stable formula for the conditional distribution of the residual service time in the Mn/PH/1 queue","authors":"Yutaka Sakuma, Yan Linn Aung","doi":"10.1016/j.rinam.2025.100603","DOIUrl":"10.1016/j.rinam.2025.100603","url":null,"abstract":"<div><div>In this paper, we consider an <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>P</mi><mi>H</mi><mo>/</mo><mn>1</mn></mrow></math></span> queue, where arriving customers decide whether to join the queue or not join based on the queue length at arrival instants. Kerner (2008, <em>Stochastic Models</em>) studies the <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>G</mi><mo>/</mo><mn>1</mn></mrow></math></span> queue, and derives a recursive formula for the Laplace-Stieltjes transform (LST, for short) of the conditional distribution of the server’s residual service time, given the queue length at arrival instants. This paper aims to analyze the <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>P</mi><mi>H</mi><mo>/</mo><mn>1</mn></mrow></math></span> queue in a much simpler way than the previous studies, and to show that our LST of the conditional distribution of the server’s residual service time is given in a more numerically stable form than that of the previous studies, specifically by avoiding the indeterminate form such as <span><math><mrow><mn>0</mn><mo>/</mo><mn>0</mn></mrow></math></span>. We then use the formula to compute the customers joining probabilities in Nash equilibrium.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100603"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144500879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-08-16DOI: 10.1016/j.rinam.2025.100626
Emer Lopera , Leandro Recôva , Adolfo Rumbos
In this paper, we study a class of problems proposed by Servadei and Valdinoci (2013); namely, (1)where is an open bounded set with Lipschitz boundary, , , with for , and is a non-local integrodifferential operator with homogeneous Dirichlet boundary condition. By computing the critical groups of the associated energy functional for problem (1) at the origin and at infinity, respectively, we prove that problem (1) has three nontrivial solutions for the case and two nontrivial solutions for the case where is the first eigenvalue of the operator . Finally, assuming that the nonlinearity is odd in the second variable, we prove the existence of an unbounded sequence of weak solutions of problem (1) for the case . We use variational methods and infinite-dimensional Morse theory to obtain the results.
{"title":"Multiplicity results for non-local operators of elliptic type","authors":"Emer Lopera , Leandro Recôva , Adolfo Rumbos","doi":"10.1016/j.rinam.2025.100626","DOIUrl":"10.1016/j.rinam.2025.100626","url":null,"abstract":"<div><div>In this paper, we study a class of problems proposed by Servadei and Valdinoci (2013); namely, <span><span><span>(1)</span><span><math><mfenced><mrow><mtable><mtr><mtd><mo>−</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>K</mi></mrow></msub><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>λ</mi><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd><mtd><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo><mtext>for</mtext><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>;</mo></mtd></mtr><mtr><mtd><mi>u</mi></mtd><mtd><mo>=</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>∖</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> is an open bounded set with Lipschitz boundary, <span><math><mrow><mi>λ</mi><mo>∈</mo><mi>R</mi></mrow></math></span>, <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>¯</mo></mover><mo>×</mo><mi>R</mi><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> for <span><math><mrow><mi>x</mi><mo>∈</mo><mi>Ω</mi></mrow></math></span>, and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> is a non-local integrodifferential operator with homogeneous Dirichlet boundary condition. By computing the critical groups of the associated energy functional for problem <span><span>(1)</span></span> at the origin and at infinity, respectively, we prove that problem <span><span>(1)</span></span> has three nontrivial solutions for the case <span><math><mrow><mi>λ</mi><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> and two nontrivial solutions for the case <span><math><mrow><mi>λ</mi><mo>⩾</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo></mrow></math></span> where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is the first eigenvalue of the operator <span><math><mrow><mo>−</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>K</mi></mrow></msub></mrow></math></span>. Finally, assuming that the nonlinearity <span><math><mi>f</mi></math></span> is odd in the second variable, we prove the existence of an unbounded sequence of weak solutions of problem <span><span>(1)</span></span> for the case <span><math><mrow><mi>λ</mi><mo>⩾</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>. We use variational methods and infinite-dimensional Morse theory to obtain the results.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100626"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144852245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-09-08DOI: 10.1016/j.rinam.2025.100635
Hamid Momeni , AllahBakhsh Yazdani Cherati , Ali Valinejad
