38 results on '"*HYPERBOLIC differential equations"'
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2. Editorial for the special issue 'Time‐delay systems: Recent trends and applications'.
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De Iuliis, Vittorio, d'Angelo, Massimiliano, Manes, Costanzo, Pepe, Pierdomenico, and Niculescu, Silviu‐Iulian
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DIRECTED graphs , *ROBUST stability analysis , *ADAPTIVE control systems , *SLIDING mode control , *BACKSTEPPING control method , *ROBUST control , *HYPERBOLIC differential equations , *FUNCTIONAL differential equations - Abstract
This document is an editorial for a special issue of the International Journal of Robust & Nonlinear Control on the topic of time-delay systems. It provides a historical overview of functional differential equations and highlights the contributions of Nikolay Nikolayevich Krasovskii and Aristide Halanay to the field. The special issue includes 21 research papers covering stability, robustness, observer theory, predictors, and delay-based approaches to partial differential equation models. Each paper is briefly described, covering topics such as stabilization of uncertain systems, sampled-data control for nonlinear systems, observer theory for differential delay systems, sensor fault estimation, and control approaches for vehicular platoons. The summaries aim to provide library patrons with a quick understanding of the papers' contents and their relevance to their research interests. [Extracted from the article]
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- 2024
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3. Tackling critical cases of the difference operator stability occurring in applications described by 1D distributed parameters.
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Rasvan, Vladimir
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DIFFERENCE operators , *FUNCTIONAL differential equations , *HYPERBOLIC differential equations , *PARTIAL differential equations , *HYDRAULIC engineering - Abstract
The article starts from the challenge of the critical cases in difference operator stability for neutral functional differential equations (NFDE). Such cases occur in the NFDE associated to 1D$$ 1D $$ hyperbolic partial differential equations (PDE) dynamics in mechanical and hydraulic engineering. For some of such applications it resulted that the aforementioned critical (nonasymptotic) stability is connected to the character and level of the energy losses. It is shown that suitable choice of the losses to be taken into account can remove the critical stability properties and give the difference operator the asymptotic stability thus ensuring asymptotic stability for the system's dynamics and also other asymptotic properties. [ABSTRACT FROM AUTHOR]
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- 2024
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4. Stabilization of age‐structured chemostat hyperbolic PDE with actuator dynamics.
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Haacker, Paul‐Erik, Karafyllis, Iasson, Krstić, Miroslav, and Diagne, Mamadou
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HYPERBOLIC differential equations , *CHEMOSTAT , *PARTIAL differential equations , *DILUTION , *ACTUATORS , *POPULATION density - Abstract
For population systems modeled by age‐structured hyperbolic partial differential equations (PDEs), we redesign the existing feedback laws, designed under the assumption that the dilution input is directly actuated, to the more realistic case where dilution is governed by actuation dynamics (modeled simply by an integrator). In addition to the standard constraint that the population density must remain positive, the dilution dynamics introduce constraints of not only positivity of dilution, but possibly of given positive lower and upper bounds on dilution. We present several designs, of varying complexity, and with various measurement requirements, which not only ensure global asymptotic (and local exponential) stabilization of a desired positive population density profile from all positive initial conditions, but do so without violating the constraints on the dilution state. To develop the results, we exploit the relation between first‐order hyperbolic PDEs and an equivalent representation in which a scalar input‐driven mode is decoupled from input‐free infinite‐dimensional internal dynamics represented by an integral delay system. [ABSTRACT FROM AUTHOR]
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- 2024
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5. Asymptotic expansion for convection-dominated transport in a thin graph-like junction.
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Mel'nyk, Taras and Rohde, Christian
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HYPERBOLIC differential equations , *DIFFUSION coefficients - Abstract
We consider for a small parameter > 0 a parabolic convection–diffusion problem with Péclet number of order ( − 1) in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order () connected through a domain (node) of diameter (). Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as → 0 , i.e. when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems (= 0) are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed. [ABSTRACT FROM AUTHOR]
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- 2024
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6. Nonlinear Inverse Problems for First Order Hyperbolic Equations.
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Kozhanov, Alexandr and Zhalgassova, Korkem A.
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NONLINEAR equations , *EQUATIONS , *HYPERBOLIC differential equations , *INVERSE problems - Abstract
We study nonlinear inverse problems for first order hyperbolic equations. We find a solution and an unknown coefficient depending on the time variable. For additional conditions we consider integral and boundary overdetermination conditions. We establish the existence of a regular solution, i.e., the solution possesses all generalized derivatives in the sense of Sobolev entering the equation. [ABSTRACT FROM AUTHOR]
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- 2024
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7. Adaptive control for synchronization of time‐delayed complex networks with multi‐weights based on semi‐linear hyperbolic PDEs.
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Yang, Chengyan and Qiu, Jianlong
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ADAPTIVE control systems , *HYPERBOLIC differential equations , *PARTIAL differential equations , *SYNCHRONIZATION - Abstract
Summary: This paper studies the adaptive synchronization of complex spatio‐temporal networks modeled by semi‐linear hyperbolic partial differential equations (CSTNSLHPDEs) as well as considering time‐invariant and time‐varying delays in a one‐dimensional space. Firstly, a distributed adaptive controller is proposed, where different nodes are with different adaptive gains. Secondly, four cases, CSTNSLHPDEs with time‐invariant delays and one single weight, with time‐invariant delays and multi‐weights, with time‐varying delays and one single weight, and with time‐varying delays and multi‐weights, are successively analyzed, and synchronization conditions of these four cases are obtained by using the proposed distributed adaptive controller. In the end, examples illustrate the effectiveness of the proposed distributed adaptive controller. [ABSTRACT FROM AUTHOR]
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- 2024
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8. Numerical Recovering of Space-Dependent Sources in Hyperbolic Transmission Problems.
