Inhomogeneous Initial Conditions Research Articles

Multiscale homogenization for dual porosity time-dependent Darcy-Brinkman/Darcy coupling and its application to the lymphnode.

We study the coupling between time-dependent Darcy-Brinkman and the Darcy equations at the microscale subjected to inhomogeneous body forces and initial conditions to describe a double porosity problem. We derive the homogenized governing equations for this problem using the asymptotic homogenization technique, and as macroscopic results, we obtain a coupling between two Darcy equations, one of which with memory effects, with mass exchange between phases. The memory effects are a consequence of considering the time dependence in the Darcy-Brinkman equation, and they allow us to study in more detail the role of time in the problem under consideration. After the formulation of the model, we solve it in a simplified setting and we use it to describe the movement of fluid within a vascularized lymph node.

Model reduction for second-order systems with inhomogeneous initial conditions

In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position or the initial velocity and the output when no input is applied. Based on this superposition of systems, our goal is to propose model reduction schemes that allow to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored second-order Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems.

Starting inflation from inhomogeneous initial conditions with momentum

We investigate the circumstances under which cosmic inflation can arise from very inhomogeneous initial conditions using numerical relativity simulations. Previous studies have not considered cases with non-zero momentum density due to technical challenges with solving the coupled Einstein constraint equations. Here we address these, introducing and comparing several different ways of constructing cosmological initial conditions with inhomogeneous scalar field and time derivative profiles. We evolve such initial conditions with large inhomogeneities in both single- and two-field inflationary models. We study cases where the initial gradient and kinetic energy are much larger than the inflationary energy scale, and black holes can form, as well as cases where the initial scalar potential energy is comparable, as in scenarios where inflation occurs at nearly Planckian densities, finding large-field inflation to be generally robust. We consider examples of initial conditions where a large scalar field velocity towards non-inflationary values would prevent inflation from occurring in the homogeneous case, finding that the addition of large gradients in the scalar field can actually dilute this effect, with the increased expansion and non-vanishing restoring force leading to inflation.

Solving System of Higher-Order Linear Differential Equations On The Level of Operators

In this paper, I present a method for solving the system of higher-order linear differential equations (HLDEs) with inhomogeneous initial conditions on the level of operators. Using proposed method, we compute the matrix Green’s operator as well as the vector Green’s function of a fully-inhomogeneous initial value problems (IVPs). Examples are discussed to demonstrate the proposed method.

Incommensurate multi‐term fractional differential equations with variable coefficients with respect to functions

Linear fractional differential equations with variable coefficients and various types of fractional derivatives can be solved explicitly, using either fixed‐point theorems and successive approximations or Green's functions. In this paper, we consider a multi‐term fractional differential equation with continuous variable coefficients and incommensurate fractional orders, and we construct an explicit solution by a direct method of successive approximations, which is made rigorous by checking the solution function explicitly via substitution. The results are obtained for the most general initial value problem of an inhomogeneous equation with inhomogeneous initial conditions, the first time that an explicit analytical solution has been found for this general problem. Special cases are provided to illustrate the main results, and connections are discussed with previous results in the literature.

О линеаризации гиперболических уравнений с интегральной нагрузкой в главной части с помощью априорной оценки их решений

A priori estimates are established for solutions of one-dimensional inhomogeneous hyperbolic equations with an integral load in the main part, which has the form a(s) = sp, for p = 1, 0.5 and -1, with inhomogeneous initial and homogeneous boundary conditions. Here s is the integral over the space variable of the square of the modulus of the derivative of the solution of the equation with respect to x. Examples of linearization of loaded equations by substituting the right-hand sides of the estimates for a(s) are given.

Gravitational dynamics in Higgs inflation: Preinflation and preheating with an auxiliary field

The dynamics of both the preinflationary and the preheating epochs for a model consisting of a Higgs inflaton plus an additional auxiliary field are studied in full General Relativity. The minimally coupled auxiliary field allows for parametric-type resonances that successfully transfer energy from the inflaton condensate to particle excitations in both fields. Depending on the interaction strengths of the fields, the broad resonance periods lead to structure formation consisting of large under/over-densities, and possibly the formation of compact objects. Moreover, when confronting the same model to multi-field inhomogeneous preinflation, the onset of inflation is shown to be a robust outcome. At relatively large Higgs values, the non-minimal coupling acts as a stabilizer, protecting the dynamics of the inflaton, and significantly reducing the impact of perturbations in other fields and matter sectors. These investigations further confirm the robustness of Higgs inflation to multi-field inhomogeneous initial conditions, while putting in evidence the formation of complex structures during the reheating.

The final Kasner regime inside black holes with scalar or vector hair

The final (close to the singularity) dynamical behavior of the metric inside black holes with massive charged scalar or vector hair is analyzed for general anisotropic and inhomogeneous initial conditions. These solutions are relevant to a holographic realization of superconductivity. It is shown that the dynamics falls within the scope of the “cosmological billiard” description and that in both cases, the corresponding hyperbolic billiard region has infinite volume so that the system ultimately settles down to a final Kasner regime. For massive vector hair, the conclusion holds because the longitudinal mode plays the same role as a scalar field. There exists, however, a measure-zero subset of solutions characterized by vanishing longitudinal modes that exhibit a chaotic behavior with an infinite number of BKL oscillations as one goes to the singularity.