This paper investigates data-driven solutions and parameter discovery to (2+ 1)-dimensional coupled nonlinear Schrödinger equations with variable coefficients (VC-CNLSEs), which describe transverse effects in optical fiber systems under perturbed dispersion and nonlinearity. By setting different forms of perturbation coefficients, we aim to recover the dark and anti-dark one- and two-soliton structures by employing an enhanced physics-based deep neural network algorithm, namely a physics-informed neural network (PINN). The enhanced PINN algorithm leverages the locally adaptive activation function mechanism to improve convergence speed and accuracy. In the lack of data acquisition, the PINN algorithms will enhance the capability of the neural networks by incorporating physical information into the training phase. We demonstrate that applying PINN algorithms to (2+ 1)-dimensional VC-CNLSEs requires distinct distributions of physical information. To address this, we propose a region-specific weighted loss function with the help of residual-based adaptive refinement strategy. In the meantime, we perform data-driven parameter discovery for the model equation, classified into two categories: constant coefficient discovery and variable coefficient discovery. For the former, we aim to predict the cross-phase modulation constant coefficient under varying noise intensities using enhanced PINN with a single neural network. For the latter, we employ a dual-network strategy to predict the dynamic behavior of the dispersion and nonlinearity perturbation functions. Our study demonstrates that the proposed framework holds significant potential for studying high-dimensional and complex solitonic dynamics in optical fiber systems.
{"title":"Enhanced PINNs for data-driven solitons and parameter discovery for (2+ 1)-dimensional coupled nonlinear Schrödinger systems","authors":"Hamid Momeni , AllahBakhsh Yazdani Cherati , Ali Valinejad","doi":"10.1016/j.rinam.2025.100635","DOIUrl":"10.1016/j.rinam.2025.100635","url":null,"abstract":"<div><div>This paper investigates data-driven solutions and parameter discovery to (2+ 1)-dimensional coupled nonlinear Schrödinger equations with variable coefficients (VC-CNLSEs), which describe transverse effects in optical fiber systems under perturbed dispersion and nonlinearity. By setting different forms of perturbation coefficients, we aim to recover the dark and anti-dark one- and two-soliton structures by employing an enhanced physics-based deep neural network algorithm, namely a physics-informed neural network (PINN). The enhanced PINN algorithm leverages the locally adaptive activation function mechanism to improve convergence speed and accuracy. In the lack of data acquisition, the PINN algorithms will enhance the capability of the neural networks by incorporating physical information into the training phase. We demonstrate that applying PINN algorithms to (2+ 1)-dimensional VC-CNLSEs requires distinct distributions of physical information. To address this, we propose a region-specific weighted loss function with the help of residual-based adaptive refinement strategy. In the meantime, we perform data-driven parameter discovery for the model equation, classified into two categories: constant coefficient discovery and variable coefficient discovery. For the former, we aim to predict the cross-phase modulation constant coefficient under varying noise intensities using enhanced PINN with a single neural network. For the latter, we employ a dual-network strategy to predict the dynamic behavior of the dispersion and nonlinearity perturbation functions. Our study demonstrates that the proposed framework holds significant potential for studying high-dimensional and complex solitonic dynamics in optical fiber systems.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100635"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-07-12DOI: 10.1016/j.rinam.2025.100614
Fredrik Armerin
We derive an explicit formula for the moment generating function of a Brownian motion with drift reflected from above in one barrier. Some other properties of this stochastic process are also reported.
我们导出了一个带有漂移的布朗运动的力矩生成函数的显式公式。本文还报道了这一随机过程的其他一些性质。
{"title":"The moment generating function of a reflected Brownian motion with drift","authors":"Fredrik Armerin","doi":"10.1016/j.rinam.2025.100614","DOIUrl":"10.1016/j.rinam.2025.100614","url":null,"abstract":"<div><div>We derive an explicit formula for the moment generating function of a Brownian motion with drift reflected from above in one barrier. Some other properties of this stochastic process are also reported.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100614"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144604876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we construct a system dynamics model to study the sustainable evolution of glacier ecotourism systems under environmental change. We calculate based on the carbon-temperature equilibrium and based on the reproduction number method in epidemiological models, and prove that the zero equilibrium is globally asymptotically stable when and , and the error dynamics with respect to the positive equilibrium are globally uniformly ultimately bounded when both and . Empirical validation based on data from the Mendenhall Glacier is conducted to support the theoretical analysis.
