We derive the solution of the one dimensional wave equation for the Dirichlet and Robin initial-boundary value problems (IBVPs) formulated on the half line and the finite interval, with nonhomogeneous boundary conditions. Although explicit formulas already exist for these problems, the unified transform method provides a convenient framework for deriving different representations of the solutions for these and other types of IBVPs. Specifically, it provides solution formulas in the Fourier space or solutions which constitute the extension of the classical formula of d’Alembert of the initial value problem on the full line. We also derive the solution of the forced wave equation on the half line.
{"title":"Extensions of the d’Alembert formulae to the half line and the finite interval obtained via the unified transform","authors":"A. S. Fokas, K. Kalimeris","doi":"10.1093/imamat/hxac030","DOIUrl":"https://doi.org/10.1093/imamat/hxac030","url":null,"abstract":"\u0000 We derive the solution of the one dimensional wave equation for the Dirichlet and Robin initial-boundary value problems (IBVPs) formulated on the half line and the finite interval, with nonhomogeneous boundary conditions. Although explicit formulas already exist for these problems, the unified transform method provides a convenient framework for deriving different representations of the solutions for these and other types of IBVPs. Specifically, it provides solution formulas in the Fourier space or solutions which constitute the extension of the classical formula of d’Alembert of the initial value problem on the full line. We also derive the solution of the forced wave equation on the half line.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46109417","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}
Gossip algorithms and their accelerated versions have been studied exclusively in discrete time on graphs. In this work, we take a different approach, and consider the scaling limit of gossip algorithms in both large graphs and large number of iterations. These limits lead to well-known partial differential equations (PDEs) with insightful properties. On lattices, we prove that the non-accelerated gossip algorithm of (??) converges to the heat equation, and the accelerated Jacobi polynomial iteration of (??) converges to the Euler–Poisson–Darboux (EPD) equation — a damped wave equation. Remarkably, with appropriate parameters, the fundamental solution of the EPD equation has the ideal gossip behaviour: a uniform density over an ellipsoid, whose radius increases at a rate proportional to $t$ — the fastest possible rate for locally communicating gossip algorithms. This is in contrast with the heat equation where the density spreads on a typical scale of $sqrt {t}$. Additionally, we provide simulations demonstrating that the gossip algorithms are accurately approximated by their limiting PDEs.
{"title":"Acceleration of Gossip Algorithms through the Euler–Poisson–Darboux Equation","authors":"Raphaël Berthier, Mufan (Bill) Li","doi":"10.1093/imamat/hxac029","DOIUrl":"https://doi.org/10.1093/imamat/hxac029","url":null,"abstract":"Gossip algorithms and their accelerated versions have been studied exclusively in discrete time on graphs. In this work, we take a different approach, and consider the scaling limit of gossip algorithms in both large graphs and large number of iterations. These limits lead to well-known partial differential equations (PDEs) with insightful properties. On lattices, we prove that the non-accelerated gossip algorithm of (??) converges to the heat equation, and the accelerated Jacobi polynomial iteration of (??) converges to the Euler–Poisson–Darboux (EPD) equation — a damped wave equation. Remarkably, with appropriate parameters, the fundamental solution of the EPD equation has the ideal gossip behaviour: a uniform density over an ellipsoid, whose radius increases at a rate proportional to $t$ — the fastest possible rate for locally communicating gossip algorithms. This is in contrast with the heat equation where the density spreads on a typical scale of $sqrt {t}$. Additionally, we provide simulations demonstrating that the gossip algorithms are accurately approximated by their limiting PDEs.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"65 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138538076","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}
