Artículos (Ecuaciones Diferenciales y Análisis Numérico)
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Artículo Self-Adjusting Multi-Rate Runge-Kutta Methods: Analysis and Efficient Implementation in An Open Source Framework(Springer, 2025-09-13) Bachmann, Bernhard; Bonaventura, Francesco; Casella, Francesco; Fernández García, Soledad; Gómez Mármol, María Macarena; Hannebohm, Philip; Ecuaciones Diferenciales y Análisis Numérico; FQM120: Modelado Matemático y Simulación de Sistemas MedioambientalesWe present an approach for the efficient implementation of self-adjusting multi-rate Runge-Kutta methods and we introduce a novel stability analysis, that covers the multi-rate extensions of all standard Runge-Kutta methods and allows to assess the impact of different interpolation methods for the latent variables and of the use of an arbitrary number of sub-steps for the active variables. The stability analysis applies successfully to the model problem typically used in the literature for multi-rate methods. Furthermore, we also propose a physically motivated model problem that can be used to assess stability to problems with purely imaginary eigenvalues and in situations closer to those arising in applications. Finally, we present an efficient implementation of multi-rate Runge-Kutta methods in the framework of the OpenModelica open-source modelling and simulation software. Results of several numerical experiments, performed with this implementation of the proposed methods, demonstrate the efficiency gains deriving from the use of the proposed multi-rate approach for physical modelling problems with multiple time scales.Artículo On the Navier condition for viscous fluids in rough domains(Springer, 2012) Casado Díaz, Juan; Luna Laynez, Manuel; Suárez Grau, Francisco Javier; Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Ciencia e Innovación (MICIN). España; FQM309: Control y Homogeneización de Ecuaciones en Derivadas ParcialesArtículo Effects of rough boundary on the heat transfer in a thin film flow(Elsevier, 2013-06-13) Pazanin, Igor; Suárez Grau, Francisco Javier; Ecuaciones Diferenciales y Análisis Numérico; Ministry of Science, Education and Sports, Republic of Croatia; Ministerio de Economía y Competitividad (MINECO). España; FQM309: Control y Homogeneización de Ecuaciones en Derivadas ParcialesIn this Note a heat flow through a rough thin domain filled with fluid (lubricant) is studied. Domain's thickness is considered as the small parameter $\varepsilon$, while the roughness is defined by a periodical function with period of order $\varepsilon^2$. We assume that the lubricant is cooled by the exterior medium and we describe the heat exchange on the rough part of the boundary by the Newton's cooling law. Depending on the magnitude of the heat transfer coefficient with respect to $\varepsilon$, we obtain three different macroscopic models via formal asymptotic analysis. We identify the critical case explicitly acknowledging both roughness-induced effects and the effects of surrounding medium on the heat transfer at main order. We illustrate the obtained results by some numerical simulations.Artículo The homogenization of elliptic partial differential systems on rugous domains with variable boundary conditions(Cambridge University Press, 2013) Casado Díaz, Juan; Luna Laynez, Manuel; Suárez Grau, Francisco Javier; Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Ciencia e Innovación (MICIN). España; FQM309: Control y Homogeneización de Ecuaciones en Derivadas ParcialesArtículo Estimates for the asymptotic expansion of a viscous fluid satisfying Navier's law on a rugous boundary(Wiley, 2011) Casado Díaz, Juan; Luna Laynez, Manuel; Suárez Grau, Francisco Javier; Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Ciencia e Innovación (MICIN). España; FQM309: Control y Homogeneización de Ecuaciones en Derivadas ParcialesIn a previous paper, we have studied the asymptotic behavior of a viscous fluid satisfying Navier's law on a periodic rugous boundary of period ε and amplitude δε, with δε/ε tending to zero. In the critical size, δε∼ε3/2, in order to obtain a strong approximation of the velocity and the pressure it is necessary to consider a boundary layer term in the corresponding ansatz. The purpose of this paper is to estimate the approximation given by this ansatz.Artículo Asymptotic behavior of a viscous fluid with slip boundary conditions on a slightly rough wall(World Scientific Publishing, 2010) Casado Díaz, Juan; Luna Laynez, Manuel; Suárez Grau, Francisco Javier; Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Ciencia e Innovación (MICIN). España; FQM309: Control y Homogeneización de Ecuaciones en Derivadas ParcialesArtículo Uniform Bounds with Difference Quotients for Proper Orthogonal Decomposition Reduced Order Models of the Burgers