Discrete orthogonal polynomials reduced models based on shift-transformation and discrete Walsh functions

This paper investigates model order reduction (MOR) methods of discrete-time systems via discrete orthogonal polynomials in the time domain. First, this system is expanded under discrete orthogonal polynomials, and the expansion coefficients are computed from a linear equation. Then the reduced-order system is produced by using the orthogonal projection matrix that is defined in terms of the expansion coefficients, which can match the first several expansion coefficients of the original output, and preserve the asymptotic stability and bounded-input/bounded-output (BIBO) stability. We also study the output error between the original system and its reduced-order system. Besides, the MOR method using discrete Walsh functions is proposed for discrete-time systems with inhomogeneous initial conditions. Finally, three numerical examples are given to illustrate the feasibility of the proposed methods.

Model order reduction of port-Hamiltonian systems with inhomogeneous initial conditions via approximate finite-time Gramians

Based on the approximate finite-time Gramians, this paper studies model order reduction method of port-Hamiltonian systems with inhomogeneous initial conditions. The approximate controllability and observability Gramians on the finite-time interval [T1,T2](0≤T1<T2<∞) can be obtained by the shifted Legendre polynomials and the reduced port-Hamiltonian system is constructed by the union of dominant eigenspaces. Since the port-Hamiltonian system is square, the cross Gramian on the time interval [T1,T2] can also be approximated by using the shifted Legendre polynomials. Then, the truncated singular value decomposition of the approximate finite-time cross Gramian is carried out to obtain the projection matrix. Finally, the proposed methods are verified by two numerical examples.

Dynamical attractors in contracting spacetimes dominated by kinetically coupled scalar fields

We present non-perturbative numerical relativity simulations of slowly contracting spacetimes in which the scalar field driving slow contraction is coupled to a second scalar field through an exponential non-linear σ model-type kinetic interaction. These models are important because they can generate a nearly scale-invariant spectrum of super-Hubble density fluctuations fully consistent with cosmic microwave background observations. We show that the non-linear evolution rapidly approaches a homogeneous, isotropic and flat Friedmann-Robertson-Walker (FRW) geometry for a wide range of inhomogeneous and anisotropic initial conditions. Ultimately, we find, the kinetic coupling causes the evolution to deflect away from flat FRW and towards a novel Kasner-like stationary point, but in general this occurs on time scales that are too long to be observationally relevant.

Time domain model order reduction of discrete-time bilinear systems with Charlier polynomials

This paper investigates time domain model order reduction of discrete-time bilinear systems with inhomogeneous initial conditions. The state of the system is approximated by the power series associated with the Charlier polynomials and the recurrence relation of the expansion coefficients is derived. The expansion coefficients are orthogonalized to construct the projection matrix by the modified multi-order Arnoldi method. The output of the resulting reduced order system maintains a certain number of expansion coefficients of the original output, and the error estimation of the reduced order system is briefly discussed. Due to the fact that the projection matrix involves the information of initial conditions, the proposed method can well reduce discrete-time bilinear systems with inhomogeneous initial conditions. Two numerical examples are employed to illustrate the effectiveness of the proposed method.

Well-posed variational formulations of Friedrichs-type systems

All finite element methods, as well as much of the Hilbert-space theory for partial differential equations, rely on variational formulations, that is, problems of the type: find $u\in V$ such that $a(v,u) = l(v)$ for each $v\in L$, where $V, L$ are Sobolev spaces. However, for systems of Friedrichs type, there is a sharp disparity between established well-posedness theories, which are not variational, and the very successful discontinuous Galerkin methods that have been developed for such systems, which are variational. In an attempt to override this dichotomy, we present, through three specific examples of increasing complexity, well-posed variational formulations of boundary and initial--boundary-value problems of Friedrichs type. The variational forms we introduce are generalizations of those used for discontinuous Galerkin methods, in the sense that inhomogeneous boundary and initial conditions are enforced weakly through integrals in the variational forms. In the variational forms we introduce, the solution space is defined as a subspace $V$ of the graph space associated with the differential operator in question, whereas the test function space $L$ is a tuple of $L^2$ spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions.

Inhomogeneous preinflation across Hubble scales in full general relativity

We use the 3+1 formalism of numerical relativity to investigate the robustness of Starobinsky and Higgs inflation to inhomogeneous initial conditions, in the form of either field gradient or kinetic energy density. Sub-Hubble and Hubble-sized fluctuations generically lead to inflation after an oscillatory phase between gradient and kinetic energies. Hubble-sized inhomogeneities also produce contracting regions that may end up forming primordial black holes, subsequently diluted by inflation. We analyse the dynamics of the preinflation era and the generation of vector and tensor fluctuations. Our analysis further supports the robustness of inflation to any size of inhomogeneity, in the field, velocity or equation of state. At large field values, the preinflation dynamics only marginally depends on the field potential and it is expected that such behaviour is universal and applies to any inflation potential of plateau-type, favoured by CMB observations after Planck.

