Artículos (Ecuaciones Diferenciales y Análisis Numérico)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/10834

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Some control results for simplified one-dimensional models of fluid-solid interaction
(World Scientific, 2005-05-10) Doubova Krasotchenko, Anna; Fernández Cara, Enrique; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
We analyze the null controllability of a one-dimensional nonlinear system which models the interaction of a fluid and a particle. This can be viewed as a first step in the control analysis of fluid-solid systems. The fluid is governed by the Burgers equation and the control is exerted at the boundary points. We present two main results: the global null controllability of a linearized system and the local null controllability of the nonlinear original model. The proofs rely on appropriate global Carleman inequalities, observability estimates and fixed point arguments.
Rotated weights in global Carleman estimates applied to an inverse problem for the wave equation
(IOS Publishing, 2006-01-30) Doubova Krasotchenko, Anna; Osses, Axel; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
In this paper, we establish geometrical conditions in order to solve an inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data from a single time-dependent Neumann boundary measurement on a suitable part of the boundary. We prove the uniqueness and the stability results for this problem when a Neumann measurement is only located on a part of the boundary satisfying a rotated exit condition. The strategy consists of introducing an angle-type dependence in the weight functions used to obtain global Carleman estimates for the wave equation and combination of several of these estimates and then apply it to the inverse problem.
Reconstruction of degenerate conductivity region for parabolic equations
(IOP Publishing, 2024-03-15) Cannarsa, Piermarco; Doubova Krasotchenko, Anna; Yamamoto, Masahiro; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.
Inverse Problems for One-Dimensional Fluid-Solid Interaction Models
(Springer, 2024-09-10) Apraiz, J.; Doubova Krasotchenko, Anna; Fernández Cara, Enrique; Yamamoto, Masahiro; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
We consider a one-dimensional fluid-solid interaction model governed by the Burgers equation with a time varying interface. We discuss the inverse problem of determining the shape of the interface from Dirichlet and Neumann data at one endpoint of the spatial interval. In particular, we establish unique results and some conditional stability estimates. For the proofs, we use and adapt some lateral estimates that, in turn, rely on appropriate Carleman and interpolation inequalities.
Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation
(IOP Science, 2021-11-04) Cannarsa, Piermarco; Doubova Krasotchenko, Anna; Yamamoto, Masahiro; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
We consider the inverse problem of identification of degenerate diffusion coefficient of the form xαa(x) in a one dimensional parabolic equation by some extra data. We first prove by energy methods the uniqueness and Lipschitz stability results for the identification of a constant coefficient a and the power α by knowing interior data at some time. On the other hand, we obtain the uniqueness result for the identification of a general diffusion coefficients a(x) and also the power α form boundary data on one side of the space interval. The proof is based on global Carleman estimates for a hyperbolic problem and an inversion of the integral transform similar to the Laplace transform. Finally, the theoretical results are satisfactory verified by numerically experiments.
Extinction-time for stochastic population models
(Elsevier, 2014-09-23) Doubova Krasotchenko, Anna; Vadillo, Fernando; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
The analysis of interacting population models is the subject of much interest in mathematical ecology. Moreover, the persistence and extinction of these models is one of the most interesting and important topics, because it provides insight into their behavior. The mean extinction-time for stochastic population models considered in this paper depends on the initial population size and satisfies a stationary partial differential equation, related to the backward Kolmogorov differential equation, a linear second-order partial differential equation with variable coefficients. In this communication we review several papers where we have proposed some numerical techniques in order to estimate the mean extinction-time for stochastic population models. Besides, we will compare the theoretical predictions and numerical simulations for stochastic differential equations (SDEs). This work can be viewed as a unified review of the contributions de la Hoz and Vadillo (2012), de la Hoz et al. (2014) and Doubova and Vadillo (2014).
Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional
(AIMS Press, 2024-02-26) Liu, Ruonan; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown.
Generalized φ-Pullback Attractors for Evolution Processes and Application to a Nonautonomous Wave Equation
(Springer, 2024-03-27) Bertolan, Matheus C.; Caraballo Garrido, Tomás; Pecorari Neto, Carlos; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this work we define the generalized φ-pullback attractors for evolution processes in complete metric spaces, which are compact and positively invariant families, that pullback attract bounded sets with a rate determined by a decreasing function φ that vanishes at infinity, called decay function. We find conditions under which a given evolution process has a generalized φ-pullback attractor, both in the discrete and in the continuous cases. We present a result for the special case of generalized polynomial pullback attractors, and apply it to obtain such an object for a nonautonomous wave equation.