{"title":"Stability analysis of a dynamical model for sustainable Glacier ecotourism","authors":"Jianbang He , Jiyue Zhang , Mazheze Xu , Zhongxiang Chen","doi":"10.1016/j.rinam.2025.100636","DOIUrl":"10.1016/j.rinam.2025.100636","url":null,"abstract":"<div><div>In this paper, we construct a system dynamics model to study the sustainable evolution of glacier ecotourism systems under environmental change. We calculate <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> based on the carbon-temperature equilibrium and <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> based on the reproduction number method in epidemiological models, and prove that the zero equilibrium is globally asymptotically stable when <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>C</mi></mrow></msub><mo><</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>≤</mo><mn>1</mn></mrow></math></span>, and the error dynamics with respect to the positive equilibrium are globally uniformly ultimately bounded when both <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>C</mi></mrow></msub><mo><</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>></mo><mn>1</mn></mrow></math></span>. Empirical validation based on data from the Mendenhall Glacier is conducted to support the theoretical analysis.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100636"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-08-22DOI: 10.1016/j.rinam.2025.100629
Juan Diego Pulgarín Rivera , Daniel Turizo , Elias D. Nino-Ruiz , Oscar Danilo Montoya
This paper proposes an A-stable one-stage Rosenbrock method for the solution of Ordinary Differential Equations (ODEs). In this method, Jacobians are approximated via complex step finite differences. An asymptotically accurate estimator of the truncation error is also provided. This error estimator can be employed to control step sizes and to perform extrapolation, which increases the accuracy of the method and yields L-stability. Numerical experiments are conducted to assess the performance of the proposed method. ODE solvers and several stiff ODE problems from the current literature are employed as references during experiments. Experimental results reveal that the proposed method exhibits superior performance with respect to the other compared methods, especially for crude error tolerances.
{"title":"Improved Rosenbrock method with error estimator and Jacobian approximation using complex step","authors":"Juan Diego Pulgarín Rivera , Daniel Turizo , Elias D. Nino-Ruiz , Oscar Danilo Montoya","doi":"10.1016/j.rinam.2025.100629","DOIUrl":"10.1016/j.rinam.2025.100629","url":null,"abstract":"<div><div>This paper proposes an A-stable one-stage Rosenbrock method for the solution of Ordinary Differential Equations (ODEs). In this method, Jacobians are approximated via complex step finite differences. An asymptotically accurate estimator of the truncation error is also provided. This error estimator can be employed to control step sizes and to perform extrapolation, which increases the accuracy of the method and yields L-stability. Numerical experiments are conducted to assess the performance of the proposed method. ODE solvers and several stiff ODE problems from the current literature are employed as references during experiments. Experimental results reveal that the proposed method exhibits superior performance with respect to the other compared methods, especially for crude error tolerances.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100629"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144889781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-08-31DOI: 10.1016/j.rinam.2025.100633
Jiaoxia Huang , Yonghui Qin
The differential operator eigenvalue problems often arise in the field of physics and engineering, such as solid band structure, electron orbitals of atoms or molecules, and quantum bound states. In this paper, the spectral Galerkin method based on a least squares setting is developed for solving the differential operator eigenvalue problems. The proposed scheme leads to a global symmetric positive definite algebraic eigenvalue problem. Two kinds of Schur complement methods are given to deal with the corresponding algebraic equation. Namely, the global block matrix can be decomposed into a local matrix eigenvalue problem. Numerical results are given to verify the effectiveness and high-order accuracy of the proposed scheme. The proposed methods are also effective for solving the three-dimensional problem. We also consider the applications of the proposed methods to solve the eigenvalue problems with a parameter and the -differential operator eigenvalue problems
