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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
E. Luckins, James M. Oliver, C. Please, Benjamin M. Sloman, A. Valderhaug, R. V. Van Gorder
Modelling the production of silicon in a submerged arc furnace (SAF) requires accounting for the wide range of timescales of the different physical and chemical processes: the electric current which is used to heat the furnace varies over a timescale of around $10^{-2},$ s, whereas the flow and chemical consumption of the raw materials in the furnace occurs over several hours. Models for the silicon furnace generally either include only the fast-timescale, or only the slow-timescale processes. In a prior work, we developed a model incorporating effects on both the fast and slow timescales, and used a multiple-timescales analysis to homogenise the fast variations, deriving an averaged model for the slow evolution of the raw materials. For simplicity, in the previous work we focussed on the electrical behaviour around the base of a single electrode, and prescribed the current in this electrode to be sinusoidal, with given amplitude. In this paper, we extend our previous analysis to include the full electrical system, modelled using an equivalent circuit system. In this way, we demonstrate how the two furnace-modelling approaches (on the fast and slow timescales) may be combined in a computationally efficient way. Our previously derived model for the arc resistance is based on the assumption that the dominant heat loss from the arc is by radiation (we will refer to this as the radiation model). Alternative arc models include the empirical Cassie and Mayr models, which are commonly used in the SAF literature. We compare these various arc models, explore the dependence of the solution of our model on the model parameters and compare our solutions with measurements from an operational silicon furnace. In particular, we show that only the radiation arc model has a rising current-voltage characteristic at high currents. Simulations of the model show that there is an upper limit on the length of the furnace arc, above which all the current bypasses the arc and flows through the surrounding material.
{"title":"Modelling alternating current effects in a submerged arc furnace","authors":"E. Luckins, James M. Oliver, C. Please, Benjamin M. Sloman, A. Valderhaug, R. V. Van Gorder","doi":"10.1093/imamat/hxac012","DOIUrl":"https://doi.org/10.1093/imamat/hxac012","url":null,"abstract":"\u0000 Modelling the production of silicon in a submerged arc furnace (SAF) requires accounting for the wide range of timescales of the different physical and chemical processes: the electric current which is used to heat the furnace varies over a timescale of around $10^{-2},$ s, whereas the flow and chemical consumption of the raw materials in the furnace occurs over several hours. Models for the silicon furnace generally either include only the fast-timescale, or only the slow-timescale processes. In a prior work, we developed a model incorporating effects on both the fast and slow timescales, and used a multiple-timescales analysis to homogenise the fast variations, deriving an averaged model for the slow evolution of the raw materials. For simplicity, in the previous work we focussed on the electrical behaviour around the base of a single electrode, and prescribed the current in this electrode to be sinusoidal, with given amplitude. In this paper, we extend our previous analysis to include the full electrical system, modelled using an equivalent circuit system. In this way, we demonstrate how the two furnace-modelling approaches (on the fast and slow timescales) may be combined in a computationally efficient way. Our previously derived model for the arc resistance is based on the assumption that the dominant heat loss from the arc is by radiation (we will refer to this as the radiation model). Alternative arc models include the empirical Cassie and Mayr models, which are commonly used in the SAF literature. We compare these various arc models, explore the dependence of the solution of our model on the model parameters and compare our solutions with measurements from an operational silicon furnace. In particular, we show that only the radiation arc model has a rising current-voltage characteristic at high currents. Simulations of the model show that there is an upper limit on the length of the furnace arc, above which all the current bypasses the arc and flows through the surrounding material.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44163717","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 examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation for solutions to half-line and finite-interval initial-boundary value problems as integrals of kernels with explicit spatial and temporal dependence. These solution representations are defined within the spatial domain of the problem. We obtain the extension of these representation formulae via Taylor series outside these spatial domains and find the extension of the initial condition that gives rise to a whole-line initial-value problem solved by the extended solution. In general, the extended initial condition is not differentiable or continuous unless the boundary and initial conditions satisfy compatibility conditions. We analyze dissipative and dispersive problems, and problems with continuous and discrete spatial variables.
{"title":"The analytic extension of solutions to initial-boundary value problems outside their domain of definition","authors":"Matthew Farkas, J. Cisneros, B. Deconinck","doi":"10.1093/imamat/hxad007","DOIUrl":"https://doi.org/10.1093/imamat/hxad007","url":null,"abstract":"\u0000 We examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation for solutions to half-line and finite-interval initial-boundary value problems as integrals of kernels with explicit spatial and temporal dependence. These solution representations are defined within the spatial domain of the problem. We obtain the extension of these representation formulae via Taylor series outside these spatial domains and find the extension of the initial condition that gives rise to a whole-line initial-value problem solved by the extended solution. In general, the extended initial condition is not differentiable or continuous unless the boundary and initial conditions satisfy compatibility conditions. We analyze dissipative and dispersive problems, and problems with continuous and discrete spatial variables.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49047344","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 look at the periodic behaviour of the Earth’s glacial cycles and the transitions between different periodic states when either external parameters (such as $omega $) or internal parameters (such as $d$) are varied. We model this using the PP04 model of climate change. This is a forced discontinuous Filippov (non-smooth) dynamical system. When periodically forced this has coexisting periodic orbits. We find that the transitions in this system are mainly due to grazing events, leading to grazing bifurcations. An analysis of the grazing bifurcations is given and the impact of these on the domains of attraction and regions of existence of the periodic orbits is determined under various changes in the parameters of the system. Grazing transitions arise for general variations in the parameters (both internal and external) of the PP04 model. We find that the grazing transitions between the period orbits resemble those of the Mid-Pleistocene-Transition.