Active-dissipative evolution equations emerge in a variety of physical and technological applications including liquid film flows, flame propagation, epitaxial film growth in materials manufacturing, to mention a few. They are characterised by three main ingredients: a term producing growth (active), a term providing damping at short length scales (dissipative), and a nonlinear term that transfers energy between modes and crucially produces a nonlinear saturation. The manifestation of these three mechanisms can produce large-time spatiotemporal chaos as evidenced by the Kuramoto-Sivashinsky equation (negative diffusion, fourth order dissipation, and a Burgers nonlinearity), which is arguably the simplest partial differential equation to produce chaos. The exact form of the terms (and in particular their Fourier symbol) determines the type of attractors that the equations possess. The present study considers the spatial analyticity of solutions under the assumption that the equations possess a global attractor. In particular we investigate the spatial analyticity of solutions of a class of one-dimensional evolutionary pseudo-differential equations with Burgers nonlinearity, which are periodic in space, thus generalising the Kuramoto-Sivashinsky equation motivated by both applications and their fundamental mathematical properties. Analyticity is examined by utilising a criterion involving the rate of growth of suitable norms of the $n$-th spatial derivative of the solution, with respect to the spatial variable, as $n$ tends to infinity. An estimate of the rate of growth of the $n$-th spatial derivative is obtained by fine-tuning the spectral method, developed elsewhere. We prove that the solutions are analytic if $gamma $, the order of dissipation of the pseudo-differential operator, is higher than one. We also present numerical evidence suggesting that this is optimal, i.e., if $gamma $ is not larger that one, then the solution is not in general analytic. Extensive numerical experiments are undertaken to confirm the analysis and also to compute the band of analyticity of solutions for a wide range of active-dissipative terms and large spatial periods that support chaotic solutions. These ideas can be applied to a wide class of active-dissipative-dispersive pseudo-differential equations.
{"title":"Optimal analyticity estimates for non-linear active-dissipative evolution equations","authors":"D. Papageorgiou, Y. Smyrlis, R. Tomlin","doi":"10.1093/imamat/hxac028","DOIUrl":"https://doi.org/10.1093/imamat/hxac028","url":null,"abstract":"\u0000 Active-dissipative evolution equations emerge in a variety of physical and technological applications including liquid film flows, flame propagation, epitaxial film growth in materials manufacturing, to mention a few. They are characterised by three main ingredients: a term producing growth (active), a term providing damping at short length scales (dissipative), and a nonlinear term that transfers energy between modes and crucially produces a nonlinear saturation. The manifestation of these three mechanisms can produce large-time spatiotemporal chaos as evidenced by the Kuramoto-Sivashinsky equation (negative diffusion, fourth order dissipation, and a Burgers nonlinearity), which is arguably the simplest partial differential equation to produce chaos. The exact form of the terms (and in particular their Fourier symbol) determines the type of attractors that the equations possess. The present study considers the spatial analyticity of solutions under the assumption that the equations possess a global attractor. In particular we investigate the spatial analyticity of solutions of a class of one-dimensional evolutionary pseudo-differential equations with Burgers nonlinearity, which are periodic in space, thus generalising the Kuramoto-Sivashinsky equation motivated by both applications and their fundamental mathematical properties. Analyticity is examined by utilising a criterion involving the rate of growth of suitable norms of the $n$-th spatial derivative of the solution, with respect to the spatial variable, as $n$ tends to infinity. An estimate of the rate of growth of the $n$-th spatial derivative is obtained by fine-tuning the spectral method, developed elsewhere. We prove that the solutions are analytic if $gamma $, the order of dissipation of the pseudo-differential operator, is higher than one. We also present numerical evidence suggesting that this is optimal, i.e., if $gamma $ is not larger that one, then the solution is not in general analytic. Extensive numerical experiments are undertaken to confirm the analysis and also to compute the band of analyticity of solutions for a wide range of active-dissipative terms and large spatial periods that support chaotic solutions. These ideas can be applied to a wide class of active-dissipative-dispersive pseudo-differential equations.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45254697","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 derive conditions for a one-term fourth-order Sturm–Liouville operator on a finite interval with one singular endpoint to have essential spectrum equal to $[0,infty )$ or $varnothing $. Of particular usefulness are Kummer–Liouville transformations which have been a valuable tool in the study of second-order equations. Applications to a mechanical beam with a thickness tapering to zero at one of the endpoints are considered. When the thickness $2h$ satisfies $c_1x^{nu }leq h(x)leq c_2x^{nu }$, we show that the essential spectrum is empty if and only if $nu < 2$. As a final application, we consider a tapered beam on a Winkler foundation and derive sufficient conditions on the beam thickness and the foundational rigidity to guarantee the essential spectrum is equal to $[0,infty )$.