{"title":"The spectral Galerkin method for the differential operator eigenvalue problems based on a least-squares form and its Schur complement type implementation methods","authors":"Jiaoxia Huang , Yonghui Qin","doi":"10.1016/j.rinam.2025.100633","DOIUrl":"10.1016/j.rinam.2025.100633","url":null,"abstract":"<div><div>The differential operator eigenvalue problems often arise in the field of physics and engineering, such as solid band structure, electron orbitals of atoms or molecules, and quantum bound states. In this paper, the spectral Galerkin method based on a least squares setting is developed for solving the differential operator eigenvalue problems. The proposed scheme leads to a global symmetric positive definite algebraic eigenvalue problem. Two kinds of Schur complement methods are given to deal with the corresponding algebraic equation. Namely, the global block matrix can be decomposed into a local matrix eigenvalue problem. Numerical results are given to verify the effectiveness and high-order accuracy of the proposed scheme. The proposed methods are also effective for solving the three-dimensional problem. We also consider the applications of the proposed methods to solve the eigenvalue problems with a parameter and the <span><math><mrow><mi>g</mi><mi>r</mi><mi>a</mi><mi>d</mi><mrow><mo>(</mo><mi>d</mi><mi>i</mi><mi>v</mi><mo>)</mo></mrow></mrow></math></span>-differential operator eigenvalue problems</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100633"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144920161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the Galerkin method for obtaining approximate solutions to linear Fredholm integral equations of the second kind. The finite element solution is represented as a linear combination of basis functions, and the construction of suitable basis functions plays a crucial role in the accuracy of the approximation. We propose an optimal interpolation formula that exactly reproduces the functions and , and derive basis functions from its coefficients. This interpolation formula is constructed within the Hilbert space . To evaluate the effectiveness of the proposed approach, we solve several integral equations using the Galerkin method with two types of basis functions: the newly constructed exponential basis and classical piecewise linear basis functions. Numerical experiments are presented to compare the accuracy of these approaches. Graphs and tables illustrate the approximation errors, demonstrating that both basis functions achieve an error order of , with the optimal interpolation-based basis yielding superior accuracy in certain cases.
{"title":"The numerical solution of a Fredholm integral equation of the second kind using the Galerkin method based on optimal interpolation","authors":"Samandar Babaev , Abdullo Hayotov , Asliddin Boltaev , Surayyo Mirzoyeva , Malika Mirzaeva","doi":"10.1016/j.rinam.2025.100607","DOIUrl":"10.1016/j.rinam.2025.100607","url":null,"abstract":"<div><div>In this paper, we study the Galerkin method for obtaining approximate solutions to linear Fredholm integral equations of the second kind. The finite element solution is represented as a linear combination of basis functions, and the construction of suitable basis functions plays a crucial role in the accuracy of the approximation. We propose an optimal interpolation formula that exactly reproduces the functions <span><math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>x</mi></mrow></msup></math></span>, and derive basis functions from its coefficients. This interpolation formula is constructed within the Hilbert space <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mrow></msubsup></math></span>. To evaluate the effectiveness of the proposed approach, we solve several integral equations using the Galerkin method with two types of basis functions: the newly constructed exponential basis and classical piecewise linear basis functions. Numerical experiments are presented to compare the accuracy of these approaches. Graphs and tables illustrate the approximation errors, demonstrating that both basis functions achieve an error order of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>h</mi><mo>)</mo></mrow></mrow></math></span>, with the optimal interpolation-based basis yielding superior accuracy in certain cases.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100607"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-07-22DOI: 10.1016/j.rinam.2025.100617
Bing Tan, Yingzhe Fan
This paper investigates the global well-posedness of the Cauchy problem for the Vlasov–Fokker–Planck equation coupled with the incompressible Euler system around a normalized global Maxwellian in a periodic spatial domain. The system describes the interaction between a fluid governed by Euler equations and a particle distribution evolving under the VFP dynamics, with coupling through a drag force. We establish the existence and uniqueness of global mild solutions for small initial data in a low regularity function space by employing Fourier analysis.
Compare to the Navier–Stokes–Vlasov-Fokker–Planck system (Tan and Fan, 2023) where velocity dissipation estimates can be directly derived from the viscous term, the Vlasov–Euler–Fokker–Planck system lacks such direct accessibility to velocity dissipation due to its inherent structural differences. To overcome this obstacle, we need to exploit the macroscopic dissipation inherent in the macroscopic equation. Then the dissipation of velocity is indirectly captured by combining the macroscopic dissipation of and the linear dissipation of within the equation. Finally the uniform energy functionals of the solution can be obtained by utilizing the refined energy estimate.
