Wave Partial Differential Equation Research Articles

Boundedness of pseudo-differential operators via coupled fractional Fourier transform

In this article, we obtained some fruitful results of the coupled fractional Fourier transform and its kernel. We defined pseudo-differential operators related to coupled fractional Fourier transform on Schwartz spaces and it is shown that their composition is again a pseudo-differential operator. It made further discussion on the composition of two pseudo-differential operators on Sobolev spaces and derived certain norm inequalities on it. Further, we have successfully applied some of the results of the coupled fractional Fourier transform to investigate the solution of nth-order linear non-homogeneous partial differential equation and wave equation. Lastly, we present some examples involving graphs and tables to illustrate the validity of our theoretical findings.

Generalized homotopy perturbation approach: an application to wave partial differential equations

Generalized homotopy perturbation approach: an application to wave partial differential equations

Robust boundary tracking control of wave PDE: Insight on forcing slow-aseismic response

The output boundary tracking problem is addressed for a wave partial differential equation (PDE) with uncertain multiplicative parameters and additive boundary and in-domain terms. To solve this problem, a homogeneous boundary feedback controller is designed using collocated boundary measurements only. Global exponential Input-to-State Stability (eISS) of the closed-loop system and its insensitivity to matched disturbances are additionally achieved. The ultimate bound estimate is diminished by tuning of the controller gains and/or by an appropriate pre-selection of a reference signal to follow. The proposed homogeneity-based strategy is further evaluated in a simplified earthquake model for controlled dissipation of its stored energy and producing slow-aseismic response. Such a case study possesses important applications in earthquake prevention, renewable energy production and storage. Numerical simulations are additionally conducted to support the robustness and the smoothness of the resulting closed-loop system depending on the homogeneous control parameter selection.

Exact and Approximate Solutions for Linear and Nonlinear Partial Differential Equations via Laplace Residual Power Series Method

The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein–Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach’s reliability, capability, and efficiency.

Model decomposition-based fault-tolerant control for interconnected wave partial differential equation systems

The fault-tolerant control (FTC) issue for a class of interconnected wave-distributed parameter systems is addressed by using the inclusion principle. Both subsystem faults and coupling faults are considered. First, the original system is expanded into an equivalent decoupled system. The main structure of the expanded system consists of multiple pair-wise subsystems. Second, the pair-wise fault-tolerant controllers are designed to stabilise the pair-wise subsystems, which can compensate for the effects of subsystem faults and major coupling faults. Third, a coordinated fault-tolerant controller with a suitable coordinated compensator for the expanded system is constructed according to the pair-wise fault-tolerant controllers of all subsystems. The coordinated compensator is used to compensate for the effects of other minor interconnections and coupling faults, and the coordinated controller is designed for the expanded system. Furthermore, the fault-tolerant controller is obtained to guarantee asymptotical stability of the faulty interconnected wave system by satisfying the contraction condition. The numerical experiments are presented to illustrate the effectiveness of the FTC method.

One-Way Wave Operator

The second-order partial differential wave Equation (Cauchy’s first equation of motion), derived from Newton’s force equilibrium, describes a standing wave field consisting of two waves propagating in opposite directions, and is, therefore, a “two-way wave equation”. Due to the second order differentials analytical solutions only exist in a few cases. The “binomial factorization” of the linear second-order two-way wave operator into two first-order one-way wave operators has been known for decades and used in geophysics. When the binomial factorization approach is applied to the spatial second-order wave operator, this results in complex mathematical terms containing the so-called “Dirac operator” for which only particular solutions exist. In 2014, a hypothetical “impulse flow equilibrium” led to a spatial first-order “one-way wave equation” which, due to its first order differentials, can be more easily solved than the spatial two-way wave equation. To date the conversion of the spatial two-way wave operator into spatial one-way wave operators is unsolved. By considering the one-way wave operator containing a vector wave velocity, a “synthesis” approach leads to a “general vector two-way wave operator” and the “general one-way/two-way equivalence”. For a constant vector wave velocity the equivalence with the d’Alembert operator can be achieved. The findings are transferred to commonly used mechanical and electromagnetic wave types. The one-way wave theory and the spatial one-way wave operators offer new opportunities in science and engineering for advanced wave and wave field calculations.