{"title":"Singular fourth-order Sturm–Liouville operators and acoustic black holes","authors":"B. Belinskiy, D. Hinton, R. Nichols","doi":"10.1093/imamat/hxac021","DOIUrl":"https://doi.org/10.1093/imamat/hxac021","url":null,"abstract":"\u0000 We derive conditions for a one-term fourth-order Sturm–Liouville operator on a finite interval with one singular endpoint to have essential spectrum equal to $[0,infty )$ or $varnothing $. Of particular usefulness are Kummer–Liouville transformations which have been a valuable tool in the study of second-order equations. Applications to a mechanical beam with a thickness tapering to zero at one of the endpoints are considered. When the thickness $2h$ satisfies $c_1x^{nu }leq h(x)leq c_2x^{nu }$, we show that the essential spectrum is empty if and only if $nu < 2$. As a final application, we consider a tapered beam on a Winkler foundation and derive sufficient conditions on the beam thickness and the foundational rigidity to guarantee the essential spectrum is equal to $[0,infty )$.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49351707","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}
Based on the inverse scattering transformation, we carry out spectral analysis of the $4times 4$ matrix spectral problems related to the Hermitian symmetric space Fokas-Lenells equation, by which the solution of the Cauchy problem of the Hermitian symmetric space Fokas-Lenells equation is transformed into the solution of a Riemann-Hilbert problem. The nonlinear steepest descent method is extended to study the Riemann-Hilbert problem, from which the various Deift-Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann-Hilbert problems and strict error estimates, we obtain explicitly the long-time asymptotics of the Cauchy problem of the Hermitian symmetric space Fokas-Lenells equation with the aid of the parabolic cylinder function. Keywords: Hermitian symmetric space Fokas-Lenells equation; Nonlinear steepest descent method; Spectral analysis; Long-time asymptotics.
{"title":"The Hermitian symmetric space Fokas-Lenells equation: spectral analysis and long-time asymptotics","authors":"Xianguo Geng, Kedong Wang, Mingming Chen","doi":"10.1093/imamat/hxac025","DOIUrl":"https://doi.org/10.1093/imamat/hxac025","url":null,"abstract":"Based on the inverse scattering transformation, we carry out spectral analysis of the $4times 4$ matrix spectral problems related to the Hermitian symmetric space Fokas-Lenells equation, by which the solution of the Cauchy problem of the Hermitian symmetric space Fokas-Lenells equation is transformed into the solution of a Riemann-Hilbert problem. The nonlinear steepest descent method is extended to study the Riemann-Hilbert problem, from which the various Deift-Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann-Hilbert problems and strict error estimates, we obtain explicitly the long-time asymptotics of the Cauchy problem of the Hermitian symmetric space Fokas-Lenells equation with the aid of the parabolic cylinder function. Keywords: Hermitian symmetric space Fokas-Lenells equation; Nonlinear steepest descent method; Spectral analysis; Long-time asymptotics.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"38 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138538075","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}
Inspired by the turf-ball interaction in golf, this paper seeks to understand the bounce of a ball that can be modelled as a rigid sphere and the surface as supplying a viscoelastic contact force in addition to Coulomb friction. A general formulation is proposed that models the finite time interval of bounce from touch-down to lift-off. Key to the analysis is understanding transitions between slip and roll during the bounce. Starting from the rigid-body limit with a an energetic or Poisson coefficient of restitution, it is shown that slip reversal during the contact phase cannot be captured in this case, which result generalises to the case of pure normal compliance. Yet, the introduction of linear tangential stiffness and damping, does enable slip reversal. This result is extended to general weakly nonlinear normal and tangential compliance. An analysis using Filippov theory of piecewise-smooth systems leads to an argument in a natural limit that lift-off while rolling is non-generic and that almost all trajectories that lift off, do so under slip conditions. Moreover, there is a codimension-one surface in the space of incoming velocity and spin which divides balls that lift off with backspin from those that lift off with topspin. The results are compared with recent experimental measurements on golf ball bounce and the theory is shown to capture the main features of the data.
