Linear Hyperbolic Partial Differential Equations Research Articles

An efficient method for solving nonlocal initial-boundary value problems for linear and nonlinear first-order hyperbolic partial differential equations

In this paper, we present a new approach to solve nonlocal initial-boundary value problems of linear and nonlinear hyperbolic partial differential equations of first-order subject to initial and nonlocal boundary conditions of integral type. We first transform the given nonlocal initial-boundary value problems into local initial-boundary value problems. Then we apply a modified Adomian decomposition method, which permits convenient resolution of these problems. Moreover, we prove this decomposition scheme applied to such nonlocal problems is convergent in a suitable Hilbert space, and then extend our discussion to include systems of first-order linear equations and other related nonlocal initial-boundary value problems.

Flow Model of Pure Water in a Pipeline

Potable water in Tunisia has well defined characteristics. Thus, the Water researches and Technologies Centre-Borj Cedria mainly processing Natural Waters Laboratory, has worked on the physic-chemical quality of the water. As the Higher Institute of Agronomy-chott mariem have studied the phenomena associated with flows. For this work shows that natural water flows that can be generated in a pipeline. During operation of agricultural pumps (centrifugal pumps), the speed of the electric motor increases from zero to the permanent regime speed. This change in regime influences the flow of the hydraulic installation and forced it to follow the starting dynamic law of the motor. Under the effect of the friction force, due to the fluid viscosity, the transient regime is dumped until reaching the normal operating conditions. In this work, we study this phenomenon and see the influence of the pump startup on the hydraulic behaviour of one dimensional flow throw a cylindrical pipe of linear elastic behaviour. The pipe is connected to constant level reservoir at the downstream end. The problem is governed by a two coupled linear hyperbolic partial differential equations which are the equations of continuity and motion. The mathematical model is solved by the method of characteristics where at the upstream end, that is, at the pump station, the boundary condition is given by the differential equation for speed change of the pump motor. A theoretical relationship is introduced to express the motor torque in terms of the time. At the downstream end the discharging reservoir is assumed to be at a constant level. The computed head and discharge time curves, caused by the starting of the pump, are plotted at some cross sections of the pipe. The results show that the evolution of the hydraulic variables is well influenced by the applied motor torque.

A new approach to numerical solution of second-order linear hyperbolic partial differential equations arising from physics and engineering

This article presents a new reliable solver based on polynomial approximation, using the Euler polynomials to construct the approximate solutions of the second-order linear hyperbolic partial differential equations with two variables and constant coefficients. Also, a formula expressing explicitly the Euler expansion coefficients of a function with one or two variables is proved. Another explicit formula, which expresses the two dimensional Euler operational matrix of differentiation is also given. Application of these formulae for reducing the problem to a system of linear algebraic equations with the unknown Euler coefficients, is explained. Hence, the result system can be solved and the unknown Euler coefficients can be found approximately. Illustrative examples with comparisons are given to confirm the reliability of the proposed method. The results show the efficiency and accuracy of the present work.

Study of Flow Problems and Those of Soil Compaction in an Agricultural Farm

In agricultural land can be observed problems regarding water and soil. The problems of water flow can cause breaks in pipes, and the problems associated with soil because of the passage of vehicles and lead to compaction, engender changes in soil parameter directly affects the infiltration water and fruit yield regression plants. In this paper, we will study numerically the flow in pipes and soil compaction. So, a first part examines the phenomena of demurrage pumps; when starting a pump, the speed of the electric motor rises from zero up to the permanent regime speed. This change in regime influences the flow of the hydraulic installation and forced it to follow the starting dynamic law of the motor. Under the effect of the friction force, due to the fluid viscosity, the transient regime is dumped until reaching the normal operating conditions. In this work, we study this phenomenon and see the influence of the pump startup on the hydraulic behaviour of one dimensional flow throw a cylindrical pipe of linear elastic behaviour. The pipe is connected to constant level reservoir at the downstream end. The problem is governed by a two coupled linear hyperbolic partial differential equations which are the equations of continuity and motion. The mathematical model is solved by the method of characteristics where at the upstream end, that is, at the pump station, the boundary condition is given by the differential equation for speed change of the pump motor. A theoretical relationship is introduced to express the motor torque in terms of the time. At the downstream end the discharging reservoir is assumed to be at a constant level. The computed head and discharge time curves, caused by the starting of the pump, are plotted at some cross sections of the pipe. The results show that the evolution of the hydraulic variables is well influenced by the applied motor torque. In a second part of the numerical study of soil compaction is developed.