Scaled Consensus Over a Network of Wave Equations

In this article, the scaled consensus issue for a network of wave partial differential equations is investigated in a directed communication setting, where the agent states approach the dictated ratios. To this end, a distributed protocol is proposed for the considered system. Then, a Lyapunov function is constructed to ensure scaled consensus, wherein the properties of algebraic connectivity are employed to handle the asymmetric interaction matrix. Using tools from semigroup theory, the well-posedness of the underlying system is guaranteed. Moreover, we extend the derived results to a leader-following scaled consensus scenario. Numerical examples are presented to support the proposed setup.

Numerical Solutions of Time-Fractional Nonlinear Water Wave Partial Differential Equation via Caputo Fractional Derivative: An Effective Analytical Method and Some Applications

Numerical Solutions of Time-Fractional Nonlinear Water Wave Partial Differential Equation via Caputo Fractional Derivative: An Effective Analytical Method and Some Applications

Robust Fault-Tolerant Control of Wave Equation Without Estimations of Plant and Failure Parameters

This paper describes a novel robust fault-tolerant control for a vibrating string (captured by a wave partial differential equation) with constraint to be clamped at the left end and free at the right one. The control input is a force at the right hand, which might be affected by an actuator failure modelled by a general multiplicative type with unknown values. Except for the actuator failure, some physical parameters of the plant are also unavailable. We propose a pure output feedback control law without any estimation terms to guarantee the prescribed performance constraint of the state on the boundary of the closed-loop system and the uniform boundedness of the state in the whole domain. By applying the backstepping technique to the boundary of the system, together with a barrier Lyapunov function, we develop a non-adaptive control method capable of dealing with the underlying problem. Effective control performance in the wave equation with the designed control strategy is verified via Lyapunov analysis and numerical simulation.

Stabilization of a Class of Nonlinear ODE/Wave PDE Cascaded Systems

We investigate stabilization of a class of cascaded systems of nonlinear ordinary differential equation (ODE)/wave partial differential equation (PDE) with time-varying propagation speed based on a two-step PDE backstepping transformation. A time-varying propagation velocity of wave PDE leads to two difficulties. One is how to prove the well-posedness and uniqueness of the time-varying kernel PDEs in the first-step backstepping transformation, the other is how to construct a backstepping transform to map the original system into a suitable target system during the second-step transformation. We prove that there exists a unique continuous <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> matrix-valued solution to the time-varying kernel PDEs, and design a predictor control for the original cascaded system. An example is provided to illustrate the feasibility of the proposed design.

Extremum seeking boundary control for PDE–PDE cascades

The goal of this paper is to shed light on new opportunities for extremum seeking through extensions to locally quadratic nonlinear maps with actuator dynamics modeled by cascades of partial differential equations (PDEs). First, we deal with PDEs with input delays such as, for example, the notoriously difficult problem of a wave PDE with input delay where, if the delay is left uncompensated, an arbitrarily short delay destroys the closed-loop stability. Then, we move forward to an even more challenging class of problems for parabolic–hyperbolic cascades of PDEs, coping with a heat equation at the input of a wave PDE. The treatment of such systems with PDE–PDE cascades is performed by means of boundary control. The proposed approach yields (small-gain) loops that make the control design constructive and enables stability analysis with quantitative estimates. Local exponential stability and convergence to a small neighborhood of the unknown extremum point are rigorously guaranteed. This result is achieved by using backstepping transformation and averaging in infinite dimensions. Although we keep the presentation using the classical Gradient extremum seeking, the generalization of the results for the Newton-based approach is also indicated. A numerical example is given to illustrate the effectiveness of the proposed extremum seeking boundary control for compensation of PDE–PDE cascades.

Numerical discretization and fast approximation of a variably distributed-order fractional wave equation

We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model, e.g., the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size σ is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom N is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting σ = O(N−1) leads to O(N3) operations of evaluating the temporal discretization coefficients. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only O(logN) terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of generating coefficients from O(N3) to O(N2 logN). Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.