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Koleva, Miglena N. and Vulkov, Lubin G.
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HYPERBOLIC differential equations , *FINITE difference method , *INVERSE problems , *DIFFERENTIAL operators , *SOBOLEV spaces , *THEORY of wave motion - Abstract
A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, using solution point constraints. Applying a transform method, we reduce the inverse problems to direct ones, which are studied for well-posedness in special weighted Sobolev spaces. This means that the inverse problem is said to be well posed in the sense of Tikhonov (or conditionally well posed). The main aim of this study is to develop a finite difference method for solution of the transformed hyperbolic problems with a non-local differential operator and initial conditions. Numerical test examples are also analyzed. [ABSTRACT FROM AUTHOR]
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- 2024
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9. Prescribed‐time constrained tracking control of a class of 2 × 2 hyperbolic PDE systems with actuator dynamics.
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Xiao, Yu, Xu, Xiaodong, and Dubljevic, Stevan
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HYPERBOLIC differential equations , *PARTIAL differential equations , *ORDINARY differential equations , *SYSTEM dynamics , *NONLINEAR differential equations , *TRACKING algorithms , *ADAPTIVE control systems - Abstract
This article studies the problem of prescribed‐time constrained tracking control for a class of 2×2$$ 2\times 2 $$ hyperbolic partial differential equation (PDE) systems with actuator dynamics, which are described by a set of nonlinear ordinary differential equations (ODEs). Since the control input only appears in the ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite challenging. The most important is that for the control of such ODE controlled PDE systems we mainly make the following two contributions: (1) the controlled output of the PDE system tracks the reference signal within the prescribed time; (2) the controlled output and all the actuator states are constrained. It is the first time that such a prescribed‐time constrained tracking control problem is addressed for the PDE‐ODE coupled system considered in this article. Through rigorous theoretical proof, it is demonstrated that all the system states and control signals are bounded and sufficiently continuous by configuring appropriate design parameters. Finally, the performance is investigated via numerical simulation. [ABSTRACT FROM AUTHOR]
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- 2024
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10. Explicit and exact travelling wave solutions for Hirota equation and computerized mechanization.
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Li, Bacui, Wang, Fuzhang, and Nadeem, Sohail
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NONLINEAR differential equations , *PARTIAL differential equations , *MECHANIZATION , *NONLINEAR evolution equations , *EQUATIONS , *PHENOMENOLOGICAL theory (Physics) , *HYPERBOLIC differential equations - Abstract
By using the power-exponential function method and the extended hyperbolic auxiliary equation method, we present the explicit solutions of the subsidiary elliptic-like equation. With the aid of the subsidiary elliptic-like equation and a simple transformation, we obtain the exact solutions of Hirota equation which include bright solitary wave, dark solitary wave, bell profile solitary wave solutions and Jacobian elliptic function solutions. Some of these solutions are found for the first time, which may be useful for depicting nonlinear physical phenomena. This approach can also be applied to solve the other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]
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- 2024
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11. Existence and uniqueness for a coupled parabolic-hyperbolic model of MEMS.
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Gimperlein, Heiko, He, Runan, and Lacey, Andrew A.
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REYNOLDS equations , *HOLDER spaces , *MICROELECTROMECHANICAL systems , *PARABOLIC operators , *HYPERBOLIC differential equations - Abstract
Local wellposedness for a nonlinear parabolic-hyperbolic coupled system modeling Micro-Electro-Mechanical System (MEMS) is studied. The particular device considered is a simple capacitor with two closely separated plates, one of which has motion modeled by a semilinear hyperbolic equation. The gap between the plates contains a gas and the gas pressure is taken to obey a quasilinear parabolic Reynolds' equation. Local-in-time existence of strict solutions of the system is shown, using well-known local-in-time existence results for the hyperbolic equation, then Hölder continuous dependence of its solution on that of the parabolic equation, and finally getting local-in-time existence for a combined abstract parabolic problem. Semigroup approaches are vital for the local-in-time existence results. [ABSTRACT FROM AUTHOR]
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- 2024
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12. Recovering a Rapidly Oscillating Lower-Order Coefficient and a Source in a Hyperbolic Equation from Partial Asymptotics of a Solution.
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Levenshtam, V. B.
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HYPERBOLIC differential equations , *EVOLUTION equations , *INVERSE problems , *PERTURBATION theory , *EQUATIONS , *CAUCHY problem - Abstract
We consider the Cauchy problem for a one-dimensional hyperbolic equation whose lower-order coefficient and right-hand side oscillate in time with a high frequency and the amplitude of the lower-order coefficient is small. Under study is the reconstruction of the cofactors of these rapidly oscillating functions independent of the space variable from a partial asymptotics of a solution at some point of the space. The classical theory of inverse problems examines the numerous problems of determining unknown sources, and coefficients without rapid oscillations for various evolutionary equations, where the exact solution to the direct problem appears in the additional overdetermination condition. Equations with rapidly oscillating data are often encountered in modeling the physical, chemical, and other processes that occur in media subjected to high-frequency electromagnetic, acoustic, vibrational, and others fields, which demonstrates the topicality of perturbation theory problems on the reconstruction of unknown functions in high-frequency equations. We give some nonclassical algorithm for solving such problems that lies at the junction of asymptotic methods and inverse problems. In this case the overdetermination condition involves a partial asymptotics of solution of a certain length rather than the exact solution. [ABSTRACT FROM AUTHOR]
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- 2024
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13. A Stability Estimate for a Solution to an Inverse Problem for a Nonlinear Hyperbolic Equation.
