Hyperbolic Equations Research Articles

Optimal stability results and nonlinear duality for L∞ entropy and L1 viscosity solutions

We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality:(⋆)∫|Stu0−Stv0|φ0dx≤∫|u0−v0|Gtφ0dx,∀φ0≥0,∀u0,∀v0, where St is the entropy solution semigroup of the anisotropic degenerate parabolic equation∂tu+divF(u)=div(A(u)Du), and where we look for the smallest semigroup Gt satisfying (⋆). This amounts to finding an optimal weighted L1 contraction estimate for St. Our main result is that Gt is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation∂tφ=supξ⁡{F′(ξ)⋅Dφ+tr(A(ξ)D2φ)}. Since weighted L1 contraction results are mainly used for possibly nonintegrable L∞ solutions u, the natural spaces behind this duality are L∞ for St and L1 for Gt. We therefore develop a corresponding L1 theory for viscosity solutions φ. But L1 itself is too large for well-posedness, and we rigorously identify the weakest L1 type Banach setting where we can have it – a subspace of L1 called Lint∞. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018].

Nonlinear Inverse Problems for First Order Hyperbolic Equations

Nonlinear Inverse Problems for First Order Hyperbolic Equations

Design of hyperbolic quasi-zero stiffness metastructures coupled with nonlinear stiffness for low-frequency vibration isolation

Quasi-zero stiffness (QZS) metastructures with simple structures exhibit a high low-frequency vibration isolation performance, but the bearing capacity and working regions are limited due to the linear positive stiffness elements. In this study, a novel hyperbolic QZS metastructure coupled with nonlinear stiffness is designed by combining a pair of cosine-shaped beams and arc-shaped beams for low-frequency vibration isolation. Finite element models (FEM) and experimental analysis are utilized to examine the static characteristics of the metastructure, revealing a wide QZS region and significant load-bearing capacity. The metastructure can achieve effective vibration isolation properties through reasonably designing the unit cells. The influence of geometrical parameters on the QZS region is investigated to optimize and realize desirable QZS features. The dynamic equation and response of the vibration isolation system are determined using the harmonic balance method. Moreover, the corresponding displacement transmissibility is confirmed by numerical simulation. The results of the dynamic sweep frequency experiment further verify the validity of the theoretical analysis model. The hyperbolic QZS metastructure system possesses good vibration isolation performance due to the introduction and compensation of a nonlinear positive stiffness arc beam element. The proposed hyperbolic QZS metastructure provides a significant route for the stiffness compensation of metastructures.

A one-dimensional augmented Shallow Water Equations system for channels of arbitrary cross-section

This work provides a new formulation of the one-dimensional augmented Shallow Water Equations system for open channels and rivers with arbitrarily shaped cross sections, suitable for numerical integration when discontinuous geometry is encountered. The additional variable considered can be the bottom elevation, a reference width, a shape coefficient, or a vector containing these or other geometric parameters. The appropriate numerical method, which is well suited to coupling with the mathematical one, is a path-conservative method, capable of reconstructing the behaviour of physical and geometrical variables at the cell boundaries, where the discrete solution of hyperbolic systems of equations is discontinuous. A nonlinear path suitable for the shallow water context is adopted. The resulting model is shown to be well-balanced and accurate to the second order and is further validated against analytical solutions related to channels with power-law cross-sections, specifically for dam break patterns over a variable-width channel and the run-up dynamics of long water waves over sloping bays.

A robust numerical method based on a deep-learning operator for solving the 2D acoustic wave equation

The numerical solution of a wave equation plays a crucial role in computational geophysics problems, which forms the foundation of inverse problems and directly impacts the high-precision imaging results of earth models. However, common numerical methods often lead to significant computational and storage requirements. Due to the heavy reliance on forward-modeling methods in inversion techniques, particularly full-waveform inversion, enhancing the computational efficiency and reducing the storage demands of traditional numerical methods have become key issues in computational geophysics. We develop the deep Lax-Wendroff correction (DeepLWC) method, a deep-learning-based numerical format for solving 2D hyperbolic wave equations. DeepLWC combines the advantages of traditional numerical schemes with a deep neural network. We provide a detailed comparison of this method with the representative traditional Lax-Wendroff correction method. Our numerical results indicate that the DeepLWC significantly improves the calculation speed (by more than 10 times) and reduces the space needed for storage by more than 10,000 times compared with traditional numerical methods. In contrast to the more popular physics-informed neural network method, DeepLWC maximizes the advantages of traditional mathematical methods in solving partial differential equations and uses a new sampling approach, leading to improved accuracy and faster computations. It is particularly worth pointing out that DeepLWC introduces a novel research paradigm for numerical equation solving, which can be combined with various traditional numerical methods, enabling acceleration and reduction in the storage requirements of conventional approaches.

