Linear Hyperbolic Partial Differential Equations Research Articles

Spectral Analysis of a Class of Linear Hyperbolic Partial Differential Equations

Spectral Analysis of a Class of Linear Hyperbolic Partial Differential Equations

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.

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.

Robustification of stabilizing controllers for ODE-PDE-ODE systems: A filtering approach

In this paper, we present a filtering technique that robustifies stabilizing controllers for systems composed of heterodirectional linear first-order hyperbolic Partial Differential Equations (PDEs) interconnected with Ordinary Differential Equations (ODEs) through their boundaries. The actuation is either available through one of the ODE or at the boundary of the PDE. The proposed framework covers a broad general class of interconnected systems. Assuming that a stabilizing controller is available, we derive simple sufficient conditions under which appropriate low-pass filters can be combined with the control law to robustify the closed-loop system. Our approach is based on a rewriting of the distributed dynamics as a delay-differential algebraic equation and an analysis in the Laplace domain. The proposed technique will simplify the design of stabilizing controllers for the class of systems under consideration (which can now be done only on a case-by-case basis due to the complexity and generality of the underlying interconnections) since it dissociates the stabilization problem from the robustness aspects. Indeed, it becomes possible to use convenient (but non-robust) techniques for the stabilization of such systems (as the cancellation of the boundary coupling terms or the inversion of the ODE dynamics), knowing that the resulting control law can be made robust (to delays and uncertainties) using the proposed filtering methodology.

A novel scheme for the hyperbolic partial differential equation through Fibonacci wavelets

ABSTRACT In this study, we generated the operational matrices of integration based on the Fibonacci wavelets through the concept of linear algebra and developed the novel technique known as the Fibonacci wavelet collocation method (FWCM). The proposed approach extracts the numerical solution of linear hyperbolic partial differential equations (HPDEs). This technique is an efficient and emerging numerical algorithm that converts the considered problem into a system of equations of algebraic type. We obtained desired numerical results in solving this system of algebraic equations with the help of the Newton–Raphson technique. We solved the five problems concerning a minimum level of resolution to strengthen our results. The obtained outcome is compared with the exact and other numerical solutions available in the literature through tables and graphs. The tables and graphical representations clarify the accuracy and efficiency of the proposed technique. Convergence analysis for the proposed method is drawn in terms of the theorems.

Control strategies for transport networks under demand uncertainty

In this article, we consider transport networks with uncertain demands. Network dynamics are given by linear hyperbolic partial differential equations and suitable coupling conditions, while demands are incorporated as solutions to stochastic differential equations. For the demand satisfaction, we solve a constrained optimal control problem. Controls in terms of network inputs are then calculated explicitly for different assumptions. Numerical simulations are performed to underline the theoretical results.

Chaos of the 2D linear hyperbolic equation with general van der Pol type boundary condition

This paper is concerned with the complex dynamics of the initial-boundary value problem of a 2D linear hyperbolic partial differential equation (PDE), where the parameter α that appeared in the general van der Pol type boundary condition is given by α∈R. The whole real line is divided into three intervals of the parameter to discuss the dynamics. The existence of chaos is first established in the sense of the exponential growth of total variation when the parameter locates in the central interval, which allows it to be positive, negative, or zero. By analyzing the chaotic dynamics of the piecewise continuous map induced by the hyperbolic PDE, such a PDE is further rigorously proved to be chaos in the interval of the positive parameter that is to the right of the central interval. Finally, the asymptotic behaviors of the hyperbolic PDE are systematically presented in the rest of the whole real line; more precisely, the hyperbolic PDE is either globally stable or unbounded.

Leak detection, size estimation and localization in branched pipe flows

We design a leak detection, size estimation and localization algorithm for a branched pipe system, requiring flow and pressure measurements to be taken at the inlet and outlet boundaries, only. By showing that the pipe system model can be mapped into a system of coupled, linear hyperbolic PDEs (partial differential equations), with the parametric uncertainties caused by the leak appearing in a particular way, established methods can be applied to obtain state and parameter estimates. Analyzing the structure of the parametric uncertainties that appear in the obtained adaptive observer canonical form, we prove that total leak size can be estimated regardless of how leaks are distributed in the pipe network. Moreover, any number of point leaks can be located, provided they occur sufficiently separated in time.