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Romanov, V. G.
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INVERSE problems , *NONLINEAR equations , *HYPERBOLIC differential equations , *GEODESICS , *DIFFERENTIAL equations , *RIEMANNIAN metric , *CAUCHY problem - Abstract
We consider a hyperbolic equation with variable leading part and nonlinearity in the lower-order term. The coefficients of the equation are smooth functions constant beyond some compact domain in the three-dimensional space. A plane wave with direction falls to the heterogeneity from the exterior of this domain. A solution to the corresponding Cauchy problem for the original equation is measured at boundary points of the domain for a time interval including the moment of arrival of the wave at these points. The unit vector is assumed to be a parameter of the problem and can run through all possible values sequentially. We study the inverse problem of determining the coefficient of the nonlinearity on using this information about solutions. We describe the structure of a solution to the direct problem and demonstrate that the inverse problem reduces to an integral geometry problem. The latter problem consists of constructing the desired function on using given integrals of the product of this function and a weight function. The integrals are taken along the geodesic lines of the Riemannian metric associated with the leading part of the differential equation. We analyze this new problem and find some stability estimate for its solution, which yields a stability estimate for solutions to the inverse problem. [ABSTRACT FROM AUTHOR]
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- 2024
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14. Identities for Measures of Deviation from Solutions to Parabolic-Hyperbolic Equations.
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Repin, S. I.
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CAUCHY problem , *HYPERBOLIC differential equations , *EQUATIONS - Abstract
Integral identities that are fulfilled for measure of difference between the exact solution of a parabolic-hyperbolic equation and any functions from a corresponding energy class are proved. These identities make it possible to derive two-sided a posteriori estimates for approximate solution to the corresponding Cauchy problem. The left-hand side of such an estimate is a natural measure of deviation from the solution, and the right-hand side depends on the problem data and the approximate solution itself and, therefore, it can be explicitly calculated. These estimates can be used to control the accuracy of approximate solutions and to compare solutions to Cauchy problems with different initial conditions. These estimates also allow one to quantitatively assess the effects occurring due to inaccuracies in the initial data and in the coefficients of the equation. [ABSTRACT FROM AUTHOR]
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- 2024
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15. Numerical solution to inverse coefficient problem for hyperbolic equation under overspecified condition of general integral type.
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Qahtan, Jehan A. and Hussein, M. S.
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HYPERBOLIC differential equations , *INVERSE problems , *FINITE difference method , *TIKHONOV regularization , *EQUATIONS , *INTEGRALS - Abstract
We investigate solving the inverse coefficient issue for a hyperbolic equation under overspecified condition of general integral type. Crank–Nicolson finite difference method (FDM) combined with the trapezoidal rule quadrature has been used for direct problem. While, the inverse problem was reformulated as nonlinear regularized least-square optimization problem with simple bound and solved efficiently by MATLAB subroutine lsqnonlin from optimization toolbox. Since problem under investigation is generally ill-posed, small error in the input data lead to huge error in the output, then Tikhonov's regularization technique is applied to obtain a regularized stable results. [ABSTRACT FROM AUTHOR]
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- 2024
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16. On the application of maximum-entropy-inspired multi-Gaussian moment closure for multi-dimensional non-equilibrium gas kinetics.
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Brooks, K. A., Groth, C. P. T., and Laurent, F.
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MAXIMUM entropy method , *HYPERBOLIC differential equations , *PARTIAL differential equations , *KERNEL functions , *MOMENTS method (Statistics) , *GASES - Abstract
Maximum-entropy moment closures for describing non-equilibrium rare fied gaseous flow shave previously been shown to provide accurate and computationally efficient descriptions of transition-regime flows. Unfortunately, for high-order variants of these closures above second order in velocity space, there are no analytical closures for the systems of hyperbolic partial differential equations (PDEs) which govern the transport ofthe macroscopic moment quantities and instead approximate closures have been sought. In this study, abi-Gaussian approximation for then umber density function (NDF) is considered both for approximating the NDF and closing moment fluxes of the resulting fourth-order 14-moment maximum-entropy closure associated with fully three-dimensional kinetic theory. Prior investigations of the bi-Gaussian approximation applied to simplified one-dimensional univariate kinetic theory has yielded excellent results when compared to the actual maximum-entropy solutions as well a similar interpolative-based maximum-entropy-based (IBME) closure. In the one-dimensional univariate case, the bi-Gaussian closure is equivalent to the so-called extended quadrature method of moments (EQMOM) with anormal or Gaussian kernel basis function. A potential benefit of the bi-Gaussian approach proposed herein is that an essentially closed-form analytical expression results for the NDF. In this study, the extension of the bi-Gaussian closure to the multi-dimensional case is considered and compared to the equivalent multi-dimensional IBME closure. The approximate form for the NDF and closing fluxes interms of the relevant moments are derived and the validity and hyperbolicity of the closure for the space of realizable predicted moments are all explored and compared to those of the IBME closure. It is shown that the bi-Gaussian closure in the multi-dimensional case unfortunately suffers from several de ficiencies: firstly, the valid region of realizable moment space for the bi-Gaussian closure is a small subset of the full realizable 14-moment space; and secondly, the closure and moment equation eigen structure for solutions associated with zero heat fluxbe come undefined. The findings here in suggest that the proposed bi-Gaussian closure may not be agood choice for practical multi-dimensional rare fied flow predictions despite the promising results exhibited in the one-dimensional case. [ABSTRACT FROM AUTHOR]
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- 2024
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17. Modified iterated Crank-Nicolson method with improved accuracy for advection equations.