Reduced-Order Aggregate Model for Large-Scale Converters With Inhomogeneous Initial Conditions in DC Microgrids

In practical microgrids, the inhomogeneous initial values are widely appeared due to soft-starting operation. If traditional model order reduction approaches are applied, the input-output maps error between the original system and reduced-order system is large. To address this problem, this paper proposes a reduced-order aggregate model based on balanced truncation approach to provide the preprocessing approach for the real-time simulation of large-scale converters with inhomogeneous initial conditions in DC microgrid. Firstly, the standard linear time-invariant model with inhomogeneous initial conditions is established through non-leader multiagents concept. To end this, it is convenient for scholars to build complex system modeling with switched topology. Furthermore, the full system is divided into two components, i.e., the unforced component with nontrivial initial conditions and forced component with null initial conditions. Moreover, this paper presents an aggregated approach that involves independent reducing component responses and combining reducing component responses. Based on this, the input-output maps error is reduced. Then, the approximated error estimate of the reduced-order aggregate model regarding large-scale converters in DC microgrid is first provided, which provides prior knowledge and theoretical basis for DC microgrid designers. Finally, the simulation results illustrate the accuracy of the proposed approach.

Analytical Approaches to the Analysis of Unsteady Heat Conduction for Partially Bounded Regions

A mathematical theory is developed for the construction of integral transforms for the following partially bounded regions: a space with an internal cylindrical cavity in the cylindrical coordinates (radial heat flux); a space with an internal spherical cavity in the spherical coordinates (central symmetry); and a space bounded by a planar surface in the Cartesian coordinates. Expressions are proposed for the integral transforms, Laplace operator images, and inversions for images. The formulated approach differs from the classical theory of differential equations of mathematical physics for the construction of integral transforms with a continuous spectrum of eigenvalues based on the corresponding singular Sturm–Liouville problems. The proposed method is based on the operational solutions of the initial boundary-value problems of unsteady heat conduction with an inhomogeneous initial function and homogeneous boundary conditions. The formulated approach makes it possible to develop the Green’s function method and to construct integral representations of the analytical solutions of the boundary-value problems simultaneously based on the Green’s functions and inhomogeneities in the main equation and boundary conditions of the problem. The proposed functional relations can be used in numerous particular cases of practical thermal physics. Examples of the application of the obtained results in some fields of science and technology are presented.

Model order reduction for bilinear control systems with inhomogeneous initial conditions

This work focuses on the model order reduction problem for bilinear control systems with nonzero initial conditions. Based on the Volterra series analysis, the system response can be decomposed into three parts. The first two parts are the zero input response and zero initial condition response of the system. The third part describes the response which couples the effect of the nonzero initial condition and the nonzero input. The system corresponding to the third part is a bilinear control system with a special time-varying input coefficient matrix. We show that such a system is equivalent to a time-invariant bilinear control system, and conventional model reduction methods can be applied to reduce it. We propose to reduce each of the component responses independently and then combine them to approximate the full system response. This method is of high flexibility and shows promising results.

РОЗВ’ЯЗУВАНIСТЬ ОСЕРЕДНЕНИХ ЗАДАЧ У ЗГОРТКАХ ДЛЯ СЛАБО ПОРИСТИХ СЕРЕДОВИЩ

Initial boundary value problems for nonstationary equations of diffusion and filtration in weakly porous media are considered. Assertions about the solvability of such problems and the corresponding homogenized problems with convolutions are given. These statements are proved for general initial data and inhomogeneous initial conditions and are generalizations of classical results on the solvability of initial-boundary value problems for the heat equation. The proofs use the methods of a priori estimates and the well-known Agranovich–Vishik method, developed to study parabolic problems of general type.

Structure‐preserved MOR method for coupled systems via orthogonal polynomials and Arnoldi algorithm

This study focuses on the topic of model order reduction (MOR) for coupled systems with inhomogeneous initial conditions and presents an MOR method by general orthogonal polynomials with Arnoldi algorithm. The main procedure is to use a series of expansion coefficients vectors in the space spanned by orthogonal polynomials that satisfy a recursive formula to generate a projection based on the multiorder Arnoldi algorithm. The resulting model not only match desired number of expansion coefficients but also has the same coupled structure as the original system. Moreover, the stability is preserved as well. The error bound between the outputs is well-discussed. Finally, numerical results show that the authors’ method can deal well with those systems with inhomogeneous initial conditions in the views of accuracy and computational cost.

Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation

In this article we study balanced model reduction of linear control systems using the singular perturbation approximation. Balanced model reduction techniques have been successfully applied to systems with homogeneous initial conditions, with one of their most important features being a priori L2 and H∞ bounds for the approximation error. The main focus of this work is to derive an L2 error bound for the singular perturbation approximation for system with inhomogeneous initial conditions, extending related work on balanced truncation. This L2 error bound measures the difference between the input-output maps of the original and of the reduced initial value systems. The advantages and flexibility of this approach are demonstrated with a variety of numerical examples.