Asymptotic stability of evolution systems of probability measures of stochastic discrete modified Swift–Hohenberg equations
(Springer, 2023-06-22) Wang, Fengling; Caraballo Garrido, Tomás; Li, Yangrong; Wang, Renhai; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
This paper is concerned with the asymptotic stability of evolution systems of probability measures for non-autonomous stochastic discrete modified Swift–Hohenberg equations driven by local Lipschitz nonlinear noise. We first show the existence of evolution systems of probability measures of the original equation. Then, using the theoretical results in Wang et al. (Proc Am Math Soc 151:2449–2458, 2023), it is proved that the evolution system of probability measures of the limit equation is the limit of the evolution system of probability measures when the noise intensity tends to a certain value.
Sufficient and necessary criteria for backward asymptotic autonomy of pullback attractors with applications to retarded sine-Gordon lattice systems
(American Institute of Physics, 2024-05-16) Yang, Shuang; Caraballo Garrido, Tomás; Zhang, Qiangheng; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper, we investigate the backward asymptotic autonomy of pullback attractors for asymptotically autonomous processes. Namely, time-components of the pullback attractors semi-converge to the global attractors of the corresponding limiting semigroups as the time-parameter goes to negative infinity. The present article is divided into two parts: theories and applications. In the theoretical part, we establish a sufficient and necessary criterion with respect to the backward asymptotic autonomy via backward compactness of pullback attractors. Moreover, this backward asymptotic autonomy is considered by the periodicity of pullback attractors. As for the applications part, we apply the abstract results to non-autonomous retarded sine-Gordon lattice systems. By backward uniform tail-estimates of solutions, we prove the existence of a pullback and global attractor for such lattice systems such that the backward asymptotic autonomy is satisfied. Furthermore, it is also fulfilled under the assumptions of the periodicity for the non-delay forcing and the convergence for processes.
Necessary and Sufficient Conditions for the Existence of Positive Periodic Solutions for Neural Networks with Time-Varying and Distributed Delays
(Springer, 2022-01-11) Benhadri, Mimia; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper, the existence of positive periodic solutions of neural networks with time-varying delays is discussed by using the fixed point theory on cones. Some necessary and sufficient conditions guaranteeing the existence of one positive periodic solution of the considered system are established. Finally, we exhibit an example to verify the applicability of our abstract results.
Dynamics and Wong-Zakai Approximations of Stochastic Nonlocal PDEs with Long Time Memory
(Springer, 2024-06-07) Xu, Jiaohui; Caraballo Garrido, Tomás; Valero, José; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper, a combination of Galerkin’s method and Dafermos’ transformation is first used to prove the existence and uniqueness of solutions for a class of stochastic nonlocal PDEs with long time memory driven by additive noise. Next, the existence of tempered random attractors for such equations is established in an appropriate space for the analysis of problems with delay and memory. Eventually, the convergence of solutions of Wong-Zakai approximations and upper semicontinuity of random attractors of the approximate random system, as the step sizes of approximations approach zero, are analyzed in a detailed way
Tempered fractional Sobolev spaces
(Elsevier, 2024-08-02) Wei, Zhiqiang; Wang, Yejuan; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
The fractional Laplacian operator is the infinitesimal generator of the isotropic -stable Lévy process on , which is the scaling limit of the Lévy flight with isotropic power law measure . However, their second and higher moments are divergent, leading to the difficulty in the modeling of practical physical processes. Replacing by the measure of an isotropic tempered power law with the tempering exponent λ (i.e., ), the tempered fractional Laplacian operator was introduced in [17] as the infinitesimal generator of the tempered Lévy process. In this paper, guided by the fractional Sobolev spaces corresponding to the fractional Laplacian operator , we deal with the tempered fractional Sobolev spaces associated with the tempered fractional Laplacian . First, the definition of the tempered fractional Sobolev spaces is given via the Gagliardo approach, and some of their basic properties were studied. Subsequently, we focus on the Hilbert case based on the Fourier transform. In particular, we deal with its relation with the tempered fractional Sobolev space and analyze their role in the trace theory, overcoming the challenges posed by the tempering exponent λ. Then we investigate the asymptotic behavior of and that appear in the definition of the tempered fractional Laplacian operator . Moreover, we show continuous and compact embeddings investigating the problem of the extension domains and the generalized Hölder regularity results. As an application of the tempered fractional Sobolev spaces, we prove that the process defined by the tempered Lévy process is a solution of some PDE with tempered fractional Lapalcian operator.
Exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain
(AMS, 2024-09-20) Hu, Wenjie; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite dimensional subspaces of a Banach space. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation.
