A nonlinear conjugate gradient method is derived for the inverse problem of identifying a treatment parameter in a nonlinear model of reaction-diffusion type corresponding to the evolution of brain tumors under therapy. The treatment parameter is reconstructed from additional information about the tumour taken at a fixed instance of time. Well-posedness of the direct problems used in the iterative method is outlined as well as uniqueness of a solution to the inverse problem. Moreover, the parameter identification is recast as the minimization of a Tikhonov type functional and the existence of a minimizer to this functional is shown. Finite difference discretization of the space and time derivatives are employed for the numerical implementation. Numerical simulations on full 3-dimensional brain data is included showing that information about a spacewise dependent treatment parameter can be recovered in a stable way.
{"title":"Identifying a response parameter in a model of brain tumor evolution under therapy","authors":"","doi":"10.1093/imamat/hxad013","DOIUrl":"https://doi.org/10.1093/imamat/hxad013","url":null,"abstract":"\u0000 A nonlinear conjugate gradient method is derived for the inverse problem of identifying a treatment parameter in a nonlinear model of reaction-diffusion type corresponding to the evolution of brain tumors under therapy. The treatment parameter is reconstructed from additional information about the tumour taken at a fixed instance of time. Well-posedness of the direct problems used in the iterative method is outlined as well as uniqueness of a solution to the inverse problem. Moreover, the parameter identification is recast as the minimization of a Tikhonov type functional and the existence of a minimizer to this functional is shown. Finite difference discretization of the space and time derivatives are employed for the numerical implementation. Numerical simulations on full 3-dimensional brain data is included showing that information about a spacewise dependent treatment parameter can be recovered in a stable way.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61109929","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}
It is natural that mosquitoes move toward high human population density and environmental heterogeneity plays a pivotal role on disease transmission, and thus we formulate and analyze a mosquito-borne disease model with chemotaxis and spatial heterogeneity. The global existence and boundedness of solutions are proven to guarantee the solvability of the model and is challenging due to the model complexity. Under appropriate conditions, we demonstrate the disease-free equilibrium is globally asymptotically stable provided that the basic reproduction number $mathcal {R}_0$ is less than one, and the system is uniformly persistent and admits at least one endemic equilibrium if $mathcal {R}_0$ is greater than one. Furthermore, we numerically explore the impacts of chemotactic effect, spatial heterogeneity and dispersal rates of infected individuals to provide a clear picture on disease severity. In particular, the mosquito chemotaxis causes disease mild in some regions but severe in others, which suggests developing targeted strategies to control mosquitoes in specific locations and achieves a deep understanding on the chemotaxis.
{"title":"Global threshold dynamics of a spatial chemotactic mosquito-borne disease model","authors":"Kai Wang, Hao Wang, Hongyong Zhao","doi":"10.1093/imamat/hxad009","DOIUrl":"https://doi.org/10.1093/imamat/hxad009","url":null,"abstract":"\u0000 It is natural that mosquitoes move toward high human population density and environmental heterogeneity plays a pivotal role on disease transmission, and thus we formulate and analyze a mosquito-borne disease model with chemotaxis and spatial heterogeneity. The global existence and boundedness of solutions are proven to guarantee the solvability of the model and is challenging due to the model complexity. Under appropriate conditions, we demonstrate the disease-free equilibrium is globally asymptotically stable provided that the basic reproduction number $mathcal {R}_0$ is less than one, and the system is uniformly persistent and admits at least one endemic equilibrium if $mathcal {R}_0$ is greater than one. Furthermore, we numerically explore the impacts of chemotactic effect, spatial heterogeneity and dispersal rates of infected individuals to provide a clear picture on disease severity. In particular, the mosquito chemotaxis causes disease mild in some regions but severe in others, which suggests developing targeted strategies to control mosquitoes in specific locations and achieves a deep understanding on the chemotaxis.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46031581","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}
L. Bourgeois, S. Fliss, Jean-François Fritsch, C. Hazard, A. Recoquillay
This paper is dedicated to an acoustic scattering problem in a two-dimensional partially open waveguide, in the sense that the left part of the waveguide is closed, that is with a bounded cross-section, while the right part is bounded in the transverse direction by some Perfectly Matched Layers that mimic the situation of an open waveguide, that is with an unbounded cross-section. We prove well-posedness of such scattering problem in the Fredholm sense (uniqueness implies existence) and exhibit the asymptotic behaviour of the solution in the longitudinal direction with the help of the Kondratiev approach. Having in mind the numerical computation of the solution, we also propose some transparent boundary conditions in such longitudinal direction, based on Dirichlet-to-Neumann operators. After proving that such artificial conditions actually enable us to approximate the exact solution, some numerical experiments illustrate the quality of such approximation.