Bernoulli Matrix Approach for Solving Two Dimensional Linear Hyperbolic Partial Differential Equations with Constant Coefficients

The purpose of this study is to give a Bernoulli polynomial appro ximat ion for thesolution of hyperbolic partial differential equations with three variables and constant coefficients. For this purpose, a Bernoulli matrix approach is intro- duced. This method is based on taking the truncated Bernoulli expansions of the functions in the partial d ifferential equations. After replacing the approximations of functions in the basic equation, we deal with a linear algebraic equation. Hence, the result matrix equation can be solved and the unknown Bernoulli coefficients can be found approximately. The efficiency of the proposed approach is dem onstrated with one exampl e.

Inertial waves in a rotating annulus with inclined inner cylinder: comparing the spectrum of wave attractor frequency bands and the eigenspectrum in the limit of zero inclination

We investigate theoretically inertial waves inside a liquid confined between two co-rotating coaxial cylinders of finite length. We consider the case of small viscosity and high angular velocity (i.e., small Ekman numbers), a parameter range of interest for many geophysical applications. In this case, inertial waves propagating in the container show multiple reflections at the walls before the waves can be damped by weak diffusion. We allow for the inner cylinder wall to be parallel or inclined with respect to the annulus’ vector of rotation (truncated cone). For the limit of zero viscosity, the wave propagation is governed by a boundary value problem that is composed of a linear second-order hyperbolic partial differential equation and the impermeability boundary conditions. For the special case of vertical cylinder walls (no inclination of the inner cylinder), this boundary value problem is separable, the corresponding eigenmodes can analytically be found and they are regular. However, when the inner cylinder wall is inclined, the hyperbolicity of the governing equation leads to internal shear layers (corresponding to singularities for the inviscid case). The geometrical structure of the shear layers can be explained by inertial waves, trapped on limit cycles denoted as wave attractors. The shape of the limit cycles depends on the wave frequency. In fact, the spectrum of regular modes, existing for the case of vertical cylinder walls, vanishes almost completely when the inner wall is inclined. Instead of a spectrum of discrete frequencies and regular eigenmodes, a spectrum of wave attractor frequency bands and singular eigenmodes exist. The question addressed here is whether the spectrum of wave attractor intervals collapses to the discrete frequency spectrum when the inclination angle of the inner cylinder goes to zero. To answer this question, the attractor frequency intervals are evaluated numerically for a series of decreasing cylinder inclination angles and are compared with the analytically found eigenspectrum for the case of zero inclination. Goal is to better understand the asymptotic behavior of the problem for decreasing inclination angles. This understanding helps to interpret results from laboratory experiments with geometries that differ from the perfect annulus with parallel cylinder walls.

Heuristic Dynamic Programming Algorithm for Optimal Control Design of Linear Continuous-Time Hyperbolic PDE Systems

This work considers the optimal control problem of linear continuous-time hyperbolic partial differential equation (PDE) systems with partially unknown system dynamics. To respect the infinite-dimensional nature of the hyperbolic PDE system, the problem can be reduced to finding a solution of the space-dependent Riccati differential equation (SDRDE), which requires the full system model. Therefore, a heuristic dynamic programming (HDP) algorithm is proposed to achieve online optimal control of the hyperbolic PDE system, which online collects data accrued along system trajectories and learns the solution of the SDRDE without requiring the internal system dynamics. The convergence of HDP algorithm is established by showing that the HDP algorithm generates a nondecreasing sequence which uniformly converges to the solution of the SDRDE. For implementation purposes, the HDP algorithm is realized by developing an approximate approach based on the method of weighted residuals. Finally, the application on a steam-jacketed tubular heat exchanger demonstrates the effectiveness of the developed control approach.

REGULARITY AND GENERALIZED POLYNOMIAL CHAOS APPROXIMATION OF PARAMETRIC AND RANDOM SECOND-ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

Initial boundary value problems of linear second-order hyperbolic partial differential equations whose coefficients depend on countably many random parameters are reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space. This parametric family is approximated by Galerkin projection onto finitely supported polynomial systems in the parameter space. We establish uniform stability with respect to the support of the resulting coupled hyperbolic systems, and provide sufficient smoothness and compatibility conditions on the data for the solution to exhibit analytic, respectively, Gevrey regularity with respect to the countably many parameters. Sufficient conditions for the p-summability of the generalized polynomial chaos expansion of the parametric solution in terms of the countably many input parameters are obtained and rates of convergence of best N-term polynomial chaos type approximations of the parametric solution are given. In addition, regularity both in space and time for the parametric family of solutions is proved for data satisfying certain compatibility conditions. The results allow obtaining convergence rates and stability of sparse space-time tensor product Galerkin discretizations in the parameter space.