Chaotic Dynamics of a 2D Hyperbolic PDE with the Boundary Conditions of Superlinear Type

The study of the chaotic dynamics of partial differential equation (PDE) system has become the focus of dynamical system. While important progress has been made for the research of chaos theory of one-dimensional wave PDE, the understanding of chaotic vibrations for two-dimensional (2D) hyperbolic PDE is still incomplete. This paper is concerned with a 2D hyperbolic PDE system governed by a linear equation with two suplinear boundary conditions. Based on the chaotic mapping theory and method of characteristics, sufficient conditions for the existence of nonisotropic chaotic vibrations are obtained for such system under three different system parameters. In addition, three numerical examples are provided to illustrate the effectiveness of our theoretical results.

Boundary Control of Nonlinear ODE/Wave PDE Systems With a Spatially Varying Propagation Speed

We consider the boundary control of a nonlinear ordinary differential equation (ODE) actuated through a wave equation whose propagation speed is spatially varying. The ODE state is driven by the uncontrolled boundary of the wave equation. We design a nonlinear backstepping compensator to enable global asymptotic stability of the closed-loop system. We deduce the controller design and the stability proof by introducing a two-step backstepping transformation. The first transformation recasts the original system into a coupled 2×2 first-order hyperbolic system with spatially varying coefficients cascading into a nonlinear ODE. The second transformation is used in the design of a compensator for the resulting cascaded system. Our design offers a global stability result that is guaranteed assuming that the spatially varying propagation speed is continuously differentiable and positive. Moreover, for nonlinear systems, our result is the first contribution enabling actual compensation of actuator delays governed by a coupled first-order hyperbolic partial differential equation (PDE) induced by a wave PDE dynamics with a spatially varying propagation speed. The validity of the proposed controller is illustrated by the benchmark system controlled via a cable.

Extremum Seeking Feedback With Wave Partial Differential Equation Compensation

Abstract This paper addresses the compensation of wave actuator dynamics in scalar extremum seeking (ES) for static maps. Infinite-dimensional systems described by partial differential equations (PDEs) of wave type have not been considered so far in the literature of ES. A distributed-parameter-based control law using back-stepping approach and Neumann actuation is initially proposed. Local exponential stability as well as practical convergence to an arbitrarily small neighborhood of the unknown extremum point is guaranteed by employing Lyapunov–Krasovskii functionals and averaging theory in infinite dimensions. Thereafter, the extension for wave equations with Dirichlet actuation, antistable wave PDEs as well as the design for the delay-wave PDE cascade are also discussed. Numerical simulations illustrate the theoretical results.

The fractional Kelvin-Voigt model for circumferential guided waves in a viscoelastic FGM hollow cylinder

Compared to the traditional integer order viscoelastic model, a fractional order derivative viscoelastic model is shown to be more accurate. A thorough knowledge of the dispersive characteristics of such model is very essential to the application of guided wave testing technique. In this paper, the guided waves in a fractional Kelvin-Voigt viscoelastic FGM hollow cylinder with material changing in the thickness direction are investigated. The Weyl definition of fractional order derivatives and the extended Legendre polynomial approach are employed for the derivations of the governing equations. The presented approach has the advantage that the solution of the complex partial differential wave equations with variable coefficients is reduced to an eigenvalue problem, which overcomes the shortcomings of the existing iterative methods such as the Newton downhill method and improves computation efficiency. The previous methods for dealing with viscoelastic guided wave transform the wave equations into a matrix determinant problem solved by the iterative methods, which has a very slow calculation speed. Comparisons with the related studies are conducted to validate the correctness of the presented approach, and the convergence of the approach is discussed. The full three dimensional spectrum, phase dispersion curves and attenuation curves are illustrated for various fractional order viscoelastic FGM hollow cylinders. The influences of fractional order, grade field and radius-thickness ratio on dispersion and attenuation curves are illustrated. The difference of the dispersion characteristics between the viscoelastic model and the elastic one is discussed. The influences of fractional order on displacement distributions are also studied.

