A comparison of analytical solutions of nonlinear complex generalized Zakharov dynamical system for various definitions of the differential operator

<abstract><p>This paper considers deriving new exact solutions of a nonlinear complex generalized Zakharov dynamical system for two different definitions of derivative operators called conformable and $ M- $ truncated. The system models the spread of the Langmuir waves in ionized plasma. The extended rational $ sine-cosine $ and $ sinh-cosh $ methods are used to solve the considered system. The paper also includes a comparison between the solutions of the models containing separately conformable and $ M- $ truncated derivatives. The solutions are compared in the $ 2D $ and $ 3D $ graphics. All computations and representations of the solutions are fulfilled with the help of Mathematica 12. The methods are efficient and easily computable, so they can be applied to get exact solutions of non-linear PDEs (or PDE systems) with the different types of derivatives.</p></abstract>

A Concise Review of State Estimation Techniques for Partial Differential Equation Systems

While state estimation techniques are routinely applied to systems represented by ordinary differential equation (ODE) models, it remains a challenging task to design an observer for a distributed parameter system described by partial differential equations (PDEs). Indeed, PDE systems present a number of unique challenges related to the space-time dependence of the states, and well-established methods for ODE systems do not translate directly. However, the steady progresses in computational power allows executing increasingly sophisticated algorithms, and the field of state estimation for PDE systems has received revived interest in the last decades, also from a theoretical point of view. This paper provides a concise overview of some of the available methods for the design of state observers, or software sensors, for linear and semilinear PDE systems based on both early and late lumping approaches.

Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design

Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first-order reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two- and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structure-preserving WENO-like schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.

Quantized State Feedback-Based H∞ Control for Nonlinear Parabolic PDE Systems via Finite-Time Interval

In this paper, the finite-time H∞ control problem of nonlinear parabolic partial differential equation (PDE) systems with parametric uncertainties is studied. Firstly, based on the definition of the quantiser, static state feedback controller and dynamic state feedback controller with quantization are presented, respectively. The finite-time H∞ control design strategies are subsequently proposed to analyze the nonlinear parabolic PDE systems with respect to the effect of quantization. And by constructing appropriate Lyapunov functionals for the studied systems, sufficient conditions for the existence of the feedback control gains and the quantizer’s adjusting parameters which guarantee the prescribed attenuation level of H∞ performance are expressed as nonlinear matrix inequalities. Then, by using some inequalities and decomposition technic, the nonlinear matrix inequalities are transformed to standard linear matrix inequalities (LMIs). Moreover, the optimal H∞ control performances are pursued by solving optimization problems subject to the LMIs. Finally, to illustrate the feasibility and effectiveness of the finite-time H∞ control design strategies, an application to the catalytic rod in a reactor is explored.