Linear Hyperbolic Equations Research Articles

Qualitative analysis of fourth-order hyperbolic equations

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

A corrected Raviart–Thomas Spectral Difference scheme stable for arbitrary order of accuracy on triangular and tetrahedral meshes

A theoretical stability estimate is presented for the Raviart–Thomas Spectral Difference scheme for triangular and tetrahedral meshes. It reveals the difficulty of guaranteeing the existence of stable flux points for an arbitrary order of accuracy. To obtain a numerical method stable for any order of accuracy and any space dimension, the present article introduces a new correction term of the flux approximation to cancel the numerical error caused by the Riemann solver exactly. This modification enables the introduction of a new family of so-called Corrected Raviart–Thomas Spectral Difference stable for any polynomial order of approximation. Stability estimates are provided for the L2 norm and a weighted broken Sobolev norm. Numerical experiments in 2D and 3D are conducted for solving linear and non-linear hyperbolic equations with a polynomial order up to p=8 for triangular elements and up to p=5 for tetrahedral elements. They confirm the accuracy, stability, and robustness of the newly introduced numerical scheme. The new method is shown to have a computational cost similar to the Raviart–Thomas Spectral Difference with – for a given mesh – a higher accuracy.

A reduction procedure for determining exact solutions of second order hyperbolic equations

In this paper we develop a systematic reduction procedure for determining intermediate integrals of second order hyperbolic equations so that exact solutions of the second order PDEs under interest can be obtained by solving first order PDEs. We give some conditions in order that such a procedure holds and, in particular, we characterize classes of linear second order hyperbolic equations for which the general solution can be found.

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

Positivity-preserving discontinuous Galerkin scheme for linear hyperbolic equations with characteristics-informed augmentation

This paper presents a positivity-preserving discontinuous Galerkin (DG) scheme for the linear hyperbolic problem with variable coefficients on structured Cartesian domains. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the unmodulated augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. A simple scaling limiter can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients. To improve efficiency, an inexact augmented basis function can be obtained rather than a analytic non-polynomial solution to the auxiliary problem from the method of characteristics.

Representation formulae for linear hyperbolic curvature flows

In [19], Smoczyk showed that parabolic expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we obtain representation formula for the corresponding linear hyperbolic evolution equation for the support function of locally convex curves of general nonzero rotation number, finding some natural similarities and differences in behaviour of solutions as compared with the parabolic case. As applications, we consider linear hyperbolic flows that preserve length or generalised enclosed area of the evolving curve, flow of curves with Neumann boundary conditions inside cones and hyperbolic approaches to the Yau problem of flowing one curve to another. We also consider radial graphs evolving by similar linear equations and linear higher order hyperbolic equations. Our results may be compared with the corresponding parabolic cases consider by the first author, Schrader and Wheeler, in [17].

Asymptotic Errors in the Superconvergence of Discontinuous Galerkin Methods Based on Upwind-Biased Fluxes for 1D Linear Hyperbolic Equations

In this paper, we study the superconvergence of the semi-discrete discontinuous Galerkin (DG) method for linear hyperbolic equations in one spatial dimension. The asymptotic errors in cell averages, downwind point values, and the postprocessed solution are derived for the initial discretization by Gaussian projection (for periodic boundary condition) or Cao projection Cao et al. (SIAM J. Numer. Anal. 5, 2555–2573 (2014)) (for Dirichlet boundary condition). We proved that the error constant in the superconvergence of order 2k+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2k+1$$\\end{document} for DG methods based on upwind-biased fluxes depends on the parity of the order k. The asymptotic errors are demonstrated by various numerical experiments for scalar and vector hyperbolic equations.

Extension of Haar wavelet technique for numerical solution of three-dimensional linear and nonlinear telegraph equations

In this manuscript, a hybrid numerical technique is presented for solving three-dimensional hyperbolic telegraph equations. The proposed technique is based on the Haar wavelet collocation method and the finite difference method. In this technique, the space derivatives are estimated by truncated Haar wavelet series, while the time derivatives are approximated using finite difference method. The method is applied to some linear and non-linear three-dimensional hyperbolic telegraph equations. In the linear case, the resulting algebraic system is solved by the Gauss elimination technique. Whereas in the nonlinear case, the quasi-Newton linearization technique is utilized to remove the nonlinearity of the problem. The maximum absolute errors, root mean square errors and computational rate of convergence are calculated for a different number of collocation points at different time steps of the final time t=1. The stability analysis for the proposed method is also discussed. Overall, the recently devised numerical method boasts straightforward implementation, rapid convergence, and enhanced accuracy in addressing both linear and nonlinear problems.