Online policy iteration algorithm for optimal control of linear hyperbolic PDE systems

In this paper, the linear quadratic (LQ) optimal control problem is considered for a class of linear distributed parameter systems described by first-order hyperbolic partial differential equations (PDEs). Reinforcement learning (RL) technique is introduced for adaptive optimal control design from the design-then-reduce (DTR) framework. Initially, a policy iteration (PI) algorithm is proposed, which learns the solution of the space-dependent Riccati differential equation (SDRDE) online without requiring the internal system dynamics of the PDE system. To prove its convergence, the PI algorithm is shown to be equivalent to an iterative procedure of a sequence of space-dependent Lyapunov differential equations (SDLDEs). Then, the convergence is established by showing that the solutions of SDLDEs are a monotone non-increasing sequence that converges to the solution of the SDRDE. For implementation purpose, an online least-square method is developed for the approximation of the solutions of the SDLDEs. Finally, the proposed design method is applied to the distributed control of a steam-jacketed tubular heat exchanger to illustrate its effectiveness.

Numerical Solution of Linear HPDEs Via Bernoulli Operational Matrix of Differentiation and Comparison with Taylor Matrix Method

In this paper a new matrix approach for solving linear hyperbolic partial differential equations (HPDEs) is presented. The method is based on the Bernoulli expansion of two-variable functions, which consists of the matrix representation of expressions in the considered HPDE. Also, a new operational matrix of differentiation is introduced, which consists of nonzero elements under its diagonal, meanwhile the operational matrix of differentiation of other polynomial bases (such as Chebyshev, Legendre, etc.) is a strictly (upper or lower) triangular matrix. In the proposed method, HPDE together with the initial and boundary conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Bernoulli coefficients. Combining these matrix equations and then solving the system yields the Bernoulli coefficients of the approximated solution. Illustrative examples are included to demonstrate the validity and applicability of the technique. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.

Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters

This paper is concerned with the problem of robustly stochastically exponential stability and stabilization for a class of distributed parameter systems described by uncertain linear first-order hyperbolic partial differential equations (FOHPDEs) with Markov jumping parameters, for which the manipulated input is distributed in space. Based on an integral-type stochastic Lyapunov functional (ISLF), the sufficient condition of robustly stochastically exponential stability with a given decay rate is first derived in terms of spatial differential linear matrix inequalities (SDLMIs). Then, an SDLMI approach to the design of robust stabilizing controllers via state feedback is developed from the resulting stability condition. Furthermore, using the finite difference method and the standard linear matrix inequality (LMI) optimization techniques, recursive LMI algorithms for solving the SDLMIs in the analysis and synthesis are provided. Finally, a simulation example is given to demonstrate the effectiveness of the developed design method.

Exponential Stability of Switched Linear Hyperbolic Initial-Boundary Value Problems

We consider the initial-boundary value problem governed by systems of linear hyperbolic partial differential equations in the canonical diagonal form and study conditions for exponential stability when the system discontinuously switches between a finite set of modes. The switching system is fairly general in that the system matrix functions as well as the boundary conditions may switch in time. We show how the stability mechanism developed for classical solutions of hyperbolic initial boundary value problems can be generalized to the case in which weaker solutions become necessary due to arbitrary switching. We also provide an explicit dwell-time bound for guaranteeing exponential stability of the switching system when, for each mode, the system is exponentially stable. Our stability conditions only depend on the system parameters and boundary data. These conditions easily generalize to switching systems in the nondiagonal form under a simple commutativity assumption. We present tutorial examples to illustrate the instabilities that can result from switching.

Lyapunov functions for switched linear hyperbolic systems

Systems of conservation laws and systems of balance laws are considered in this paper. These kinds of infinite dimensional systems are described by a linear hyperbolic partial differential equation with and without a linear source term. The dynamics and the boundary conditions are subject to a switching signal that is a piecewise constant function. By means of Lyapunov techniques some sufficient conditions are given for the exponential stability of the switching system, uniformly for all switching signals. Different cases are considered depending on the presence or not of the linear source term in the hyperbolic equation, and depending on the dwell time assumption on the switching signals. Some numerical simulations are also given to illustrate some main results, and to motivate this study.