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Tran, Qiqi and Liu, Jinjie
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CRANK-nicolson method , *HYPERBOLIC differential equations , *PARTIAL differential equations , *GROSS-Pitaevskii equations , *ADVECTION , *ADVECTION-diffusion equations , *BURGERS' equation - Abstract
The iterated Crank-Nicolson (ICN) method is a successful numerical algorithm for solving partial differential equations. When unequal weights are used in the ICN predictor-corrector process, the convergence rate is reduced to first order. In this paper, we propose two modified ICN algorithms that achieve second order convergence rate, based on two different ways of choosing the weights. The first approach employs geometrically averaged weights in two consecutive iterations, and the second one uses arithmetically averaged weights for two consecutive time steps. The stability and second order accuracy of our methods are verified using stability and truncation error analysis and numerically demonstrated by solving linear and semi-linear hyperbolic partial differential equations, Burgers' equation, and the Gross-Pitaevskii equation. [ABSTRACT FROM AUTHOR]
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- 2024
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18. Explicit P 1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.
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Beilina, Larisa and Ruas, Vitoriano
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MAXWELL equations , *FINITE differences , *WAVE equation , *HYPERBOLIC differential equations , *ELECTROMAGNETISM , *ELECTRIC fields , *EQUATIONS - Abstract
In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell's equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell's equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell's equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space. [ABSTRACT FROM AUTHOR]
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- 2024
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19. Applications of Space–Time Elements.
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Epstein, Marcelo, Soleimani, Kasra, and Sudak, Leszek
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PARABOLIC differential equations , *HYPERBOLIC differential equations , *SPACETIME , *DIFFERENTIAL operators , *LINEAR operators - Abstract
The potential of a finite-element technique based on an egalitarian meshing of the space–time domain D of physical problems described by parabolic or hyperbolic differential equations is explored. A least-squares minimization technique is applied in the meshed domain D to obtain stiffness-like matrices associated with various linear differential operators. Applications discussed include problems of boundary growth, and diffusive coalescence, in which D cannot be regarded as the Cartesian product of two independent domains in space and time. [ABSTRACT FROM AUTHOR]
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- 2024
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20. Approximation of the derivatives beyond Taylor expansion.
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Xu, Qiuyan and Liu, Zhiyong
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TAYLOR'S series , *FINITE difference method , *SMOOTHNESS of functions , *RUNGE-Kutta formulas , *APPROXIMATION error , *HYPERBOLIC differential equations , *CRANK-nicolson method - Abstract
Different from the construction process of the traditional finite difference method, we derive a large class of high-accuracy methods for approximating derivatives. Since Taylor expansion is avoided, the requirement for function smoothness in the new methods is greatly reduced. We analyze the approximation errors of the proposed methods and compare their approximation effects. As a typical application, we use the proposed methods to solve Elliptic, Hyperbolic and Parabolic problems numerically. For time-dependent problems, a class of new fully implicit difference schemes and the fourth-order Runge-Kutta method are both used for discretization. When using fully implicit difference schemes, unconditional stability and convergence are proved for hyperbolic and parabolic problems with the periodic boundary conditions. A large number of numerical experiments are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]
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- 2024
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21. Critical degeneracy in a nonlinear hyperbolic equation can produce atypical instability.
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Stuart, C. A.
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NONLINEAR equations , *INITIAL value problems , *DIFFERENTIAL operators , *ACOUSTIC models , *BLACK holes , *WAVE equation , *HYPERBOLIC differential equations - Abstract
This paper deals with the initial value problem for a semilinear wave equation on a bounded domain and solutions are required to vanish on the boundary of this domain. The essential feature of the situation considered here is that the ellipticity of the spatial part of the differential operator degenerates like the square of the distance from given point in the domain and so hyperbolicity is lost at this point. The assumptions ensure that u ≡ 0 is a stationary solution of the problem and the object is to study the stability of this solution with respect to perturbations of the initial data. Stability, instability and asymptotic stability are all considered. The assumptions about the nonlinear terms ensure that the problem has a well-defined linearization at u ≡ 0. There are simple cases where this linearization is asymptotically stable but u ≡ 0 is an unstable solution of the nonlinear problem. We also establish conditions under which the stability of a stationary solution u ≢ 0 can be determined using our results. The quadratic degeneracy at a point treated here is typical of what is required in models for acoustic (or sonic) black holes. It also occurs in a simplified Wheeler–DeWitt model which we discuss in some detail. [ABSTRACT FROM AUTHOR]
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- 2024
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22. Exponential stabilisation of delayed distributed semilinear systems in Banach spaces.
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Ouarit, M. and Tsouli, A.
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BANACH spaces , *HYPERBOLIC differential equations , *COMPUTER simulation , *SEMILINEAR elliptic equations - Abstract
In this paper, we deal with the problem of stabilisation for a class of distributed semilinear systems. The considered systems present a state time delay and evolve in Banach spaces. A new stabilising feedback control is proposed. Two kinds of stabilizability, namely exponential and weak stabilizability, are investigated. An explicit estimate of the energy decay rate is given. Some illustrating applications to parabolic and hyperbolic equations with numerical simulations are considered. [ABSTRACT FROM AUTHOR]
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- 2024
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23. Diagnostics of Mathematical Models of Thermoelasticity. Part 1. Weak Solutions of Boundary-Value Problems and Formulation of the Problems of Diagnostics.
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Vikulov, A. G.
