Time-dependent Coefficients Research Articles

Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure

<abstract><p>This paper is devoted to the homogenization of a class of nonlinear nonlocal parabolic equations with time dependent coefficients in a periodic and stationary structure. In the first part, we consider the homogenization problem with a periodic structure. Inspired by the idea of Akagi and Oka for local nonlinear homogenization, by a change of unknown function, we transform the nonlinear nonlocal term in space into a linear nonlocal scaled diffusive term, while the corresponding linear time derivative term becomes a nonlinear one. By constructing some corrector functions, for different time scales $ r $ and the nonlinear parameter $ p $, we obtain that the limit equation is a local nonlinear diffusion equation with coefficients depending on $ r $ and $ p $. In addition, we also consider the homogenization of the nonlocal porous medium equation with non negative initial values and get similar homogenization results. In the second part, we consider the previous problem in a stationary environment and get some similar homogenization results. The novelty of this paper is two folds. First, for the determination equation with a periodic structure, our study complements the results in literature for $ r = 2 $ and $ p = 1 $. Second, we consider the corresponding equation with a stationary structure.</p></abstract>

Оптимизация типа Кротова когерентного и некогерентного управлений для открытых двухкубитных систем

This work considers two-qubit open quantum systems driven by coherent and incoherent controls. Incoherent control induces time-dependent decoherence rates via time-dependent spectral density of the environment which is used as a resource for controlling the system. The system evolves according to the Gorini–Kossakowski– Sudarshan–Lindblad master equation with time-dependent coefficients. For two types of interaction with coherent control, three types of objectives are considered: 1) maximizing the Hilbert–Schmidt overlap between the final and target density matrices; 2) minimizing the Hilbert–Schmidt distance between these matrices; 3) steering the overlap to a given value. For the first problem, we develop the Krotov type methods directly in terms of density matrices with or without regularization for piecewise continuous controls with constaints and find the cases where the methods produce (either exactly or with some precision) zero controls which satisfy the Pontryagin maximum principle and produce the overlap’s values close to their upper bounds. For the problems 2) and 3), we find cases when the dual annealing method steers the objectives close to zero and produces a non-zero control.

Discrete fragmentation equations with time-dependent coefficients

We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such discrete-size fragmentation models, we allow the fragmentation coefficients to vary with time. By formulating the initial-value problem for the system as a non-autonomous abstract Cauchy problem, posed in an appropriately weighted $ \ell^1 $ space, and then applying results from the theory of evolution families, we prove the existence and uniqueness of physically relevant, classical solutions for suitably constrained coefficients.

Integration of the Hirota equation with time-dependent coefficients

Integration of the Hirota equation with time-dependent coefficients

Finite Time Blow Up of Solutions for the m-Laplacian Equation With Variable Coefficients

In this work, we deal with the m- Laplacian equation with time dependent variable coefficients. Under suitable conditions on variable coefficients, we prove the blow up of solutions for finite time with negative initial energy. These results partially generalize and extend some recent ones in previous literature.

Random attractors and invariant measures for 3D stochastic globally modified Navier-Stokes equations with time-dependent delay and coefficient

In this article, we investigate the dynamics of three-dimensional, non-autonomous, and globally modified Navier-Stokes equations with time-dependent delay and additive noise. Based on the well-posedness of solutions, we demonstrate that the solution operators of such equations generate a non-autonomous random dynamical system. We further prove the existence and uniqueness of random attractors as well as the periodicity and the upper semicontinuity for this random dynamical system. We also prove the existence of invariant measures supported on the random attractors. Moreover, we rigorously find that every limit of a sequence of invariant measures must be an invariant measure of the corresponding limiting equations as the noise intensity $ \varepsilon\rightarrow0 $ or the modification parameter $ N\rightarrow N_0\in (0, +\infty) $.

High-precision solidification simulation by estimating heat transfer coefficients through data assimilation

Solidification analysis is important for accurately predicting casting processes. A high-precision casting simulation requires accurate data of the physical properties of materials and heat transfer parameters. In particular, the heat transfer coefficient is a major parameter influencing the casting simulation results. Because the heat transfer coefficient is not uniquely determined by the material, it is typically estimated by trial and error, which requires considerable time for accurate estimations. Therefore, we developed a method to quickly estimate the heat transfer coefficient in a casting process by combining data assimilation and solidification simulations. With this method, the time-dependent heat transfer coefficient can be estimated in an appropriate and easy manner. To reproduce the 3D temperature distribution in a casting with high accuracy, we developed a method to estimate multiple heat transfer coefficients simultaneously by combining data assimilation and 3D solidification simulation. Al–5 mass% Si alloy was cast into a sand mold with a casting size of 30 mm × 30 mm × 100 mm, and the temperature was measured at five points in the casting to obtain a cooling curve. Two cooling curves corresponding to locations of 5 mm from the casting wall were used to estimate the two heat transfer coefficients on the side and bottom surfaces, which were simultaneously estimated by data assimilation. The results confirmed that all the five cooling curves could be reproduced with a high accuracy and that high-precision casting simulation was possible.