{"title":"Analysis of point-contact models of the bounce of a hard spinning ball on a compliant frictional surface.","authors":"Stanislaw W. Biber, A. Champneys, R. Szalai","doi":"10.1093/imamat/hxad020","DOIUrl":"https://doi.org/10.1093/imamat/hxad020","url":null,"abstract":"\u0000 Inspired by the turf-ball interaction in golf, this paper seeks to understand the bounce of a ball that can be modelled as a rigid sphere and the surface as supplying a viscoelastic contact force in addition to Coulomb friction. A general formulation is proposed that models the finite time interval of bounce from touch-down to lift-off. Key to the analysis is understanding transitions between slip and roll during the bounce. Starting from the rigid-body limit with a an energetic or Poisson coefficient of restitution, it is shown that slip reversal during the contact phase cannot be captured in this case, which result generalises to the case of pure normal compliance. Yet, the introduction of linear tangential stiffness and damping, does enable slip reversal. This result is extended to general weakly nonlinear normal and tangential compliance. An analysis using Filippov theory of piecewise-smooth systems leads to an argument in a natural limit that lift-off while rolling is non-generic and that almost all trajectories that lift off, do so under slip conditions. Moreover, there is a codimension-one surface in the space of incoming velocity and spin which divides balls that lift off with backspin from those that lift off with topspin. The results are compared with recent experimental measurements on golf ball bounce and the theory is shown to capture the main features of the data.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44195862","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}
This work is devoted to the study of the inverse photoacoustic tomography (PAT) problem. It is an imaging technique similar to TAT studied in El Badia & Ha-Duong (2000); however, in this case, a high-frequency radiation is delivered into the biological tissue to be imaged, such as visible or near infra red light that are characterized by their high frequency compared with that of radio waves that are used in TAT. As in the case of TAT El Badia & Ha-Duong (2000), the inverse problem we are concerned in is the reconstruction of small absorbers in an open, bounded and connected domain $Omega subset{mathbb{R}}^3$. Again, we follow the algebraic algorithm, initially proposed in El Badia & Jebawy (2020), that allows us to resolve the problem from a single Cauchy data and without the knowledge of the Grüneisen’s coefficient. However, the high-frequency radiation used in this case makes some changes in the context of the problem and allows us to give our results using partial boundary observations and in both cases of constant and variable acoustic speed. Finally, we establish the corresponding Hölder stability result.