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BOUNDARY value problems , *THERMOELASTICITY , *MATHEMATICAL models , *HYPERBOLIC differential equations , *WAVE functions , *WAVE equation , *THERMAL stresses , *NONLINEAR equations - Abstract
The formulations and methods of solution of the problems of diagnostics of thermoelasticity on a stationary boundary of a solid body are considered. A weak formulation of the boundary-value problem is presented for a nonlinear hyperbolic equation (a nonstationary wave equation with density, elasticity modulus, and thermal stress depending on temperature) with mixed boundary conditions. An approximate solution of this problem in the space W 2 1 has been obtained, and an example of solution is given. Based on the weak formulation of the boundary-value problem, problems of diagnostics have been formulated in the sense of the theory of function traces. According to the theorem about the uniqueness of the relationship between the function and its trace in the space W 2 1 , a conclusion has been drawn about the need to control the convergence of the calculated wave function to the experimental one only on boundaries with unstable conditions in solving the problem of diagnostics. [ABSTRACT FROM AUTHOR]
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- 2024
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24. On Global Solutions of Hyperbolic Equations with Positive Coefficients at Nonlocal Potentials.
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Muravnik, Andrey B.
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HYPERBOLIC differential equations , *OPERATOR equations , *EQUATIONS , *INDEPENDENT variables - Abstract
We study hyperbolic equations with positive coefficients at potentials undergoing translations with respect to the spatial independent variable. The qualitative novelty of the investigation is that the real part of the symbol of the differential-difference operator contained in the equation is allowed to change its sign. Earlier, only the case where the said sign is constant was investigated. We find a condition relating the coefficient at the nonlocal term of the investigated equation and the length of the translation, guaranteeing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation. [ABSTRACT FROM AUTHOR]
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- 2024
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25. On a two-dimensional Dirichlet type problem for a linear hyperbolic equation of fourth order.
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Kiguradze, Tariel and Alhuzally, Reemah
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DIRICHLET problem , *LINEAR equations , *CONVEX domains , *HYPERBOLIC differential equations - Abstract
For a linear hyperbolic equation of fourth order, a Dirichlet type boundary problem in an orthogonally convex domain is investigated. Sharp sufficient conditions guaranteeing solvability and well-posedness of the problem under consideration are established. [ABSTRACT FROM AUTHOR]
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- 2024
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26. Global asymptotic stability for Gurtin-MacCamy's population dynamics model.
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Ma, Zhaohai and Magal, Pierre
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GLOBAL asymptotic stability , *POPULATION dynamics , *HYPERBOLIC differential equations , *PARTIAL differential equations , *NONLINEAR differential equations , *HOPF bifurcations - Abstract
In this paper, we investigate the global asymptotic stability of an age-structured population dynamics model with a Ricker's type of birth function. This model is a hyperbolic partial differential equation with a nonlinear and nonlocal boundary condition. We prove a uniform persistence result for the semiflow generated by this model. We obtain the existence of global attractors and we prove the global asymptotic stability of the positive equilibrium by using a suitable Lyapunov functional. Furthermore, we prove that our global asymptotic stability result is sharp, in the sense that Hopf bifurcation may occur as close as we want from the region global stability in the space of parameter. [ABSTRACT FROM AUTHOR]
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- 2024
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27. PDE‐constrained model predictive control of open‐channel systems.
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Xie, Yongfang, Zeng, Ningjun, Zhang, Shaohui, Cen, Lihui, and Chen, Xiaofang
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PREDICTIVE control systems , *ADJOINT differential equations , *OPTIMIZATION algorithms , *HYPERBOLIC differential equations , *PARTIAL differential equations , *CALCULUS of variations - Abstract
A PDE‐constrained model predictive control (MPC) algorithm for open‐channel systems based on the Saint‐Vevant(S‐V) equations is investigated in this paper. The S‐V equations, which precisely model the dynamics of open‐channel systems, are quasi‐linear hyperbolic partial differential equations (PDEs) without analytical solutions. Directly applying the S‐V equations to an MPC controller design becomes sophisticated. In this work, the calculus of variation is used to obtain the adjoint equations and the adjoint analysis method is utilized to deduce the gradients of the MPC optimization problem. Particularly, the physical constraints involving both the state and control variables are also considered. A gradient‐based optimization algorithm in combination with the numerical computation of Preissmann implicit scheme is proposed to solve the constrained MPC optimization problem. The control performances of the developed PDE‐constrained MPC algorithm with respect to the controlled water levels and gate openings are compared with those of the MPC controller designed for the linearized model. All the simulation tests are carried out on an aqueduct reach in Yehe Irrigation District in Hebei Province, China. The results show that the proposed PDE‐constrained MPC algorithm is a promising method in dealing with the constraints in terms of hyperbolic PDEs, control variables and state variables simultaneously. [ABSTRACT FROM AUTHOR]
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- 2024
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28. Stochastic finite volume method for uncertainty quantification of transient flow in gas pipeline networks.
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Tokareva, S., Zlotnik, A., and Gyrya, V.
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NATURAL gas pipelines , *GAS flow , *HYPERBOLIC differential equations , *PARTIAL differential equations , *RIEMANN-Hilbert problems , *FLUID flow - Abstract
We develop a weakly intrusive framework to simulate the propagation of uncertainty in solutions of generic hyperbolic partial differential equation systems on graph-connected domains with nodal coupling and boundary conditions. The method is based on the Stochastic Finite Volume (SFV) approach, and can be applied for uncertainty quantification (UQ) of the dynamical state of fluid flow over actuated transport networks. The numerical scheme has specific advantages for modeling intertemporal uncertainty in time-varying boundary parameters, which cannot be characterized by strict upper and lower (interval) bounds. We describe the scheme for a single pipe, and then formulate the controlled junction Riemann problem (JRP) that enables the extension to general network structures. We demonstrate the method's capabilities and performance characteristics using a standard benchmark test network. • Formulation of a stochastic hyperbolic PDE as a high-dimensional parametric PDE. • Novel SFV method for quantification of intertemporal uncertainty in systems of conservation laws posed on graphs. • Formulation of the stochastic junction Riemann problem to compute numerical fluxes at the nodes of the graph. • Application of the SFV method to a real-world problem of stochastic gas flows on networks. • Convergence analysis for the statistical moments of the solution. [ABSTRACT FROM AUTHOR]
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- 2024
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29. On the Riemann problem and interaction of elementary waves for two‐layered blood flow model through arteries.