The combined use of adaptive optics and nonlinear optical wavefront reversal techniques to compensate for turbulent distortions when focusing laser radiation on distant objects

Approaches to constructing a mock-up of a system for focusing laser radiation on distant objects using both adaptive optics elements and nonlinear-optical wavefront reversal methods providing compensation for turbulent distortions are considered. Numerical calculations were preliminarily performed, in which the split-step method was used as a numerical method for solving a second-order partial differential wave equation for the complex amplitude of the wave field of a laser beam. This method, combined with methods of spectral-phase Fourier transforms and statistical tests, is the most effective way to obtain reliable quantitative results for solving engineering problems of atmospheric wave optics. Quantitative data are obtained on the effect of turbulent atmospheric distortions along propagation paths on the main parameters of coherent laser beams – focusing, effective average radius, and the proportion of the beam energy in its diffraction spot. The preliminary results obtained of the system mock-up performance confirm the conclusions of the theory.

Vibration Suppression for Coupled Wave PDEs in Deep-Sea Construction

A deep-sea construction vessel (DCV) is used to install underwater parts of an offshore oil drilling platform at the designated locations on the seafloor. By using extended Hamilton's principle, a nonlinear partial differential equation (PDE) system governing the lateral-longitudinal coupled vibration dynamics of the DCV consisting of a time-varying-length cable with an attached item is derived, and it is linearized at the steady state generating a linear PDE model, which is extended to a more general system including two coupled wave PDEs connected with two interacting ordinary differential equations (ODEs) at the uncontrolled boundaries. Through a preliminary transformation, an equivalent reformulated plant is generated as a 4×4 coupled heterodirectional hyperbolic PDE-ODE system characterized by spatially varying coefficients on a time-varying domain. To stabilize such a system, an observer-based output-feedback control design is proposed, where the measurements are only placed at the actuated boundary of the PDE, namely, at the platform at the sea surface. The exponential stability of the closed-loop system, boundedness and exponential convergence of the control inputs, are proved via Lyapunov analysis. The obtained theoretical result is tested on a nonlinear model with ocean disturbances, even though the design is developed in the absence of such real-world effects.

Magnetic Resonance-Electrical Properties Tomography by Directly Solving Maxwell’s Curl Equations

Magnetic Resonance-Electrical Properties Tomography (MR-EPT) is a method to reconstruct the electrical properties (EPs) of bio-tissues from the measured radiofrequency (RF) field in Magnetic Resonance Imaging (MRI). Current MR-EPT approaches reconstruct the EP profile by solving a second-order partial differential wave equation problem, which is sensitive to noise and can induce large reconstruction artefacts near tissue boundaries and areas with inaccurate field measurements. In this paper, a novel MR-EPT approach is proposed, which is based on a direct solution to Maxwell’s curl equations. The distribution of EPs is calculated by iteratively fitting the RF field calculated by the finite-difference-time-domain (FDTD) technique to the measured values. To solve the time-consuming problem of the iterative fitting, a graphics processing unit (GPU) is used to accelerate the FDTD technique to process the field calculation kernel. The new EPT method was evaluated by investigating a numerical head phantom, and it was found that EP values can be accurately calculated and were relatively insensitive to simulated RF field errors. The feasibility of the proposed method was further validated using phantom-based experimental data collected from a 9.4 Tesla (T) Magnetic Resonance Imaging (MRI) system. The results also indicated that more accurate EP values could be reconstructed from the new method compared with conventional methods. Moreover, even in the absence of phase information of the RF field, the proposed approach is still capable of offering robust EPT solutions, thus having improved practicality for potential clinical applications.

Adaptive output-feedback control of torsional vibration in off-shore rotary oil drilling systems

Motivated by an engineering application of torsional vibration suppression of off-shore oil drilling, we design an adaptive output-feedback controller for a one-dimensional wave partial differential equation (PDE) system, where an anti-damping term with an unknown coefficient and a general harmonic disturbance with unknown amplitudes exist in the bit, which is modeled as a second-order-in-time boundary. The control input anti-collocated with this boundary subject to uncertainty, is designed by using the adaptive control method and infinite-dimensional backstepping technique. The asymptotic convergence to zero of the uncontrolled boundary states, i.e., the oscillations of the angular displacement and velocity at the bit, and the boundedness of all states in the closed-loop system, are proved via Lyapunov analysis. The effectiveness of the proposed adaptive controller is verified via numerical simulation. The results also can be applied to other applications, such as vibration control of cable elevators with uncertain cage-guide friction and cage disturbances.