Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions

In this paper, we consider the controllability and the observability of a networked system of linear hyperbolic partial differential equations with coupled boundary conditions. Using the method of characteristics, we characterize them by the controllability/observability of a discrete-time system which is defined on the boundaries of the PDE system, has low-dimension, and hence the analysis is easy for. We also show the effectiveness of our approach using an simple example.

In this paper, a domain decomposition method has been presented for solving singularly perturbed differential difference equations with delay as well as advances whose solution exhibits boundary layer behavior. By introducing a terminal point, the original problem is divided into inner and outer region problems. An implicit terminal boundary condition at the terminal point has been determined. The outer region problem with the implicit boundary condition is solved and produces an explicit boundary condition for the inner region problem. Then, the modified inner region problem (using the stretching transformation) is solved as a two-point boundary value problem. Fourth order stable central difference method has been used to solve both the inner and outer region problems. The proposed method is iterative on the terminal point. To demonstrate the applicability of the method, some numerical examples have been solved for different values of the perturbation parameter, delay and advance parameters. The stability and convergence of the scheme has also investigated

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.

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

On the variational iteration method and other iterative techniques for nonlinear differential equations

We extend recent results regarding disturbance rejection control of $n+m$ linear hyperbolic partial differential equations (PDEs) in a number of ways: 1) We allow spatially varying coefficients; 2) The disturbance is allowed to enter in the interior of the domain, and; 3) Rejection is achieved in minimum time. Additionally, we solve a tracking problem, where the tracking objective is achieved in finite, minimum time. We use a recently derived Fredholm transformation technique in addition to infinite-dimensional backstepping in our design.

Recently, the first result on backstepping-based adaptive control of a 1-D linear hyperbolic partial differential equation (PDE) with an uncertain transport speed was presented. The system also had an uncertain, constant in-domain coefficient, and the derived controller achieved convergence to zero in the $L_{\infty}$ -sense in finite time. In this paper, we extend that result to systems with a spatially varying in-domain coefficient, achieving asymptotic convergence to zero in the $L_{\infty}$ -sense. Additionally, for the case of having a constant in-domain coefficient, the new method is shown to have a slightly improved finite-time convergence time. The theory is illustrated in simulations.

A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations and switched differential-algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modelled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.

Water Hammer Stability in a Hydroelectric Plant with Surge Tank and Throttling

Effects of boundary conditions on numerical solution of implicit hyperbolic difference equations

In this work, we study the observer design problem for estimating thermoacoustic instabilities in a Rijke tube, using an in-domain point pressure measurement. Writing the system model in Riemann coordinates and, after a "folding transformation", it takes the form of 4 × 4 linear hyperbolic partial differential equations (PDEs) coupled at the boundaries with a linear ordinary differential equation (ODE). This results in a PDE-ODE observer design problem, whose output is a combination of two infinite-dimensional states measured inside the domain. As a first step in our observer design, a second "folding" transformation is applied around the measurement point, resulting in a 6 × 6 PDE-ODE system with measurements at one boundary. Then, the observer is designed as a copy of these equations plus an output injection term, which is given by a linear operator in the right-hand side of the ODE part. This operator is then chosen so that convergence of estimates is guaranteed by using stability properties of differential-delay systems. The design extends a previous result, based on backstepping, that required two measurements (both pressure and velocity). Simulation results are presented to illustrate the effectiveness of the proposed observer design.

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods

We derive a method of obtaining approximate numerical solution of linear variable-coefficient partial differential equations (PDEs) with two independent variables from Kida's optimum approximation theory. It is shown that a certain generalized filter bank implements linear PDEs. By applying generalized discrete orthogonality of Kida's optimum approximation to this filter bank, we prove that our approximate solution satisfies a given linear PDEs and the corresponding initial or boundary conditions at all given sample points, simultaneously.

A new method is introduced for studying boundary-value problems for a class of linear partial differential equations (PDEs) with variable coefficients. This method is based on ideas recently introduced by the author for the study of boundary-value problems for PDEs with constant coefficients. As illustrative examples the following boundary-value problems are solved: a Dirichlet and a Neumann problem on the half line for the time-dependent Schrödinger equation with a space-dependent potential; and a Poincaré problem on the quarter plane for a variable coefficient generalization of the Laplace equation.