- Author
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Jana, Sumita and Kuila, Sahadeb
- Subjects
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RIEMANN-Hilbert problems , *HYPERBOLIC differential equations , *PARTIAL differential equations , *SHOCK waves , *ARTERIES , *BLOOD flow - Abstract
In this paper, we focus on the Riemann problem for two‐layered blood flow model, which is represented by a system of quasi‐linear hyperbolic partial differential equations (PDEs) derived from the Euler equations by vertical averaging across each layer. We consider the Riemann problem with varying velocities and equal constant density through arteries. For instance, the flow layer close to the wall of vessel has a slower average speed than the layer far from the vessel because of the viscous effect of the blood vessel. We first establish the existence and uniqueness of the corresponding Riemann solution by a thorough investigation of the properties of elementary waves, namely, shock wave, rarefaction wave, and contact discontinuity wave. Further, we extensively analyze the elementary wave interaction between rarefaction wave and shock wave with contact discontinuity and rarefaction wave and shock wave. The global structure of the Riemann solutions after each wave interaction is explicitly constructed and graphically illustrated towards the end. [ABSTRACT FROM AUTHOR]
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- 2024
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30. An improved dynamical Poisson equation solver for self-gravity.
- Author
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Maeda, Ryunosuke, Inoue, Tsuyoshi, and Inutsuka, Shu-ichiro
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POISSON'S equation , *HYPERBOLIC differential equations , *FAST Fourier transforms , *PARTIAL differential equations , *SPEED of sound , *ELLIPTIC equations , *PHASE velocity , *GALAXY formation - Abstract
Since self-gravity is crucial in the structure formation of the Universe, many hydrodynamics simulations with the effect of self-gravity have been conducted. The multigrid method is widely used as a solver for the Poisson equation of the self-gravity; however, the parallelization efficiency of the multigrid method becomes worse when we use a massively parallel computer, and it becomes inefficient with more than 104 cores, even for highly tuned codes. To perform large-scale parallel simulations (>104 cores), developing a new gravity solver with good parallelization efficiency is beneficial. In this article, we develop a new self-gravity solver using the telegraph equation with a damping coefficient, κ. Parallelization is much easier than the case of the elliptic Poisson equation since the telegraph equation is a hyperbolic partial differential equation. We analyse convergence tests of our telegraph equations solver and determine that the best non-dimensional damping coefficient of the telegraph equations is |$\tilde{\kappa } \simeq 2.5$|. We also show that our method can maintain high parallelization efficiency even for massively parallel computations due to the hyperbolic nature of the telegraphic equation by weak-scaling tests. If the time-step of the calculation is determined by heating/cooling or chemical reactions, rather than the Courant–Friedrichs–Lewy (CFL) condition, our method may provide the method for calculating self-gravity faster than other previously known methods such as the fast Fourier transform and multigrid iteration solvers because gravitational phase velocity determined by the CFL condition using these time-scales is much larger than the fluid velocity plus sound speed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. Analysis of the Dynamic Characteristics of Conical Shells of Variable Thickness on an Elastic Bed Under Unsteady Loading.
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Lugovyi, P. Z., Meish, Yu. A., Orlenko, S. P., and Arnauta, N. V.
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CONICAL shells , *FINITE difference method , *HYPERBOLIC differential equations , *TIME integration scheme , *INTERPOLATION algorithms , *FINITE differences , *EQUATIONS of motion , *VARIATIONAL principles - Abstract
The model of Timoshenko's shell theory of shells was used to analyze the dynamic characteristics of conical shells of variable thickness on a Pasternak elastic bed under nonstationary loading. Based on the Hamilton–Ostrogradsky variational principle, the equations of motion of a conical shell of variable thickness on a Pasternak elastic bed were derived. This system of hyperbolic differential equations is solved by the finite difference method. The numerical algorithm for solving the obtained equations is based on applying the integral-interpolation method for constructing difference schemes in the spatial coordinate and an explicit finite difference scheme for integration in the time coordinate. The influence of geometric dimensions, taper angle, and elastic media on the natural frequencies and other dynamic characteristics of a conical shell of variable thickness under the action of a pulsed load is analyzed using specific examples. New mechanical effects are revealed. [ABSTRACT FROM AUTHOR]
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- 2024
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32. Modeling and Quantifying Parameter Uncertainty of Co‐Seismic Non‐Classical Nonlinearity in Rocks.