Efficient WENO Schemes for Nonuniform Grids

A set of arbitrarily high-order WENO schemes for reconstructions on nonuniform grids is presented. These non-linear interpolation methods use simple smoothness indicators with a linear cost with respect to the order, making them easy to implement and computationally efficient. The theoretical analysis to verify the accuracy and the essentially non-oscillatory properties are presented together with some numerical experiments involving algebraic problems in order to validate them. Also, these general schemes are applied for the solution of conservation laws and hyperbolic systems in the context of finite volume methods.

Explicit and exact travelling wave solutions for Hirota equation and computerized mechanization.

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.

Solution forms for generalized hyperbolic cotangent type systems of p-difference equations

Due to the recent increasing interest in hyperbolic-cotangent types of scalar-or two-dimensional systems of difference equations and treatment of some particular states. This paper presents a natural extension of the p-dimensional of four-systems of this generalized type and treats general states. Which is an extension of Stevic's work (J. Inequal. Appl., 2021, 184 (2021)). We also show these systems are solvable by using appropriate variable transformations and obtaining systems of homogeneous linear difference equations with constant coefficients. Some numerical examples of these systems are presented.

Exponential Stability of the Numerical Solution of a Hyperbolic System with Nonlocal Characteristic Velocities

In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is given. A difference scheme is constructed for the numerical solution of the considered initial boundary value problem. The definition of the exponential stability of the numerical solution in ℓ2-norm with respect to a discrete perturbation of the equilibrium state of the initial boundary value difference problem is given. A discrete Lyapunov function for a numerical solution is constructed, and a theorem on the exponential stability of a stationary solution of the initial boundary value difference problem in ℓ2-norm with respect to a discrete perturbation is proved.

Distributed Control for Non-Cooperative Systems Governed by Time-Fractional Hyperbolic Operators

This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system’s state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed.

Boundary Value Problems for a Model Hyperbolic Equation of the Third Order with Non-local Conditions of the First Kind

Considers nonlocal problems of the first kind for a model third-order hyperbolic equation. The main goal of the article is to prove the solvability of nonlocal problems of the first kind for a model third-order hyperbolic equation. Using the Riemann function method, the problem is reduced to Volterra integral equations of the second kind. Using the method of integral equations, the existence of a unique solution to nonlocal problems of the first kind is proven. The resulting solution to nonlocal problems of the first kind makes it possible to describe the process of moisture transfer in soils, heat transfer in a heterogeneous medium, and fluid filtration in porous media.

Admissibility and mean hyperbolicity for evolution equations

Mean hyperbolic systems permit non-hyperbolic behavior at certain moments during the evolution process T(t,s), owing to non-fixed error hyperbolic degree ε(t,s). Despite the coexistence of compression and expansion behaviors in generalized stable and unstable subspaces, mean hyperbolic systems admit fixed average contraction and expansion rates measured at sufficiently long time. To discuss the relationship between admissibility and mean hyperbolicity for evolution equations, our main goal is to construct admissible function classes and invariant decomposition with generalized stable and unstable subspaces. As applications, the mean hyperbolicity of damped wave equations with variable coefficients and the roughness of mean hyperbolicity are presented.

Regularity of the Hausdorff dimension of some hyperbolic systems

We study the regularity of the Hausdorff dimension for certain hyperbolic basic sets, as a function of the dynamical system. The main results ensure that the Hausdorff dimension of certain multidimensional thin hyperbolic attractors and of certain nonconformal iterated function systems varies smoothly and analytically with respect to the diffeomorphism. Similar results hold for average conformal hyperbolic sets and average conformal repellers.

Periodic Optimal Control of a Plug Flow Reactor Model with an Isoperimetric Constraint

We study a class of nonlinear hyperbolic partial differential equations with boundary control. This class describes chemical reactions of the type “A→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\rightarrow $$\\end{document} product” carried out in a plug flow reactor (PFR) in the presence of an inert component. An isoperimetric optimal control problem with periodic boundary conditions and input constraints is formulated for the considered mathematical model in order to maximize the mean amount of product over the period. For the single-input system, the optimality of a bang-bang control strategy is proved in the class of bounded measurable inputs. The case of controlled flow rate input is also analyzed by exploiting the method of characteristics. A case study is performed to illustrate the performance of the reaction model under different control strategies.

A one-dimensional computational model for blood flow in an elastic blood vessel with a rigid catheter.