Sturmian comparison theorem for hyperbolic equations on a rectangular prism

<abstract><p>In this paper, new Sturmian comparison results were obtained for linear and nonlinear hyperbolic equations on a rectangular prism. The results obtained for linear equations extended those given by Kreith [Sturmian theorems on hyperbolic equations, <italic>Proc. Amer. Math. Soc.</italic>, <bold>22</bold> (1969), 277-281] in which the Sturmian comparison theorem for linear equations was obtained on a rectangular region in the plane. For the purpose of verification, an application was described using an eigenvalue problem.</p></abstract>

Optimal control of the systems governed by linear hyperbolic integro-differential equations

Optimal control of the systems governed by linear hyperbolic integro-differential equations

Spectral Analysis of a Class of Linear Hyperbolic Partial Differential Equations

Spectral Analysis of a Class of Linear Hyperbolic Partial Differential Equations

Lyapunov-type inequalities for linear hyperbolic and elliptic equations on a rectangular domain

In the case of oscillatory potential, we present some new Lyapunov-type inequalities for linear hyperbolic and elliptic equations on a rectangular domain in ${\mathbb R}^2$. No sign restriction is imposed on the potential function. As applications of the Lyapunov-type inequalities obtained, we give some estimations for disconjugacy of hyperbolic and elliptic Dirichlet boundary value problems.

On a two-dimensional Dirichlet type problem for a linear hyperbolic equation of fourth order

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.

Lyapunov Functions for Linear Hyperbolic Systems

We develop new methods to construct a Lyapunov function for 1-D linear hyperbolic equations with variable coefficients. The main focus is on the nonstrictly hyperbolic case for which we give an example demonstrating that existing approaches cannot provide sufficient conditions for the asymptotic stability, but our approach does. Sufficient conditions for exponential <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}$</tex-math></inline-formula> -stability for a connected <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2 \times 2$</tex-math></inline-formula> system of linear 1-D hyperbolic systems are obtained. By means of examples we compare the capabilities of our approach with the existing ones.

Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients

Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients

APPLICATION OF THE FOURIER TRANSFORMATION FOR SOLVING A FIRST ORDER PARTIAL DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

This work is devoted to the solution of a linear first-order hyperbolic partial differential equation with constant coefficients. The specific objective is to prove the existence and uniqueness of the solution of the proposed PDE. The existence and uniqueness of the solution have been proved. To demonstrate the existence of the solution, the Fourier transformation was used. The variational formulation was used to prove the uniqueness of the solution. The combination of the Fourier transformation and the variational formulation yielded the expected results: the existence and uniqueness of the solution.

Adaptive Fuzzy Boundary Observer Design for Uncertain Linear Coupled Hyperbolic Partial Differential Equation Systems

Joint uncertainties and state estimation of a class of linear coupled hyperbolic partial differential equation systems in the presence of unstructured and structured uncertainties are studied in this paper. For unstructured uncertainties which are completely unknown, by employing Takagi-Sugeno fuzzy logic system to approximate the unstructured uncertainties, a novel adaptive fuzzy boundary observer is developed to estimate both unknown system states as well as unknown weights in the fuzzy logic system, and the estimation errors are ultimately bounded. Therein, in the design of the proposed observer, a set of swapping filters and infinite dimensional backstepping technique are combined. On the other hand, for structured uncertainties that can be described in a concrete parameterized form, the proposed method can easily achieve the exact estimation of weights and states to their true values. The rigorous proof is provided to show that the ultimately bounded estimation errors for the case of unstructured uncertainties and the exponential convergent estimation errors for the case of structured uncertainties can be realized. Finally, three illustrative simulations are carried out to show the feasibility and effectiveness of the developed methods in this paper.

Numerical analysis of a time filtered scheme for a linear hyperbolic equation inspired by DNA transcription modeling

Numerical analysis of a time filtered scheme for a linear hyperbolic equation inspired by DNA transcription modeling

Robust stabilization of 2 × 2 first-order hyperbolic PDEs with uncertain input delay

A backstepping-based compensator design is developed for a system of 2 × 2 first-order linear hyperbolic partial differential equations (PDE) in the presence of an uncertain long input delay at boundary. We introduce a transport PDE to represent the delayed input, which leads to three coupled first-order hyperbolic PDEs. A novel backstepping transformation, composed of two Volterra transformations and an affine Volterra transformation, is introduced for the predictive control design. The resulting kernel equations from the affine Volterra transformation are two coupled first-order PDEs and each with two boundary conditions, which brings challenges to the well-posedness analysis. We solve the challenge by using the method of characteristics and the successive approximation. To analyze the sensitivity of the closed-loop system to uncertain input delay, we introduce a neutral system which captures the control effect resulted from the delay uncertainty. It is proved that the proposed control is robust to small delay variations. Numerical examples illustrate the performance of the proposed compensator.

The Homotopy Perturbation Method for Solving Nonlocal Initial-Boundary Value Problems for Parabolic and Hyperbolic Partial Differential Equations

To obtain approximate-exact solutions to nonlocal initial-boundary value problems (IBVPs) of linear and nonlinear parabolic and hyperbolic partial differential equations (PDEs) subject to initial and nonlocal boundary conditions of integral type, the homotopy perturbation method (HPM) is utilized in this study. The HPM is used to solve the specified nonlocal IBVPs, which are then transformed into local Dirichlet IBVPs. Some examples demonstrate how accurate and efficient the HPM.&#x0D;