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Niu, Zihua, Gabriel, Alice‐Agnes, Seelinger, Linus, and Igel, Heiner
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SEISMIC waves , *MARKOV chain Monte Carlo , *ELASTIC wave propagation , *HYPERBOLIC differential equations , *SEISMIC response , *EFFECT of earthquakes on bridges , *DAMAGE models , *NONLINEAR waves - Abstract
Dynamic perturbations reveal unconventional nonlinear behavior in rocks, as evidenced by field and laboratory studies. During the passage of seismic waves, rocks exhibit a decrease in elastic moduli, slowly recovering after. Yet, comprehensive physical models describing these moduli alterations remain sparse and insufficiently validated against observations. Here, we demonstrate the applicability of two physical damage models—the internal variable model (IVM) and the continuum damage model (CDM)—to provide quantitative descriptions of nonlinear co‐seismic elastic wave propagation observations. While the IVM uses one internal variable to describe the evolution of elastic material moduli, the CDM damage variable is a mathematical representation of microscopic defects. We recast the IVM and CDM models as nonlinear hyperbolic partial differential equations and implement 1D and 2D numerical simulations using an arbitrary high‐order discontinuous Galerkin method. We verify the modeling results with co‐propagating acousto‐elastic experimental measurements. Subsequently, we infer the parameters for these nonlinear models from laboratory experiments using probabilistic Bayesian inversion and 2D simulations. By adopting the Adaptive Metropolis Markov chain Monte Carlo method, we quantify the uncertainties of inferred parameters for both physical models, investigating their interplay in 70,000 simulations. We find that the damage variables can trade off with the stress‐strain nonlinearity in discernible ways. We discuss physical interpretations of both damage models and that our CDM quantitatively captures an observed damage increase with perturbation frequency. Our results contribute to a more holistic understanding of co‐seismic damage and post‐seismic recovery after earthquakes bridging the worlds of theoretical analysis and laboratory findings. Plain Language Summary: Rocks react to earthquakes by softening when seismic waves—the energy released by earthquakes—pass through them. Observations of such rock softening during the passage of seismic waves are common both in the laboratory and in the field. Interestingly, rocks gradually harden again once the shaking stops. Different physical mechanisms have been proposed to explain the observations. In this study, we put two existing theories to the test. One assumes that a term in the internal energy of the material increases with damage accumulation, while the other incorporates the opening and closing of micro‐cracks. We implement them into a powerful simulation program called ExaHyPE. This allows us to model how nonlinear waves move through rocks. When we compare the computer simulation outcomes with real laboratory tests, we find that both models match what we see in reality. Studying thousands of simulations with different model parameters, we find some intriguing insights. For instance, the initial state of strain and the tiny cracks that open and close within the rock may be key to understanding the hardening and softening process. We hope to use these physical models in future earthquake simulations, offering more accurate predictions of how our Earth's crust reacts to earthquakes. Key Points: We analyze two physical models suitable for simulations of nonlinear elastic wave propagation observed in the laboratoryThe experimentally observed co‐seismic acoustic modulus drop can be explained with nonlinear damage modelsWe use the Markov chain Monte Carlo method to explore connections and uncertainties of nonlinear parameters [ABSTRACT FROM AUTHOR]
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- 2024
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33. "Differential Equations of Mathematical Physics and Related Problems of Mechanics"—Editorial 2021–2023.
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Matevossian, Hovik A.
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DIFFERENTIAL equations , *HYPERBOLIC differential equations , *LINEAR differential equations , *LAPLACE'S equation , *BOUNDARY value problems , *INVERSE problems , *MATHEMATICAL physics , *DIFFERENTIAL operators - Abstract
This document is an editorial for a special issue of the journal Mathematics titled "Differential Equations of Mathematical Physics and Related Problems of Mechanics." The special issue covers a range of topics related to differential equations in mathematical physics and mechanics, including wave equations, spectral theory, scattering, and inverse problems. The editorial provides a summary of the published papers in the special issue, highlighting their contributions to the field. The document emphasizes the importance of the special issue in covering both applied and fundamental aspects of mathematics, physics, and their applications in various fields. The author expresses gratitude to the authors, reviewers, assistants, associate editors, and editors for their contributions to the special issue. The report does not provide specific details about the content of the papers or the nature of the special issue. [Extracted from the article]
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- 2024
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34. A relaxation approach to modeling properties of hyperbolic–parabolic type models.
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Abreu, Eduardo, Santo, Arthur Espírito, Lambert, Wanderson, and Pérez, John
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HYPERBOLIC differential equations , *TRANSPORT equation , *DISCONTINUOUS coefficients , *PARTIAL differential equations , *NONLINEAR equations - Abstract
In this work, we propose a novel relaxation modeling approach for partial differential equations (PDEs) involving convective and diffusive terms. We reformulate the original convection–diffusion problem as a system of hyperbolic equations coupled with relaxation terms. In contrast to existing literature on relaxation modeling, where the solution of the reformulated problem converges to certain types of equations in the diffusive limit, our formalism treats the augmented problem as a system of coupled hyperbolic equations with relaxation acting on both the convective flux and the source term. Furthermore, we demonstrate that the new system of equations satisfies Liu's sub-characteristic condition. To verify the robustness of our proposed approach, we perform numerical experiments on various important models, including nonlinear convection–diffusion problems with discontinuous coefficients. The results show the promising potential of our relaxation modeling approach for both pure and applied mathematical sciences, with applications in different models and areas. • We propose a novel relaxation approach for modeling convection-diffusion problems via a hyperbolic system. • The new system satisfies Liu's sub-characteristic condition, ensuring solution stability. • We apply the relaxation modeling to both classical and non-classical models. • From numerical experiments, we verify the new system's solutions converge to those of the original equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. Output regulation and tracking for linear ODE-hyperbolic PDE–ODE systems.