{"title":"Grazing bifurcations and transitions between periodic states of the PP04 model for the glacial cycle","authors":"Chris J Budd Kgomotso S. Morupisi","doi":"10.1093/imamat/hxac013","DOIUrl":"https://doi.org/10.1093/imamat/hxac013","url":null,"abstract":"\u0000 We look at the periodic behaviour of the Earth’s glacial cycles and the transitions between different periodic states when either external parameters (such as $omega $) or internal parameters (such as $d$) are varied. We model this using the PP04 model of climate change. This is a forced discontinuous Filippov (non-smooth) dynamical system. When periodically forced this has coexisting periodic orbits. We find that the transitions in this system are mainly due to grazing events, leading to grazing bifurcations. An analysis of the grazing bifurcations is given and the impact of these on the domains of attraction and regions of existence of the periodic orbits is determined under various changes in the parameters of the system. Grazing transitions arise for general variations in the parameters (both internal and external) of the PP04 model. We find that the grazing transitions between the period orbits resemble those of the Mid-Pleistocene-Transition.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44181526","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}
Nastaran Naghshineh, W. Reinberger, N. Barlow, M. Samaha, S. J. Weinstein
We examine the power series solutions of two classical nonlinear ordinary differential equations of fluid mechanics that are mathematically related by their large-distance asymptotic behaviors in semi-infinite domains. The first problem is that of the “Sakiadis” boundary layer over a moving flat wall, for which no exact analytic solution has been put forward. The second problem is that of a static air–liquid meniscus with surface tension that intersects a flat wall at a given contact angle and limits to a flat pool away from the wall. For the latter problem, the exact analytic solution—given as distance from the wall as function of meniscus height—has long been known (Batchelor, 1967). Here, we provide an explicit solution as meniscus height vs. distance from the wall to elucidate structural similarities to the Sakiadis boundary layer. Although power series solutions are readily obtainable to the governing nonlinear ordinary differential equations, we show that—in both problems—the series diverge due to non-physical complex or negative real-valued singularities. In both cases, these singularities can be moved by expanding in exponential gauge functions motivated by their respective large distance asymptotic behaviors to enable series convergence over their full semi-infinite domains. For the Sakiadis problem, this not only provides a convergent Taylor series (and conjectured exact) solution to the ODE, but also a means to evaluate the wall shear parameter (and other properties) to within any desired precision. Although the nature of nonlinear ODEs precludes general conclusions, our results indicate that asymptotic behaviors can be useful when proposing variable transformations to overcome power series divergence. Sakiadis boundary layer; meniscus; asymptotic expansion; summation of series
{"title":"On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs","authors":"Nastaran Naghshineh, W. Reinberger, N. Barlow, M. Samaha, S. J. Weinstein","doi":"10.1093/imamat/hxad006","DOIUrl":"https://doi.org/10.1093/imamat/hxad006","url":null,"abstract":"\u0000 We examine the power series solutions of two classical nonlinear ordinary differential equations of fluid mechanics that are mathematically related by their large-distance asymptotic behaviors in semi-infinite domains. The first problem is that of the “Sakiadis” boundary layer over a moving flat wall, for which no exact analytic solution has been put forward. The second problem is that of a static air–liquid meniscus with surface tension that intersects a flat wall at a given contact angle and limits to a flat pool away from the wall. For the latter problem, the exact analytic solution—given as distance from the wall as function of meniscus height—has long been known (Batchelor, 1967). Here, we provide an explicit solution as meniscus height vs. distance from the wall to elucidate structural similarities to the Sakiadis boundary layer. Although power series solutions are readily obtainable to the governing nonlinear ordinary differential equations, we show that—in both problems—the series diverge due to non-physical complex or negative real-valued singularities. In both cases, these singularities can be moved by expanding in exponential gauge functions motivated by their respective large distance asymptotic behaviors to enable series convergence over their full semi-infinite domains. For the Sakiadis problem, this not only provides a convergent Taylor series (and conjectured exact) solution to the ODE, but also a means to evaluate the wall shear parameter (and other properties) to within any desired precision. Although the nature of nonlinear ODEs precludes general conclusions, our results indicate that asymptotic behaviors can be useful when proposing variable transformations to overcome power series divergence. Sakiadis boundary layer; meniscus; asymptotic expansion; summation of series","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46709109","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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