Artículos (Ecuaciones Diferenciales y Análisis Numérico)
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Artículo Ecuaciones diferenciales como jeroglíficos(2025-01-15) Langa Rosado, José Antonio; Lukaszewicz, Grzegorz; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas DiferencialesLa matemática supone un lenguaje que se adapta a lo real de manera sorprendente. En particular, las ecuaciones diferenciales utilizan una serie de símbolos propios, con un significado muy rico, no solo por su contenido abstracto, sino por la manera en que describen fenómenos del mundo físico o natural. En este sentido, es apropiada la metáfora de que el estudio de un sistema de ecuaciones diferenciales se asemeja a enfrentrarse a un jeroglífico. Este no solo informa de un conjunto de propiedades matemáticas del modelo, sino que es capaz de aprehender lo íntimo de lo real, con la materia, la energía y la información como sus constitu- yentes básicos, los cuales encuentran una expresión en este lenguaje tan impresionante que supone la matemática.Artículo Controllability for neutral stochastic functional integrodifferential equations with infinite delay(Sciendo, 2016-10-24) Caraballo Garrido, Tomás; Diop, Mamadou Abdoul; Mane, Aziz; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas DiferencialesIn this work, we study the controllability for a class of nonlinear neutral stochastic functional integrodifferential equations with infinite delay in a real separable Hilbert space. Sufficient conditions for the controllability are established by using Nussbaum fixed point theorem combined with theories of resolvent operators. As an application, an example is provided to illustrate the obtained result.Artículo Stability and convergence for a complete model of mass diffusion(Elsevier, 2011) Cabrales, R. C.; Guillén González, Francisco Manuel; Gutiérrez Santacreu, Juan Vicente; Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII); Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis NuméricoWe propose a fully discrete scheme for approximating a three-dimensional, strongly nonlinear model of mass diffusion, also called the complete Kazhikhov–Smagulov model. The scheme uses a C0 finite-element approximation for all unknowns (density, velocity and pressure), even though the density limit, solution of the continuous problem, belongs to H2. A first-order time discretization is used such that, at each time step, one only needs to solve two decoupled linear problems for the discrete density and the velocity–pressure, separately. We extend to the complete model, some stability and convergence results already obtained by the last two authors for a simplified model where λ2-terms are not considered, λ being the mass diffusion coefficient. Now, different arguments must be introduced, based mainly on an induction process with respect to the time step, obtaining at the same time the three main properties of the scheme: an approximate discrete maximum principle for the density, weak estimates for the velocity and strong ones for the density. Furthermore, the convergence towards a weak solution of the density-dependent Navier–Stokes problem is also obtained as λ→0 (jointly with the space and time parameters). Finally, some numerical computations prove the practical usefulness of the scheme.Artículo Theoretical and numerical results for some bi-objective optimal control problems(AIMS, 2019-04-14) Fernández Cara, Enrique; Marín Gayte, Irene; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo SoftwareThis article deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. More precisely, we look for Pareto equilibria associated to standard cost functionals. First, we study the linear and semilinear cases. We prove the existence of equilibria, we deduce appropriate optimality systems, we present some iterative algorithms and we establish convergence results. Then, we analyze the existence and characterization of Pareto equilibria for the Navier-Stokes equations. Here, we use the formalism of Dubovitskii and Milyutin. In this framework, we also present a finite element approximation of the bi-objective problem and we illustrate the techniques with several numerical experiments.Artículo Theoretical and numerical local null controllability of a quasi-linear parabolic equation in dimensions 2 and 3(Elsevier, 2021-01-27) Fernández Cara, Enrique; Límaco, Juan; Marín Gayte, Irene; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo SoftwareThis paper is devoted to the theoretical and numerical analysis of the null controllability of a quasi-linear parabolic equation. First, we establish a local controllability result. The proof relies on an appropriate inverse function argument. Then, we formulate an iterative algorithm for the computation of the null control and we prove a convergence result. Finally, we illustrate the analysis with some numerical experiments.Artículo Multiobjective optimal control problems. Stationary Navier-Stokes equations(Taylor & Francis, 2024-07-31) Gayte Delgado, Inmaculada; Marín Gayte, Irene; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo SoftwareThis paper deals with the solution of some multi-objective optimal control problems for stationary Navier-Stokes equations. More precisely, we look for Pareto and Nash equilibria associated to standard cost functionals. First, we prove the existence of equilibria and we deduce appropriate optimality systems. Then, we analyze the existence and characterization of Pareto and Nash equilibria for the Navier-Stokes equations. Here, we use the formalism of Dubovitskii and Milyoutin., see [13]. Finally, we also present