On the solution of hyperbolic equations using the peridynamic differential operator

Numerical solution of hyperbolic differential equations, such as the advection equation, poses challenges. Classically, this issue has been addressed by using a scheme known as the upwind scheme. It simply invokes more points from the upwind side of the flow stream when calculating derivatives. This study presents a generalized upwind scheme, referred to as directional nonlocality, for the numerical solution of linear and nonlinear hyperbolic Partial Differential Equations (PDEs) using the peridynamic differential operator (PDDO). The PDDO provides the nonlocal form of the differential equations by introducing an internal length parameter (horizon) and a weight function. The weight function controls the degree of interaction among the points within the horizon. A modification to the weight function, i.e., upwinded-weight function, accounts for directional nonlocality along which information travels. This modification results in a stable PDDO discretization of hyperbolic PDEs. Solutions are constructed in a consistent manner without special treatments through simple discretization. The capability of this approach is demonstrated by considering time dependent linear and nonlinear hyperbolic equations as well as the time invariant Eikonal equation. Numerical stability is ensured for the linear advection equation and the PD solutions compare well with the analytical/reference solutions.

Backstepping Control of a Hyperbolic PDE System with Zero Characteristic Speed

In this paper, we study the single-input boundary feedback stabilization of 3 × 3 linear hyperbolic partial differential equations (PDEs) with two counterconvecting PDEs and the third one with zero characteristic speed. We design a full-state backstepping controller which exponentially stabilizes the origin in the L2 sense. The zero transport velocity makes the previous backstepping designs inapplicable (their application would result in a controller with infinite gains). To employ backstepping in the presence of zero speed, we use an invertible Volterra transformation only for the PDEs with nonzero speeds, leaving the state of the zero-speed PDE unaltered in the target system, but making the target zero-speed PDE input-to-state stable with respect to the decoupled and stable counterconvecting nonzero-speed PDEs. In addition to achieving stabilization, we produce an explicit bound on the rate of convergence of the target system by a method of successive approximations and the use of Laplace transform. Simulation results are presented to illustrate the effectiveness of the proposed control design.

Robust state-feedback stabilization of an underactuated network of interconnected [formula omitted] hyperbolic PDE systems

We detail in this article the development of a delay-robust stabilizing state feedback control law for an underactuated network of two subsystems of heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. Only one of the two subsystems is actuated. The proposed approach is based on the backstepping methodology. A backstepping transform allows us to construct a first feedback to tackle in-domain couplings present in the actuated PDE subsystem. Then, we introduce a predictive tracking controller to stabilize the second PDE subsystem. The stabilization of this subsystem implies the stabilization of the whole network. Finally, the proposed control law is combined with a low-pass filter to become robust with respect to small delays in the control signal and uncertainties on the system parameters.

Chaotic vibrations of 3D linear hyperbolic PDEs with linear perturbations of superlinear boundary conditions

This article studies the chaotic vibrations of the infinite-dimensional dynamical systems governed by linear hyperbolic partial differential equations (PDEs) in three-dimensional (3D) space, where the boundary conditions include two linear perturbations of superlinear types. A rigorous mathematical theorem that guarantees the occurrence of chaos of such systems is obtained. The main theme of this article is the advancement of existing chaos to 3D hyperbolic PDEs. As applications, two examples are provided for showing the effectiveness of the theoretical chaotic results.