Legendre Approximation for Solving Linear HPDEs and Comparison with Taylor and Bernoulli Matrix Methods

The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.

Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs

Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporalk-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of bestN-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.

Efficient long-time computations of time-domain boundary integrals for 2D and dissipative wave equation

Linear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be reformulated as time-domain boundary integral equations. We propose an efficient implementation of a numerical discretization of such equations when the strong Huygens’ principle does not hold. For the numerical discretization, we make use of convolution quadrature in time and standard Galerkin boundary element method in space. The quadrature in time results in a discrete convolution of weights W j with the boundary density evaluated at equally spaced time points. If the strong Huygens’ principle holds, W j converge to 0 exponentially quickly for large enough j . If the strong Huygens’ principle does not hold, e.g., in even space dimensions or when some damping is present, the weights are never zero, thereby presenting a difficulty for efficient numerical computation. In this paper we prove that the kernels of the convolution weights approximate in a certain sense the time domain fundamental solution and that the same holds if both are differentiated in space. The tails of the fundamental solution being very smooth, this implies that the tails of the weights are smooth and can efficiently be interpolated. Further, we hint on the possibility to apply the fast and oblivious convolution quadrature algorithm of Schädle et al. to further reduce memory requirements for long-time computation. We discuss the efficient implementation of the whole numerical scheme and present numerical experiments.

High-order, finite-volume methods in mapped coordinates

We present an approach for constructing finite-volume methods for flux-divergence forms to any order of accuracy defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the flux averages over faces that generalize a method developed for Cartesian grids to the case of mapped grids. The second is a method for computing the averages of the metric terms on faces such that freestream preservation is automatically satisfied. We derive detailed formulas for the cases of fourth-order accurate discretizations of linear elliptic and hyperbolic partial differential equations. For the latter case, we combine the method so derived with Runge–Kutta time discretization and demonstrate how to incorporate a high-order accurate limiter with the goal of obtaining a method that is robust in the presence of discontinuities and underresolved gradients. For both elliptic and hyperbolic problems, we demonstrate that the resulting methods are fourth-order accurate for smooth solutions.

Laplace‐Type Semi‐Invariants for a System of Two Linear Hyperbolic Equations by Complex Methods

In 1773 Laplace obtained two fundamental semi‐invariants, called Laplace invariants, for scalar linear hyperbolic partial differential equations (PDEs) in two independent variables. He utilized this in his integration theory for such equations. Recently, Tsaousi and Sophocleous studied semi‐invariants for systems of two linear hyperbolic PDEs in two independent variables. Separately, by splitting a complex scalar ordinary differential equation (ODE) into its real and imaginary parts PDEs for two functions of two variables were obtained and their symmetry structure studied. In this work we revisit semi‐invariants under equivalence transformations of the dependent variables for systems of two linear hyperbolic PDEs in two independent variables when such systems correspond to scalar complex linear hyperbolic equations in two independent variables, using the above‐mentioned splitting procedure. The semi‐invariants under linear changes of the dependent variables deduced for this class of hyperbolic linear systems correspond to the complex semi‐invariants of the complex scalar linear (1 + 1) hyperbolic equation. We show that the adjoint factorization corresponds precisely to the complex splitting. We also study the reductions and the inverse problem when such systems of two linear hyperbolic PDEs arise from a linear complex hyperbolic PDE. Examples are given to show the application of this approach.

Wave-type equations of low regularity

We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second-order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.

Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems

Radial basis function (RBF) methods have been actively developed in the last decades. RBF methods are global methods which do not require the use of specialized points and that yield high order accuracy if the function is smooth enough. Like other global approximations, the accuracy of RBF approximations of discontinuous problems deteriorates due to the Gibbs phenomenon, even as more points are added. In this paper we show that it is possible to remove the Gibbs phenomenon from RBF approximations of discontinuous functions as well as from RBF solutions of some hyperbolic partial differential equations. Although the theory for the resolution of the Gibbs phenomenon by reprojection in Gegenbauer polynomials relies on the orthogonality of the basis functions, and the RBF basis is not orthogonal, we observe that the Gegenbauer polynomials recover high order convergence from the RBF approximations of discontinuous problems in a variety of numerical examples including the linear and nonlinear hyperbolic partial differential equations. Our numerical examples using multi-quadric RBFs suggest that the Gegenbauer polynomials are Gibbs complementary to the RBF multi-quadric basis.