a finite element approximation of the bi-objective problem, we illustrate the techniques with several numerical experiments and we compare the Pareto and Nash equilibria.Artículo On a certified VMS-Smagorinsky reduced basis model with LPS pressure stabilisation(Elsevier, 2022-12-05) Chacón Rebollo, Tomás; Delgado Ávila, Enrique; Gómez Mármol, María Macarena; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM120: Modelado Matemático y Simulación de Sistemas MedioambientalesIn this work we introduce a certified Reduced Basis VMS-Smagorinsky turbulence model with local projection stabilisation (LPS) on the pressure, for steady state problems. We prove its stability for Taylor-Hood discretisations of velocity-pressure or piecewise linear pressure. We construct an a posteriori error estimator for the snapshot selection with a Greedy algorithm, based on the Brezzi-Rappaz-Raviart theory of approximation of non-singular branches of non-linear PDEs. The Empirical Interpolation Method (EIM) is used for the approximation of the non-linear terms. We present some numerical tests in which we show an improved speedup on the computation of the reduced basis problem with the LPS pressure stabilisation, with respect to the method of using pressure supremizers.Artículo Mathematical modelling and computational reduction of molten glass fluid flow in a furnace melting basin(Springer, 2024-08-25) Ballarin, Francesco; Delgado Ávila, Enrique; Mola, Andrea; Rozza, Gianluigi; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM120: Modelado Matemático y Simulación de Sistemas MedioambientalesIn this work, we present the modelling and numerical simulation of a molten glass fluid flow in a furnace melting basin. We first derive a model for a molten glass fluid flow and present numerical simulations based on the finite element method (FEM). We further discuss and validate the results obtained from the simulations by comparing them with experimental results. Finally, we also present a non-intrusive proper orthogonal decomposition (POD) based on artificial neural networks (ANN) to efficiently handle scenarios which require multiple simulations of the fluid flow upon changing parameters of relevant industrial interest. This approach lets us obtain solutions of a complex 3D model, with good accuracy with respect to the FEM solution, yet with negligible associated computational times.Artículo Reduced Basis modelling of turbulence with well-developed inertial range(Elsevier, 2024-02-01) Bandera Moreno, Alejandro; Caravaca García, Cristina; Chacón Rebollo, Tomás; Delgado Ávila, Enrique; Gómez Mármol, María Macarena; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM120: Modelado Matemático y Simulación de Sistemas MedioambientalesIn this work, we introduce a Reduced Basis model for turbulence at statistical equilibrium. This is based upon an a-posteriori error estimation procedure that measures the distance from a trial solution to the K41 theory energy spectrum. We apply this general idea to build a Reduced Basis Smagorinsky turbulence model through a Greedy Algorithm. We derive some error estimates that make apparent the role of the energy spectrum in the ROM approximation. We carry on some tests for some academic unsteady 2D flows at large Reynolds number, that present well developed inertial spectrum. The methods presents a high efficiency, as the error achieved with the reduced method is 3 to 4 times the ones achieved if the exact error is used in the Greedy Algorithm.Artículo A logistic type equation in Rᴺ with a nonlocal reaction term via bifurcation method(Elsevier, 2021-01-01) Delgado Delgado, Manuel; Molina Becerra, Mónica; Suárez Fernández, Antonio; Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI); Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER); Universidad de Sevilla. TIC130: Investigación en Sistemas Dinámicos en Ingeniería; Universidad de Sevilla. FQM131: Ecuaciones Diferenciales, Simulación Numérica y Desarrollo SoftwareWe study the existence of positive solutions of a logistic equation in the entire space with a nonlocal reaction term. Mainly, we apply a bifurcation method and singular boundary equations to obtain a priori bounds of the solutions. Our results show a drastic change of behaviour of the set of positive solutions depending on the sign of the nonlocal term.Artículo 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 SoftwareWe 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.Artículo 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 SoftwareIn 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.Artículo 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 SoftwareWe 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.Artículo 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 SoftwareWe 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.Artículo 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 SoftwareWe 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.Artículo 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 SoftwareThe 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).Artículo 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 DiferencialesIn 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.Artículo 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 DiferencialesIn 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.Artículo 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 DiferencialesThis 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.Artículo 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 DiferencialesIn 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.