Time-dependent uniform upper semicontinuity of pullback attractors for non-autonomous delay dynamical systems: Theoretical results and applications
(AMS, 2024-09-10) Zhang, Qiangheng; Caraballo Garrido, Tomás; Yang, Shuang; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper we provide general results on the uniform upper semicontinuity of pullback attractors with respect to the time parameter for non-autonomous delay dynamical systems. Namely, we establish a criteria in terms of the multi-index convergence of solutions for the delay system to the non-delay one, locally pointwise convergence and local controllability of pullback attractors. As an application, we prove the upper semicontinuity of pullback attractors for a non-autonomous delay reaction-diffusion equation to the corresponding nondelay one over any bounded time interval as the delay parameter tends to zero.
Dynamics of stochastic differential equations with memory driven by colored noise
(American Institute of Physics, 2024-09-09) Liu, Ruonan; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper, we will show two approaches to analyze the dynamics of a stochastic partial differential equation (PDE) with long time memory, which does not generate a random dynamical system and, consequently, the general theory of random attractors is not applicable. On the one hand, we first approximate the stochastic PDEs by a random one via replacing the white noise by a colored one. The resulting random equation does generate a random dynamical system which possesses a random attractor depending on the covariance parameter of the colored noise. On the other hand, we define a mean random dynamical system via the solution operator and prove the existence and uniqueness of weak pullback mean random attractors when the problem is driven by a more general white noise.
Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay
(Bolyai Institute, University of Szeged, 2024-10-16) Hu, Wenjie; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
The aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a stochastic reaction-diffusion equation with delay. The stochastic equation is firstly transformed into a delayed random partial differential equation by means of a random conjugation, which is then recast into an auxiliary Hilbert space. For the obtained equation, it is firstly proved that it generates a random dynamical system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation possesses random attractors by a uniform estimate of the solution and the asymptotic compactness of the generated RDS. After establishing the variational equation in the auxiliary Hilbert space and the almost surely differentiable properties of the RDS, upper estimates of both Hausdorff and fractal dimensions of the random attractors are obtained.
Stochastic SIR epidemic model dynamics on scale-free networks
(Elsevier, 2024-10-04) Settati, A.; Caraballo Garrido, Tomás; Lahrouz, A.; Bouzalmat, I.; Assadouq, A.; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
This study introduces a stochastic SIR (Susceptible–Infectious–Recovered) model on complex networks, utilizing a scale-free network to represent inter-human contacts. The model incorporates a threshold parameter, denoted as , which plays a decisive role in determining whether the disease will persist or become extinct. When , the disease exhibits exponential decay and eventually disappear. Conversely, when , the disease persists. The critical case of is also examined. Furthermore, we establish a unique stationary distribution for . Our findings highlight the significance of network topology in modeling disease spread, emphasizing the role of social networks in epidemiology. Additionally, we present computational simulations that consider the scale-free network’s topology, offering comprehensive insights into the behavior of the stochastic SIR model on complex networks. These results have substantial implications for public health policy, disease control strategies, and epidemic modeling in diverse contexts.
Randomness suppress backward bifurcation in an epidemic model with limited medical resources
(Elsevier, 2024-09-30) Lahrouz, A.; Caraballo Garrido, Tomás; Bouzalmat, I.; Settati, A.; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
This paper delves into the dynamic features of a stochastic SIR epidemic model featuring a perturbed transmission rate influenced by white noise. Our primary aim is to unravel the intricate interplay between restricted medical resources, their supply efficiency, and environmental stochasticity, shedding light on their collective impact on the transmission dynamics of infectious diseases. Our findings bring to light a notable distinction from the deterministic counterpart of the model. Specifically, under varying scenarios of medical resource availability and supply efficiency, the stochastic model exhibits a departure from bifurcation phenomena. This stands in contrast to the deterministic model, which is characterized by the presence of both backward bifurcation and Hopf bifurcation phenomena. To complement and validate our theoretical findings, numerical simulations are employed, providing concrete illustrations of the dynamical phenomena discussed in the paper. This research contributes to a nuanced understanding of the intricate interplay between stochastic environmental factors, medical resource constraints, and disease transmission dynamics, offering valuable insights for public health management and epidemic control strategies.
A unified analysis of mixed and stabilized finite element solutions of Navier-Stokes equations
(Elsevier, 2000) Chacón Rebollo, Tomás; Domínguez Delgado, Antonio; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII)
Abstract This paper performs a uni®ed numerical analysis of Mixed and Stabilized Finite Element numerical solutions of 2D and 3D Steady Navier±Stokes Equations. We introduce a general internal discretization of Navier±Stokes equations of which both Mixed and Stabilized Methods are particular cases. We prove convergence of Stabilized Methods with non linear stabilization coe cients, in particular for ¯ows with convection dominance. We also analyze the approximation of branches of regular solutions, in the case of convection dominance. We introduce a stabilized post-processing of Galerkin Finite Element (FE) solution of convection-dominated ¯ows. Some numerical test for nonlinear ¯ows show the good performances of this technique.

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