Strokes are one of the leading causes of death in the United States. Stroke treatment involves removal or dissolution of the obstruction (usually a clot) in the blocked artery by catheter insertion. A computer simulation to systematically plan such patient-specific treatments needs a network of about 105 blood vessels including collaterals. The existing computational fluid dynamic (CFD) solvers are not employed for stroke treatment planning as they are incapable of providing solutions for such big arterial trees in a reasonable amount of time. This work presents a novel one-dimensional mathematical formulation for blood flow modeling in an elastic blood vessel with a centrally placed rigid catheter. The governing equations are first-order hyperbolic partial differential equations, and the hypergeometric function needs to be computed to obtain the characteristic system of these hyperbolic equations. We employed the Discontinuous Galerkin method to solve the hyperbolic system and validated the implementation by comparing it against a well-established 3D CFD solver using idealized vessels and a realistic truncated arterial network. The results showed clinically insignificant differences in steady flow cases, with overall variations between 1D and 3D models remaining below 10%. Additionally, the solver accurately captured wave reflection phenomena at domain discontinuities in unsteady cases. A primary advantage of this model over 3D solvers is its ease in obtaining a discretized geometry of complex vasculatures with multiple arterial branches. Thus, the 1D computational model offers good accuracy and applicability in simulating complex vasculatures, demonstrating promising potential for investigating patient-specific endovascular interventions in strokes.

Relaxation and asymptotic expansion of controlled stiff differential equations

The control of relaxation-type systems of ordinary differential equations is investigated using the Hamilton–Jacobi–Bellman equation. Firstly, we recast the model as a singularly perturbed dynamics which we embed in a family of controlled systems. Then we study this dynamics together with the value function of the associated optimal control problem. We provide an asymptotic expansion in the relaxation parameter of the value function. We also show that its solution converges toward the solution of a Hamilton–Jacobi–Bellman equation for a reduced control problem. Such systems are motivated by semi-discretisation of kinetic and hyperbolic partial differential equations. Several examples are presented including Jin–Xin relaxation.

On a quasilinear parabolic–hyperbolic system arising in MEMS modeling

AbstractA coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions.

A numerical technique based on Legendre wavelet for linear and nonlinear hyperbolic telegraph equation

A numerical technique based on Legendre wavelet for linear and nonlinear hyperbolic telegraph equation

Collaborative positioning for emergency rescuers based on INS, GPS and ZigBee

Multi-person cooperative positioning is one of the main techniques to improve the success rate of search and rescue. Nevertheless, current integrated positioning solutions for ZigBee and inertial navigation system (INS) are insufficiently versatile and commonly necessitate pre-existing infrastructure within the area. In this study, it is not necessary to install ZigBee positioning anchors in the environment in advance, and all positioning devices were mounted on rescuers, in contrast to previous positioning methods. First, an improved time difference of arrival (TDOA) method is proposed, which weights the position coordinates to minimize positioning errors. Then, the position data of the commander and the rescuers are substituted into the hyperbolic equations of the improved TDOA method, and the initial position of the rescuers is solved by Taylor method. Finally, based on the initial position, an improved Extended Kalman filter (EKF) position update technology is proposed, which uses ZigBee positioning system to monitor the east and north position of rescuers, and uses barometers to measure the altitude of rescuers. The magnetometers are used to limit the heading angle of the rescuers, so as to realize the high-precision 3D positioning of the emergency rescuers in the harsh environment. In this paper, three different scenarios are used to verify the effectiveness of the proposed method. The experimental results demonstrate that, when compared to the inertial positioning method, the traditional TDOA method, and the cooperative positioning method based on the relative ranging constraints of UWB and INS, respectively, the average positioning accuracy of the proposed method is 68.58%, 48.71%, and 13.59% higher.

Penalty Sponge Layers (PSL) for hyperbolic systems. General formulation, well-posedness and stability

In this work we design and analyze an alternative to the Absorbing Boundary Conditions (ABC) or Perfectly Matched Layers (PML) technique to solve symmetric first order hyperbolic systems in unbounded domains. We propose a general formulation of absorbing layers that we call Penalty Sponge Layer (PSL). Well-posedness as well as stability properties are quite simply inherited from the original problem. Such a formulation is written in terms of a set of auxiliary variables similar to those encountered in the Complex Frequency Shifted PML (CFS-PML) technique, but this time they have the role of a penalty term, hence the name PSL. This second role of penalty, ensured by stretching an artificial frequency to infinity in the vicinity of the interface, makes the layer asymptotically non-reflecting. Numerical tests are performed to compare them with Standard Sponge Layers (SSL) and PMLs, and to validate their success, even in severe situations where the latter fail (e.g., the Euler system at oblique subsonic/critical or supersonic flow and anisotropic waves).