A high order discontinuous Galerkin method for the symmetric form of the anisotropic viscoelastic wave equation

We introduce a new symmetric treatment of anisotropic viscous terms in the viscoelastic wave equation. An appropriate memory variable treatment of stress-strain convolution terms, result into a symmetric system of first order linear hyperbolic partial differential equations, which we discretize using a high-order discontinuous Galerkin finite element method. The accuracy of the resulting numerical scheme is verified using convergence studies against analytical plane wave solutions and analytical solutions of the viscoelastic wave equation. Computational experiments are shown for various combinations of homogeneous and heterogeneous viscoelastic media in two and three dimensions.

A two-level parameterized Model-Order Reduction approach for time-domain elastodynamics

We present a two-level parameterized Model Order Reduction (pMOR) technique for the linear hyperbolic Partial Differential Equation (PDE) of time-domain elastodynamics. In order to approximate the frequency-domain PDE, we take advantage of the Port-Reduced Reduced-Basis Component (PR-RBC) method to develop (in the offline stage) reduced bases for subdomains; the latter are then assembled (in the online stage) to form the global domains of interest. The PR-RBC approach reduces the effective dimensionality of the parameter space and also provides flexibility in topology and geometry. In the online stage, for each query, we consider a given parameter value and associated global domain. In the first level of reduction, the PR-RBC reduced bases are used to approximate the frequency-domain solution at selected frequencies. In the second level of reduction, these instantiated PR-RBC approximations are used as surrogate truth solutions in a Strong Greedy approach to identify a reduced basis space; the PDE of time-domain elastodynamics is then projected on this reduced space. We provide a numerical example to demonstrate the computational capability and assess the performance of the proposed two-level approach.

Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations

Through the current article, a numerical technique to obtain an approximate solution of one-dimensional linear hyperbolic partial differential equations is implemented. A certain combination of the shifted Chebyshev polynomials of the fifth-kind is used as basis functions. The main idea behind the proposed technique is established on converting the governed boundary-value problem into a system of linear algebraic equations via the application of the spectral Galerkin method. The resulting linear system can be solved by expedients of the Gaussian elimination procedure. The convergence and error analysis of the shifted Chebyshev expansion are carefully investigated. Various numerical examples are given to demonstrate the power and accuracy of the given method.

Numerical analysis of a dual-phase-lag model with microtemperatures

In the last twenty years, the analysis of problems involving dual-phase-lag models has received an increasing attention. In this work, we consider the coupling between one of these models and the microtemperatures effects. In order to overcome the infinite speed paradox, two relaxation parameters are introduced for each evolution equation related to the temperature and the microtemperatures, leading to a system of linear hyperbolic partial differential equations. Its variational formulation is written in terms of the temperature acceleration and the microtemperatures acceleration. An energy decay property is proved. Next, fully discrete approximations are introduced by using the finite element method and the Euler scheme, proving a stability property and a discrete version of the energy decay, obtaining a priori error estimates and performing one- and two-dimensional numerical simulations.

An efficient higher order compact finite difference schemes for linear hyperbolic partial differential equations

To approximate the partial derivative of Linear Hyperbolic Partial Differential Equations (LPHDE), a system of compact schemes used at non-boundary nodes. For the spatial discretization of the advection term, instead of using the upwind schemes, central difference based compact 4th order and conventional 2nd order schemes are experimented. The main aim of the numerical experiments carried out is, to assess the ability of the compact 4th order scheme in capturing the convection process, in comparison with the conventional 2nd order scheme.

Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition

There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. The wave equation is a second order linear hyperbolic partial differential equation that describes the propagation of variety of waves, such as sound or water waves. In this paper we consider the convergence analysis of the explicit schemes for solving one dimensional, time-dependent wave equation with Drichlet and Neumann boundary condition. Taylor's series expansion is used to expand the finite difference approximations in the explicit scheme. We present the derivation of the schemes and develop a computer program to implement it We use spectral radius of Matrix obtained from discretization and Von Neumann stability condition to determine stability, and consistence of the method from truncated error from discretized method. Using Lax Equivalence Theorem, convergence of the methods was described by testing consistency and stability of the methods. And it is found out that the scheme is stable with the Drichlet boundary and conditionally stable with Derivative boundary condition.