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Redaud, Jeanne, Bribiesca-Argomedo, Federico, and Auriol, Jean
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HYPERBOLIC differential equations , *PARTIAL differential equations , *ORDINARY differential equations , *BACKSTEPPING control method , *CLOSED loop systems , *ADAPTIVE control systems - Abstract
This paper proposes a constructive solution to the output regulation — output tracking problem for a general class of interconnected systems. The class of systems under consideration consists of a linear 2 × 2 hyperbolic Partial Differential Equations (PDE) system coupled at both ends with Ordinary Differential Equations (ODEs). The proximal ODE system, which represents actuator dynamics, is actuated. Colocated measurements are available. The distal ODE system represents the load dynamics. The control objective is to ensure, in the presence of a disturbance signal (regulation problem), that a virtual output exponentially converges to zero. By doing so, we can ensure that a state component of the distal ODE state robustly converges towards a known reference trajectory (output tracking problem) even in the presence of a disturbance with a known structure. The proposed approach combines the backstepping methodology and frequency analysis techniques. We first map the original system to a simpler target system using an invertible integral change of coordinates. From there, we design an adequate full-state feedback controller in the frequency domain. Following a similar approach, we propose a state observer that estimates the state and reconstructs the disturbance from the available measurement. Combining the full-state feedback controller with the state estimation results in a dynamic output-feedback control law. Finally, existing filtering techniques guarantee the closed-loop system robustness properties. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. Global well-posedness of 2D Hyperbolic perturbation of the Navier–Stokes system in a thin strip.
- Author
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Aarach, Nacer
- Subjects
- *
NAVIER-Stokes equations , *LITTLEWOOD-Paley theory , *HYPERBOLIC differential equations - Abstract
In this paper, we study a hyperbolic version of the Navier–Stokes equations, obtained by using the approximation by relaxation of the Euler system, evolving in a thin strip domain. The formal limit of these equations is a hyperbolic Prandtl type equation, our goal is to prove the existence and uniqueness of a global solution to these equations for analytic initial data in the tangential variable, under a uniform smallness assumption. Then we justify the limit from the anisotropic hyperbolic Navier–Stokes system to the hydrostatic hyperbolic Navier–Stokes system with small analytic data. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. Dispersion analysis of SPH as a way to understand its order of approximation.
- Author
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Stoyanovskaya, O.P., Lisitsa, V.V., Anoshin, S.A., Savvateeva, T.A., and Markelova, T.V.
- Subjects
- *
PARTICLE size determination , *THEORY of wave motion , *DISPERSION relations , *DERIVATIVES (Mathematics) , *GAS dynamics , *HYPERBOLIC differential equations - Abstract
Smoothed Particle Hydrodynamics (SPH) is a numerical method to solve dynamical partial differential equations (PDE). The basis of the method is a ¡¡kernel-based¿¿ way to compute the spatial derivatives of a function whose values are given in moving irregularly located nodes (Lagrangian particles). Accuracy of the SPH is determined by independent parameters — the shape of the kernel, the kernel size h , the distance between the particles Δ x. Constructing high-order SPH-schemes for different types of PDE is a state-of-the-art problem of computational mathematics. For the classical SPH-approximation of one-dimensional hyperbolic equations (isothermal gas dynamics) we found that the order of approximation of smooth solution correlates to the dispersion properties of the method. To this end we analyzed the dispersion relation for the approximation and found analytical representation of the numerical wave phase velocity. Moreover, for the first time, the order of approximation with respect to Δ x / h was confirmed in computational experiments on a dynamic problem of sound wave propagation. For two kernels with 2 and 4 continuum derivatives, the second and the fourth order of approximation, respectively, was found. This finding may be generalized as follows. The solution error in the one-dimensional case for a quasi-uniformly located particles has the form O ( (h λ) η + (Δ x h) ξ) , where ξ is a parameter determined by the shape of kernel (its smoothness, i.e. the number of continuum derivatives), η is a parameter that does not depend on the shape of kernel (for classical non-negative kernels η = 2), λ is the wavelength. Our results indicates that to develop high-order SPH-schemes for hyperbolic equations besides improving the order of approximation with respect to h / λ one need to ensure the order of approximation with respect to Δ x / h. To this end kernels of which smoothness is at least 4 are necessary. • Dispersion relation for SPH allows to find the approximation order on smooth solution • Order of SPH approximation with the number of neighbors depends on the kernel • 4th order of approximation with the number of neighbors is confirmed in simulation [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. Adaptive constrained tracking control for dynamical actuator driven linear 2 × 2 hyperbolic PDE systems with nonlinear uncertainties.
- Author
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Xiao, Yu, Yuan, Yuan, Xu, Xiaodong, and Dubljevic, Stevan
- Subjects
- *
ADAPTIVE control systems , *TRACKING algorithms , *HYPERBOLIC differential equations , *PARTIAL differential equations , *ACTUATORS , *NONLINEAR systems , *BACKSTEPPING control method - Abstract
The paper develops an adaptive constrained tracking control technique for a class of 2 × 2 hyperbolic partial differential equation (PDE) systems with boundary actuator dynamics, which are described by a set of ordinary differential equations (ODEs) in the presence of unknown parametric nonlinearities. Since the control input only appears in the uncertain ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite difficult and the existing direct boundary control approaches are ineffective. Moreover, in this paper, a more challenging problem is considered such that the controlled output and the states of ODE actuators are constrained. To this end, by utilizing finite and infinite dimensional backstepping techniques, barrier Lyapunov functions (BLFs) and adaptive methods, a novel adaptive tracking control approach is proposed. It is the first time that such a constrained tracking control problem is addressed for the PDE-ODE coupled systems considered in this paper. On the basis of the presented method, the rigorous theoretical proof is provided to show that the PDE controlled output and all the states of the ODE actuator stay within the predefined compact sets. Finally, the results are illustrated via a comparative numerical simulation. [Display omitted] • For a linear 2 × 2 hyperbolic PDE systems, the high-order unknown nonlinear actuator dynamics described by a set of ODEs are considered in the boundary point of the PDE. A tracking controller with adaptive dynamic compensation mechanism is developed such that the tracking control problem is addressed for the PDE-ODE coupled system considered in this paper. • For practical requirements for state constraints such as safe operations, physical limitations and so on, the Barrier Lyapunov function is employed to constrain the spatially distributed output of infinite-dimensional system and all the actuator states. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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