Equation(Springer, 2023-03-20) Koc, Birgul; Rubino, Samuele; Chacón Rebollo, Tomás; Ecuaciones Diferenciales y Análisis NuméricoIn this paper, we prove uniform error bounds for proper orthogonal decomposition (POD) reduced ordermodeling (ROM) of Burgers equation, considering difference quotients (DQs), introduced in Kunisch and Volkwein (Numer Math 90(1):117–148, 2001). In particular, we study the behavior of the DQ ROM error bounds by considering L2(_) and H1 0 (_) POD spaces and l∞ (L2) and natural-norm errors. We present some meaningful numerical tests checking the behavior of error bounds. Based on our numerical results, DQ ROM errors are several orders of magnitude smaller than noDQ ones (in which the POD is constructed in a standard way, i.e., without the DQ approach) in terms of the energy kept by the ROM basis. Further, noDQ ROM errors have an optimal behavior, while DQ ROM errors, where the DQ is added to the POD process, demonstrate an optimality/super-optimality behavior. It is conjectured that this possibly occurs because the DQ inner products allow the time dependency in the ROM spaces to make an impact.Artículo Residual-based data-driven variational multiscale reduced order models for parameter-dependent problems(Springer, 2025-06-04) Koc, Birgul; Rubino, Samuele; Chacón Rebollo, Tomás; Iliescu, Traian; Ecuaciones Diferenciales y Análisis NuméricoIn this paper, we propose a novel residual-based data-driven closure strategy for reduced order models (ROMs) of under-resolved, convection-dominated problems. The new ROM closure model is constructed in a variational multiscale (VMS) framework by using the available full order model data and a model form ansatz that depends on the ROM residual. We emphasize that this closure modeling strategy is fundamentally different from the current data-driven ROM closures, which generally depend on the ROM coefficients. We investigate the new residual-based data-driven VMS ROM closure strategy in the numerical simulation of three test problems: (i) a one-dimensional parameter-dependent advection-diffusion problem; (ii) a two-dimensional time-dependent advection-diffusion-reaction problem with a small diffusion coefficient (ε=1e−4); and (iii) a two-dimensional flow past a cylinder at Reynolds number Re=1000. Our numerical investigation shows that the new residual-based data-driven VMS-ROM is more accurate than the standard coefficient-based data-driven VMS-ROM.Artículo Pullback asymptotic behavior and statistical solutions for lattice Klein-Gordon-Schrödinger equations with varying coefficient(American Institute of Mathematical Sciences, 2025-02-14) Zhao, Caidi; Zhuang, Rong; Caraballo Garrido, Tomás; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesIn this article, the authors investigate the pullback asymptotic behavior and statistical solutions for lattice Klein-Gordon-Schrödinger equations with varying coefficient. They first prove the global well-posedness of the addressed equations and the existence of a family of time-dependent pullback attractor for the associated process acting on the time-dependent phase spaces. Then they verify that the process possesses a family of invariant Borel probability measures with support contained in the time-dependent pullback attractor. Further, they reformulate the definition of statistical solution for the evolutionary equations on time-dependent phase spaces. As a result, they prove the existence of statistical solution for the lattice Klein-Gordon-Schrödinger equations with varying coefficient and show that it satisfies the Liouville theorem.Artículo Stability Analysis of Random Attractors for Stochastic Modified Swift–Hohenberg Equations with Delays(Springer, 2024-01-07) Zhang, Qiangheng; Caraballo Garrido, Tomás; Yang, Shuang; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesA new type of random attractors is introduced to study dynamics of a stochastic modified Swift–Hohenberg equation with a general delay. A compact, pullback attracting and dividedly invariant set is called a backward attractor, while the criteria for its existence are established in terms of increasing dissipation and backward asymptotic compactness of a cocycle. If the delay term in the equation is Lipschitz continuous such that the Lipschitz bound and the external force are backward limitable, then we prove the existence of a backward attractor, which further leads to the longtime stability as well as the existence of a pullback attractor, where the pullback attractor and the backward attractor are shown to be random and dividedly random, respectively. Two examples of the delay term are provided to illustrate variable and distributed delays without restricting the upper bound of Lipschitz bounds.Artículo Positivity of periodic solutions for neutral-type impulsive neural networks with distributed delays