本文致力于逆光声层析成像(PAT)问题的研究。这是一种与El Badia & Ha-Duong(2000)研究的TAT类似的成像技术;然而，在这种情况下，高频辐射被传送到要成像的生物组织中，例如可见光或近红外光，与TAT中使用的无线电波相比，它们的特点是频率高。在TAT El Badia和Ha-Duong(2000)的案例中，我们所关注的逆问题是在一个开放的、有界的和连通的域$Omega 子集{mathbb{R}}^3$中重建小吸收体。同样，我们遵循最初在El Badia和Jebawy(2020)中提出的代数算法，该算法使我们能够从单个柯西数据中解决问题，而不需要知道尼森系数。然而，在这种情况下使用的高频辐射在问题的背景下做出了一些改变，并允许我们使用部分边界观测以及在恒定和变声速两种情况下给出我们的结果。最后建立了相应的Hölder稳定性结果。
{"title":"On an inverse photoacoustic tomography problem of small absorbers with inhomogeneous sound speed","authors":"Hanin AL Jebawy, A. El Badia","doi":"10.1093/imamat/hxac017","DOIUrl":"https://doi.org/10.1093/imamat/hxac017","url":null,"abstract":"\u0000 This work is devoted to the study of the inverse photoacoustic tomography (PAT) problem. It is an imaging technique similar to TAT studied in El Badia & Ha-Duong (2000); however, in this case, a high-frequency radiation is delivered into the biological tissue to be imaged, such as visible or near infra red light that are characterized by their high frequency compared with that of radio waves that are used in TAT. As in the case of TAT El Badia & Ha-Duong (2000), the inverse problem we are concerned in is the reconstruction of small absorbers in an open, bounded and connected domain $Omega subset{mathbb{R}}^3$. Again, we follow the algebraic algorithm, initially proposed in El Badia & Jebawy (2020), that allows us to resolve the problem from a single Cauchy data and without the knowledge of the Grüneisen’s coefficient. However, the high-frequency radiation used in this case makes some changes in the context of the problem and allows us to give our results using partial boundary observations and in both cases of constant and variable acoustic speed. Finally, we establish the corresponding Hölder stability result.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46915124","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}
Fine-grain patterns produced by juxtacrine signalling have previously been studied using static monolayers as cellular domains. However, analytic results are usually restricted to a few cells due to the algebraic complexity of non-linear dynamical systems. Motivated by concentric patterning of Notch expression observed in the mammary gland, we combine concepts from graph and control theory to represent cellular connectivity and behaviour. The resulting theoretical framework allows us to exploit the symmetry of multicellular bilayer structures in 2D and 3D, thereby deriving analytical conditions that drive the dynamical system to form laminar patterns, consistent with the formation of cell polarity by activator localization. Critically, the patterning conditions are independent of the precise dynamical details, thus the framework allows for generality in understanding the influence of cellular geometry and signal polarity on patterning using lateral-inhibition systems. Applying the analytic conditions to mammary organoids suggests that intense cell signalling polarity is required for the maintenance of stratified cell types within a static bilayer using a lateral-inhibition mechanism. Furthermore, by employing 2D and 3D cell-based models, we highlight that the cellular polarity conditions derived from static domains can generate laminar patterning in dynamic environments. However, they are insufficient for the maintenance of patterning when subjected to substantial morphological perturbations. In agreement with the mathematical implications of strict signalling polarity induced on the cells, we propose an adhesion-dependent Notch-Delta biological process that has the potential to initiate bilayer stratification in a developing mammary organoid.