Diffusive size-structured population model with time-varying diffusion rate

<p style='text-indent:20px;'>This work is devoted to the study of a size-structured population model with a time-varying diffusion rate. Due to the seasonal variation, it is natural to consider the time-varying diffusion rate. Moreover, introducing the time-varying diffusion rate makes the model more challenging and requires the results of evolution operators for the analysis. Under some assumptions on the time-dependent diffusion coefficient, the existence and uniqueness of mild solution is shown. We apply the method of characteristics and evolution operators to derive our results. Positivity and boundedness of mild solution is also shown. Some examples are provided to illustrate the theoretical findings. We also solve our model numerically to study the size and spatial dynamics of population density.</p>

Parabolic homogenization with an interface

This paper considers a family of second-order parabolic equations in divergence form with rapidly oscillating and time-dependent periodic coefficients and an interface between two periodic structures. Following a framework initiated by Blanc, Le Bris and Lions and a generalized two-scale expansion in divergence form of elliptic homogenization with an interface by Josien, we can determine the effective (or homogenized) equation with the coefficient matrix being piecewise constant and discontinuous across the interface. Moreover, we obtain the $O(\varepsilon)$ convergence rates in $L^{2(d+2)/{d}}_{x,t}$ with $\varepsilon$-smoothing method and the uniform interior Lipschitz estimates via compactness argument.

Dynamics and Control for Transverse Formation by a Distributed Drag Sail

The constant effective vertical measurement baseline provided by the transverse formation satellites in low orbit has a critical application prospect for ground and high-orbit target detection. However, traditional formation satellites decay owing to atmospheric perturbation, and their effective baseline is periodically time-varying. A distributed drag sail (DDS) with several louvered subsails is promoted based on the free molecular flow theory, which allows for high maneuverability in satellite-tether-sail (STS) transverse formation flying. Moreover, the fuzzy PD control is introduced into the attitude-orbit model, which attains the time-dependent PD coefficients based on the geometrical configuration, to handle coupled attitude-orbit of the DDS. The aerodynamic force vectors are distributed geometrically in a bounded two-dimensional area with a pentagonal shape, through different twist combinations of subsails. The size of the distributed area depends on the number of subsails, the length and width of DDS. Finally, it is achieved that the steering engines of DDS can work effectively in appropriate frequencies during the stable formation configuration. Binocular detection with a constant vertical measurement baseline can be provided by DDS to gain situational awareness of space targets in practical scenarios such as remote sensing applications.

Attractor of the nonclassical diffusion equation with memory on time- dependent space

<abstract><p>We consider the dynamic behavior of solutions for a nonclassical diffusion equation with memory</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{t}-\varepsilon(t) \triangle u_{t}- \triangle u-\int_{0}^{\infty}\kappa(s)\triangle u(t-s)ds+f(u) = g(x) $\end{document} </tex-math></disp-formula></p> <p>on time-dependent space for which the norm of the space depends on the time $ t $ explicitly, and the nonlinear term satisfies the critical growth condition. First, based on the classical Faedo-Galerkin method, we obtain the well-posedness of the solution for the equation. Then, by using the contractive function method and establishing some delicate estimates along the trajectory of the solutions on the time-dependent space, we prove the existence of the time-dependent global attractor for the problem. Due to very general assumptions on memory kernel $ \kappa $ and the effect of time-dependent coefficient $ \varepsilon(t) $, our result will include and generalize the existing results of such equations with constant coefficients. It is worth noting that the nonlinear term cannot be treated by the common decomposition techniques, and this paper overcomes the difficulty by dealing with it as a whole.</p></abstract>

On inverse problems for uncoupled space-time fractional operators involving time-dependent coefficients

We study the uncoupled space-time fractional operators involving time-dependent coefficients and formulate the corresponding inverse problems. Our goal is to determine the variable coefficients from the exterior partial measurements of the Dirichlet-to-Neumann map. We exploit the integration by parts formula for Riemann-Liouville and Caputo derivatives to derive the Runge approximation property for our space-time fractional operator based on the unique continuation property of the fractional Laplacian. This enables us to extend early unique determination results for space-fractional but time-local operators to the space-time fractional case.

An inverse problem of finding a time-dependent coefficient in a fractional diffusion equation

An inverse problem of finding a time-dependent coefficient in a fractional diffusion equation

BLOW-UP PROBLEMS FOR FUJITA-TYPE PARABOLIC SYSTEM INVOLVING TIME-DEPENDENT COEFFICIENTS ON GRAPHS

In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053].

Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations

<abstract><p>Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that extend a recently proposed SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above-mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamiltonian models based on the book and oscillator algebras, denoted by $ \mathfrak{b}_2 $ and $ \mathfrak{h}_4 $, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry $ \mathfrak{h}_6 $, according to the embedding chain $ \mathfrak{b}_2\subset \mathfrak{h}_4\subset \mathfrak{h}_6 $, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.</p></abstract>

Identification of a time-dependent diffusivity coefficient in heat-like space-time fractional differential equations

The goal of this research is to reveal the unknown time dependent diffusion coefficient in space-time fractional differential equations by means of fractional Taylor series method. Unlike most methods used in inverse problems, using no over-measured data is a substantial advantage of this method.
 As a result, the unknown diffusion coefficient could be determined with high precision. Illustrative examples shows that the retrieved unknown coefficient and the solution of the problem are in a high agreement with the exact solution of the corresponding the inverse problems.

Inverse boundary value problem for nonlinear diffusion equation with non-local time-integral conditions of the second kind2

This work deals with the solvability of inverse boundary value problem with time-dependent unknown coefficient for nonlinear diffusion equation. Definition of classical solution for the considered inverse boundary value problem is introduced. By the Fourier method, the problem is reduced to the system of integral equations. Using the contraction mappings method, the existence and uniqueness of the solution of the system of integral equations are proved. Finally, the existence and uniqueness of the classical solution of original problem are proved.

Holes and magnetic polarons in a triangular lattice antiferromagnet

The intricate interplay between charge motion and magnetic order in geometrically frustrated lattices is central for the properties of many two-dimensional quantum materials. The triangular lattice antiferromagnet is a canonical example of a frustrated system, and here we analyze the dynamics of a hole in such a lattice focusing on observables that have become accessible in a new generation of experiments. Using the $t\ensuremath{-}J$ model, we solve the problem exactly within linear spin wave theory in the limit of strong magnetic interactions, showing that the ground state is described by a coherent state of spin waves. The derivation highlights the crucial role played by the interaction between a static hole and the neighboring spins, which originates in the geometric frustration and has often been omitted in earlier works. Furthermore, we show that the nonequilibrium dynamics after a hole has abruptly been inserted at a lattice site is given by a coherent state with time-dependent oscillatory coefficients. Physically, this describes a burst of magnetic frustration propagating through only two-thirds of the lattice sites, since a destructive interference of spin waves leaves spins parallel to that removed by the hole unperturbed. After the wave has propagated through the lattice, the magnetization relaxes to that of the ground state. We then use our analytical solution to benchmark the widely used self-consistent Born approximation (SCBA) in the limit of strong magnetic interactions, showing that it is very accurate also for a triangular lattice. The magnetic polaron spectrum is analyzed for general magnetic interactions using the SCBA, and we compare our results with those for a square lattice.

A Vulnerability Criterion of Dynamic Instability for Ship Course-Keeping in Following Waves

_ A practical criterion which can be used for assessing the course-keeping capability of a ship in following waves, is proposed and evaluated. Presently, it accounts for regular waves and it accrued from an analytical estimation of the course instability region’s boundary by applying the method of harmonic balance. The calculation is performed with regard to the third-order yaw equation, derived from a classic sway-yaw-rudder model of ship maneuvering motions, with time-dependent coefficients at two places, which is like a Mathieu-type equation extended to third order. The proposed analytical criterion was evaluated thoroughly against simulations with regard to this sway-yaw model and it was found to be adequately accurate. A supplementary quasi-static yaw stability criterion (fitting to ship operation with frequency of encounter, with respect to the waves, close to zero) was also considered in order to determine which one yields more stringent requirements, for various operating conditions. The proposed criterion could be an extra vulnerability check for broaching-to, in the context of the Second Generation Intact Stability Criteria. Introduction The difficulties of steering of ships in following seas have received the attention of the research community since more than 70 years ago (Davidson 1948). A relevant direction of research refers to the avoidance of the broaching-to instability and, in particular, to the development of practical criteria that could ensure sufficient course-keeping capability for a ship encountering steep following waves. Some classic works on this topic, such as those of DuCane and Goodrich (1962), Wahab and Swaan (1964), and Motora et al. (1981), were focused on the quasi-static condition of a ship on the wave which could be practically realized if the ship was advancing with speed equal to the wave celerity (zero frequency of encounter). As implied, these works were essentially focused on the avoidance of a type of broaching-to instability that is preceded by the realization of surf-riding, a phenomenon where the ship is forced to move with the wave, usually riding a downslope.

Stabilization of semilinear wave equations with time-dependent variable coefficients and memory

Stabilization of semilinear wave equations with time-dependent variable coefficients and memory