and two parameters(Wiley, 2025-02-11) Benhadri, Mimia; Caraballo Garrido, Tomás; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesIn this paper, the existence of positive periodic solutions of impulsive neutral-type neural networks with distributed delays and two parameters is discussed by applying the cone-theoretic fixed point theorem. Some sufficient conditions are obtained to ensure the existence of multiple and single positive periodic solutions for the considered system. Several existence results are established. Finally, we exhibit an example to illustrate the applicability of the established results.Artículo Dynamics of non-local lattice systems in l1(Elsevier, 2025-02-14) Xu, Jiaohui; Caraballo Garrido, Tomás; Valero, José; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesIn this paper, the well-posedness and asymptotic behavior of a non-local lattice system are analyzed in the space ℓ1. In fact, the analysis is carried out in the subspace ℓ+1 formed by the nonnegative elements, remaining open the case of the whole space. The same problem has been analyzed recently in the space ℓ2 (see Y. Li et al., Communications on Pure and Applied Analysis, 23 (2024), 935-960). However, the latter does not allow us to consider non-local terms which are natural in the modeling of reaction–diffusion problems introduced by M. Chipot in the wide literature published on this problem. With the current analysis, it is possible to investigate these interesting situations.Artículo Invariant measures of stochastic Klein–Gordon–Schrödinger equations on infinite lattices(American Institute of physics, 2025-02-13) Mi, Shaoyue; Li, Dingshi; Freitas, Mirelson M.; Caraballo Garrido, Tomás; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesWe study the long-time dynamics of stochastic Klein–Gordon–Schrödinger equations driven by infinite-dimensional nonlinear noise defined on integer set. Firstly, we formulate the stochastic lattice equations as an abstract system defined in an appropriated space of square-summable sequences, and then prove the existence and uniqueness of global solutions to the abstract system. To such solutions, we establish the uniform boundedness and uniform estimates on the tails of solutions, which are necessary to ensure the tightness of a family of probability distributions. Finally, we prove the existence of invariant measures for the stochastic lattice equations using the Krylov–Bogolyubov’s method.Artículo Strong Convergence of Solutions and Attractors for Reaction-Diffusion Equations Governed by a Fractional Laplacian(Springer, 2025-02-20) Xu, Jiaohui; Caraballo Garrido, Tomás; Valero, José; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesA nonlocal reaction-diffusion equation governed by a fractional Laplace operator on a bounded domain is studied in this paper. First, the strong convergence of solutions of the equations governed by fractional Laplacian to the solutions of the classical equations governed by a standard Laplace operator is proved, when the fractional parameter grows to 1. Second, for the autonomous case, the upper semicontinuity of global attractors with respect to the attractors of the limit problem is established. Apparently, these are the first results for this kind of problems on bounded domains.Artículo A Delay Nonlocal Quasilinear Chafee–Infante Problem: An Approach via Semigroup Theory(Springer, 2025-02-19) Caraballo Garrido, Tomás; Carvalho, A.N.; Julio, Yessica; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesIn this work we study a dissipative one dimensional scalar parabolic problem with non-local nonlinear diffusion with delay. We consider the general situation in which the functions involved are only continuous and solutions may not be unique. We establish conditions for global existence and prove the existence of global attractors. All results are presented only in the autonomous since the non-autonomous case follows in the same way, including the existence of pullback attractors. A particularly interesting feature is that there is a semilinear problem (nonlocal in space and in time) from which one can obtain all solutions of the associated quasilinear problem and that for this semilinear problem the delay depends on the initial function making its study more involved.Artículo Asymptotic behavior of a stochastic differential system with infinite delay and α-stable process(American Institute of Mathematical Sciences, 2025-08-28) Huang, Hai; Caraballo Garrido, Tomás; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesThis work is devoted to the asymptotic behavior of solutions for a stochastic differential system with infinite delay and α-stable process. By applying the theory of semigroups, a fixed point theorem, and results on stochastic convolutions in [14] (see Lemma 2.1), we study, respectively, the existence, uniqueness, global attracting sets, and stability in the distribution of mild solutions for the considered equation. An example is provided at the end to illustrate the applications of the obtained results.Artículo Exponential attractors with explicit fractal dimensions for functional differential equations in Banach spaces(Springer, 2025-04-17) Hu, Wenjie; Caraballo Garrido, Tomás; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesThe aim of this paper is to propose a new method to construct exponential attractors for infinite dimensional dynamical systems in Banach spaces with explicit fractal dimensions. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalizes the method established in Hilbert spaces. The method is especially effective for functional differential equations in Banach spaces for which state decomposition of the linear part can be adopted to prove squeezing property. The theoretical results are applied to retarded functional differential equations and retarded reaction-diffusion equations for which the constructed exponential attractors possess explicit fractal dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts.Artículo Asymptotical behavior of the 2D stochastic partial dissipative Boussinesq system with memory(Elsevier, 2025-04-25) Dai, Haoran; You, Bo; Caraballo Garrido, Tomás; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesThe objective of this paper is to consider the asymptotical behavior of solutions for the two-dimensional partial dissipative Boussinesq system with memory and additive noise. We first establish the existence of a random absorbing set in the phase space. However, due to the presence of the memory term, we cannot obtain some kind of compactness of the corresponding cocycle through Sobolev compactness embedding theorem or by verifying the pullback flattening property. To overcome this difficulty, we first prove the asymptotical compactness of the velocity component of weak solutions, and then we prove the asymptotical compactness of other components based on some energy estimates and the Aubin–Lions compactness lemma, which implies the asymptotical compactness of the corresponding cocycle. Thus, the existence of a random attractor is obtained. Finally, we establish an abstract result about some kind of upper semi-continuity of the random attractor, which is applied to the two-dimensional partial dissipative Boussinesq system.Artículo Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise(Springer, 2023-12-03) Li, Yangrong; Wang, Fengling; Caraballo Garrido, Tomás; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesFor a nonautonomous random dynamical system, we introduce a concept of a pullback random bi-spatial attractor (PRBA). We prove an existence theorem of a PRBA, which includes its measurability, compactness and attraction in the regular space. We then establish the residual dense continuity of a family of PRBAs from a parameter space into the space of all compact subsets of the regular space equipped by Hausdorff metric. The abstract results are illustrated in the nonautonomous random quasilinear equation driven by nonlinear colored noise, where the size of noise belongs to (0,∞] and the infinite size corresponds to the deterministic equation. The application results are the existence and residual dense continuity of PRBAs on (0,∞] in both square and p-order Lebesgue spaces, where p>2. The lower semi-continuity of attractors in the regular space seems to be a new subject even for an autonomous deterministic system.Artículo Hausdorff Dimension of Random Attractors for a Stochastic Delayed Parabolic Equation in Banach Spaces(Springer, 2025-06-02) Hu, Wenjie; Caraballo Garrido, Tomás; Duan, Yueliang; Ecuaciones Diferenciales y Análisis Numérico; FQM314: Análisis Estocástico de Sistemas DiferencialesThe main purpose of this paper is to give an upper bound of Hausdorff dimension of random attractors for a stochastic delayed parabolic equation in Banach spaces. The estimation of dimensions of random attractors are obtained by combining the squeezing property and a covering lemma of finite subspace of Banach spaces, which generalizes the method established in Hilbert spaces. Due to the lack of smooth inner product geometry structure, we adopt the state decomposition of phase space based on the exponential dichotomy of the linear deterministic part of the studied equations instead of orthogonal projectors with finite ranks used for stochastic partial differential equations. The obtained dimension of the random attractors depends only on the inner characteristics of the studied equation, such as spectrum of the linear part and the random Lipschitz constant of the nonlinear term, while not relating to the compact embedding of the phase space to another Banach space as the existing works did.