{"title":"Polarity-driven laminar pattern formation by lateral-inhibition in 2D and 3D bilayer geometries","authors":"Joshua W Moore, T. Dale, T. Woolley","doi":"10.1093/imamat/hxac011","DOIUrl":"https://doi.org/10.1093/imamat/hxac011","url":null,"abstract":"\u0000 Fine-grain patterns produced by juxtacrine signalling have previously been studied using static monolayers as cellular domains. However, analytic results are usually restricted to a few cells due to the algebraic complexity of non-linear dynamical systems. Motivated by concentric patterning of Notch expression observed in the mammary gland, we combine concepts from graph and control theory to represent cellular connectivity and behaviour. The resulting theoretical framework allows us to exploit the symmetry of multicellular bilayer structures in 2D and 3D, thereby deriving analytical conditions that drive the dynamical system to form laminar patterns, consistent with the formation of cell polarity by activator localization. Critically, the patterning conditions are independent of the precise dynamical details, thus the framework allows for generality in understanding the influence of cellular geometry and signal polarity on patterning using lateral-inhibition systems. Applying the analytic conditions to mammary organoids suggests that intense cell signalling polarity is required for the maintenance of stratified cell types within a static bilayer using a lateral-inhibition mechanism. Furthermore, by employing 2D and 3D cell-based models, we highlight that the cellular polarity conditions derived from static domains can generate laminar patterning in dynamic environments. However, they are insufficient for the maintenance of patterning when subjected to substantial morphological perturbations. In agreement with the mathematical implications of strict signalling polarity induced on the cells, we propose an adhesion-dependent Notch-Delta biological process that has the potential to initiate bilayer stratification in a developing mammary organoid.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47240635","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}
H. L. Oliveira, S. McKee, G. Buscaglia, J. Cuminato, I. Stewart, D. Wheatley
This note extends previous work of the authors modelling the Wheatley valve by using six intersecting and contiguous ellipses to obtain a generalized mathematical representation of the Wheatley valve: this provides a number of free parameters that could be employed to obtain an optimal design. Since optimality is multi-objective with many of the objectives conflicting we focus on the stresses imposed on the valve by a constant force field. Three distinctly different designs are chosen and an analysis of the stresses is undertaken, conclusions are drawn and results are discussed.
{"title":"A Generalized mathematical representation of the shape of the Wheatley heart valve and the associated static stress fields upon opening and closing","authors":"H. L. Oliveira, S. McKee, G. Buscaglia, J. Cuminato, I. Stewart, D. Wheatley","doi":"10.1093/imamat/hxac016","DOIUrl":"https://doi.org/10.1093/imamat/hxac016","url":null,"abstract":"\u0000 This note extends previous work of the authors modelling the Wheatley valve by using six intersecting and contiguous ellipses to obtain a generalized mathematical representation of the Wheatley valve: this provides a number of free parameters that could be employed to obtain an optimal design. Since optimality is multi-objective with many of the objectives conflicting we focus on the stresses imposed on the valve by a constant force field. Three distinctly different designs are chosen and an analysis of the stresses is undertaken, conclusions are drawn and results are discussed.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45412807","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}
A simple and explicit expression of the solution of the SIR epidemiological model of Kermack and McKendrick is constructed in the asymptotic limit of large basic reproduction numbers ${mathsf R_0}$. The proposed formula yields good qualitative agreement already when ${mathsf R_0}geq 3$ and rapidly becomes quantitatively accurate as larger values of ${mathsf R_0}$ are assumed. The derivation is based on the method of matched asymptotic expansions, which exploits the fact that the exponential growing phase and the eventual recession of the outbreak occur on distinct time scales. From the newly derived solution, an analytical estimate of the time separating the first inflexion point of the epidemic curve from the peak of infections is given. Finally, we use the same method on the SEIR model and find that the inclusion of the ‘exposed’ population in the model can dramatically alter the time scales of the outbreak.
{"title":"Asymptotic solutions of the SIR and SEIR models well above the epidemic threshold","authors":"G. Kozyreff","doi":"10.1093/imamat/hxac015","DOIUrl":"https://doi.org/10.1093/imamat/hxac015","url":null,"abstract":"\u0000 A simple and explicit expression of the solution of the SIR epidemiological model of Kermack and McKendrick is constructed in the asymptotic limit of large basic reproduction numbers ${mathsf R_0}$. The proposed formula yields good qualitative agreement already when ${mathsf R_0}geq 3$ and rapidly becomes quantitatively accurate as larger values of ${mathsf R_0}$ are assumed. The derivation is based on the method of matched asymptotic expansions, which exploits the fact that the exponential growing phase and the eventual recession of the outbreak occur on distinct time scales. From the newly derived solution, an analytical estimate of the time separating the first inflexion point of the epidemic curve from the peak of infections is given. Finally, we use the same method on the SEIR model and find that the inclusion of the ‘exposed’ population in the model can dramatically alter the time scales of the outbreak.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47858522","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}
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