{"title":"Scattering in a partially open waveguide: the forward problem","authors":"L. Bourgeois, S. Fliss, Jean-François Fritsch, C. Hazard, A. Recoquillay","doi":"10.1093/imamat/hxad004","DOIUrl":"https://doi.org/10.1093/imamat/hxad004","url":null,"abstract":"\u0000 This paper is dedicated to an acoustic scattering problem in a two-dimensional partially open waveguide, in the sense that the left part of the waveguide is closed, that is with a bounded cross-section, while the right part is bounded in the transverse direction by some Perfectly Matched Layers that mimic the situation of an open waveguide, that is with an unbounded cross-section. We prove well-posedness of such scattering problem in the Fredholm sense (uniqueness implies existence) and exhibit the asymptotic behaviour of the solution in the longitudinal direction with the help of the Kondratiev approach. Having in mind the numerical computation of the solution, we also propose some transparent boundary conditions in such longitudinal direction, based on Dirichlet-to-Neumann operators. After proving that such artificial conditions actually enable us to approximate the exact solution, some numerical experiments illustrate the quality of such approximation.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48882379","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}
{"title":"Correction to: A degenerating convection–diffusion system modelling froth flotation with drainage","authors":"","doi":"10.1093/imamat/hxad001","DOIUrl":"https://doi.org/10.1093/imamat/hxad001","url":null,"abstract":"","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136297796","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}
{"title":"On global in time self-similar solutions of Smoluchowski equation with multiplicative kernel","authors":"G. Breschi, M. Fontelos","doi":"10.1093/imamat/hxad012","DOIUrl":"https://doi.org/10.1093/imamat/hxad012","url":null,"abstract":"\u0000 We study the similarity solutions (SS) of Smoluchowski coagulation equation with multiplicative kernel $K(x,y)=(xy)^{s}$ for $s<frac {1}{2}$. When $s<0$ , the SS consists of three regions with distinct asymptotic behaviours. The appropriate matching yields a global description of the solution consisting of a Gamma distribution tail, an intermediate region described by a lognormal distribution and a region of very fast decay of the solutions to zero near the origin. When $sin left ( 0,frac {1}{2}right ) $, the SS is unbounded at the origin. It also presents three regions: a Gamma distribution tail, an intermediate region of power-like (or Pareto distribution) decay and the region close to the origin where a singularity occurs. Finally, full numerical simulations of Smoluchowski equation serve to verify our theoretical results and show the convergence of solutions to the selfsimilar regime.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44728371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-13eCollection Date: 2022-12-01DOI: 10.1093/imamat/hxac027
Tyler Cassidy, Peter Gillich, Antony R Humphries, Christiaan H van Dorp
Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). In this work, we address the lack of appropriate numerical tools for gamma distributed DDEs in two ways. First, we develop a functional continuous Runge-Kutta (FCRK) method to numerically integrate the gamma distributed DDE without resorting to Erlang approximation. We prove the fourth-order convergence of the FCRK method and perform numerical tests to demonstrate the accuracy of the new numerical method. Nevertheless, FCRK methods for infinite delay DDEs are not widely available in existing scientific software packages. As an alternative approach to solving gamma distributed DDEs, we also derive a hypoexponential approximation of the gamma distributed DDE. This hypoexponential approach is a more accurate approximation of the true gamma distributed DDE than the common Erlang approximation but, like the Erlang approximation, can be formulated as a system of ODEs and solved numerically using standard ODE software. Using our FCRK method to provide reference solutions, we show that the common Erlang approximation may produce solutions that are qualitatively different from the underlying gamma distributed DDE. However, the proposed hypoexponential approximations do not have this limitation. Finally, we apply our hypoexponential approximations to perform statistical inference on synthetic epidemiological data to illustrate the utility of the hypoexponential approximation.
{"title":"Numerical methods and hypoexponential approximations for gamma distributed delay differential equations.","authors":"Tyler Cassidy, Peter Gillich, Antony R Humphries, Christiaan H van Dorp","doi":"10.1093/imamat/hxac027","DOIUrl":"10.1093/imamat/hxac027","url":null,"abstract":"<p><p>Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). In this work, we address the lack of appropriate numerical tools for gamma distributed DDEs in two ways. First, we develop a functional continuous Runge-Kutta (FCRK) method to numerically integrate the gamma distributed DDE without resorting to Erlang approximation. We prove the fourth-order convergence of the FCRK method and perform numerical tests to demonstrate the accuracy of the new numerical method. Nevertheless, FCRK methods for infinite delay DDEs are not widely available in existing scientific software packages. As an alternative approach to solving gamma distributed DDEs, we also derive a hypoexponential approximation of the gamma distributed DDE. This hypoexponential approach is a more accurate approximation of the true gamma distributed DDE than the common Erlang approximation but, like the Erlang approximation, can be formulated as a system of ODEs and solved numerically using standard ODE software. Using our FCRK method to provide reference solutions, we show that the common Erlang approximation may produce solutions that are qualitatively different from the underlying gamma distributed DDE. However, the proposed hypoexponential approximations do not have this limitation. Finally, we apply our hypoexponential approximations to perform statistical inference on synthetic epidemiological data to illustrate the utility of the hypoexponential approximation.</p>","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9850366/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10667425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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