本文研究了周期空间域上Vlasov-Fokker-Planck方程与不可压缩欧拉系统在规格化全局麦克斯韦方程组周围耦合的Cauchy问题的全局适定性。该系统描述了由欧拉方程控制的流体与在VFP动力学下演化的粒子分布之间的相互作用，并通过阻力进行耦合。利用傅里叶分析，建立了低正则性函数空间Lk1LT∞Lv2上小初始数据全局温和解的存在唯一性。与Navier-Stokes-Vlasov-Fokker-Planck系统（Tan and Fan, 2023）相比，Vlasov-Euler-Fokker-Planck系统由于其固有的结构差异，无法直接获得速度耗散估计。在Navier-Stokes-Vlasov-Fokker-Planck系统中，可以直接从粘性项中导出速度耗散估计。为了克服这个障碍，我们需要利用宏观方程中固有的宏观耗散b。然后结合方程中b的宏观耗散和u−b的线性耗散，间接捕捉速度耗散。最后利用精化的能量估计得到解的均匀能量泛函。
{"title":"Global existence for the Vlasov–Euler–Fokker–Planck system in low-regularity space","authors":"Bing Tan, Yingzhe Fan","doi":"10.1016/j.rinam.2025.100617","DOIUrl":"10.1016/j.rinam.2025.100617","url":null,"abstract":"<div><div>This paper investigates the global well-posedness of the Cauchy problem for the Vlasov–Fokker–Planck equation coupled with the incompressible Euler system around a normalized global Maxwellian in a periodic spatial domain. The system describes the interaction between a fluid governed by Euler equations and a particle distribution evolving under the VFP dynamics, with coupling through a drag force. We establish the existence and uniqueness of global mild solutions for small initial data in a low regularity function space <span><math><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>T</mi></mrow><mrow><mi>∞</mi></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></math></span> by employing Fourier analysis.</div><div>Compare to the Navier–Stokes–Vlasov-Fokker–Planck system (Tan and Fan, 2023) where velocity dissipation estimates can be directly derived from the viscous term, the Vlasov–Euler–Fokker–Planck system lacks such direct accessibility to velocity dissipation due to its inherent structural differences. To overcome this obstacle, we need to exploit the macroscopic dissipation <span><math><mi>b</mi></math></span> inherent in the macroscopic equation. Then the dissipation of velocity is indirectly captured by combining the macroscopic dissipation of <span><math><mi>b</mi></math></span> and the linear dissipation of <span><math><mrow><mi>u</mi><mo>−</mo><mi>b</mi></mrow></math></span> within the equation. Finally the uniform energy functionals of the solution can be obtained by utilizing the refined energy estimate.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100617"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on the portfolio optimization problem in the presence of the European options in the illiquid market. To do this, we extract the features of the market data using the statistical test to design a general financial model. After that, applying the dynamic replicating portfolio strategy, we derive a comprehensive partial integral differential equation for European option pricing in the illiquid market where the jump part of the model follows the empirical distribution. Since the structure of the equation is complex, we use the finite difference method to solve it. Furthermore, we apply the MCVaR portfolio optimization model with the short selling constraint to obtain the optimal portfolio strategy according to the risk tolerance amounts of the investors. Finally, we find the optimal portfolio under different amounts of the model’s parameters based on the S&P market data.
{"title":"Portfolio optimization in the illiquid market using the empirical distribution","authors":"Pouya Fakhraeipour, Farshid Mehrdoust, Alireza Najafi","doi":"10.1016/j.rinam.2025.100611","DOIUrl":"10.1016/j.rinam.2025.100611","url":null,"abstract":"<div><div>This paper focuses on the portfolio optimization problem in the presence of the European options in the illiquid market. To do this, we extract the features of the market data using the statistical test to design a general financial model. After that, applying the dynamic replicating portfolio strategy, we derive a comprehensive partial integral differential equation for European option pricing in the illiquid market where the jump part of the model follows the empirical distribution. Since the structure of the equation is complex, we use the finite difference method to solve it. Furthermore, we apply the MCVaR portfolio optimization model with the short selling constraint to obtain the optimal portfolio strategy according to the risk tolerance amounts of the investors. Finally, we find the optimal portfolio under different amounts of the model’s parameters based on the S&P market data.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100611"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144500878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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