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ABSTRACT This work investigates the adaptive constrained control design for a parallel‐flow heat exchanger represented by a system of two coupled linear first‐order hyperbolic partial differential equations (PDEs). This system incorporates structured uncertainty involving unknown in‐domain parameters that characterize neglected dynamics in the heat exchanger model. These parameters may encompass unmodeled heat transfer phenomena, variations in fluid properties, and modeling simplifications. The objective is to regulate the internal fluid outlet temperature to track a desired reference trajectory by adjusting the external fluid velocity. Due to inherent physical constraints, this manipulated variable is upper and lower‐bounded. Accordingly, the control problem is bounded and bilinear. Using the set‐invariance principle and an energy‐like framework, we first develop a bounded state‐feedback controller. Then, since the measurements are considered only at the boundaries, we propose an adaptive boundary observer using a swapping scheme and a recursive least squares identifier. The proposed adaptive observer provides online estimates of the distributed state and the unknown parameters. Next, the state‐feedback controller is associated with the boundary observer and parameter identifier, and the exponential stability of the closed‐loop system is guaranteed using Lyapunov's stability theory. Finally, we provide numerical simulations to demonstrate the efficiency of the proposed control scheme. [ABSTRACT FROM AUTHOR]

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L 2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L 2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L -traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods. [ABSTRACT FROM AUTHOR]

A large Courant–Friedrichs–Lewy (CFL) algorithm is presented for the explicit, finite volume solution of hyperbolic systems of conservation laws, with a focus on the shallow water equations. The Riemann problems used in the flux computation are determined using averaging kernels that extend over several computational cells. The usual CFL stability constraint is replaced with a constraint involving the kernel support size. This makes the method unconditionally stable with respect to the size of the computational cells, allowing the computational mesh to be refined locally to an arbitrary degree without altering solution stability. The practical implementation of the method is detailed for the shallow water equations with topographical source term. Computational examples report applications of the method to the linear advection, Burgers and shallow water equations. In the case of sharp bottom discontinuities, the need for improved, well‐balanced discretisations of the geometric source term is acknowledged. [ABSTRACT FROM AUTHOR]

The initial-boundary and the inverse coefficient problems for the semilinear hyperbolic equation with strong damping are considered in this study. The conditions for the existence and uniqueness of solutions in Sobolev spaces to these problems have been established. The inverse problem involves determining the unknown time-dependent parameter in the right-hand side function of the equation using an additional integral type overdetermination condition. [ABSTRACT FROM AUTHOR]

This study investigates the controllability of a Volterra evolution equation with impulsive terms and nonlocal initial conditions. With the aid of the resolvent operator generated by the linear part of the equation, mild solutions can be defined. Notably, the resolvent operator lacks compactness and equicontinuity. Additionally, the compactness of the impulsive and nonlocal functions is not required. Sufficient conditions for controllability are obtained through measures of noncompactness in Banach spaces. Functional differential equations and hyperbolic partial differential equations can be solved with these results. An example is given to illustrate the validity of the presented results. [ABSTRACT FROM AUTHOR]

ABSTRACT A long‐standing issue in the parallel‐in‐time community is the poor convergence of standard iterative parallel‐in‐time methods for hyperbolic partial differential equations (PDEs), and for advection‐dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two‐level variant of the iterative parallel‐in‐time method of multigrid reduction‐in‐time (MGRIT). This closed‐form theory allows for new insights into the poor convergence of MGRIT for advection‐dominated PDEs when using the standard approach of rediscretizing the fine‐grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse‐grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady‐state advection‐dominated PDEs. We apply this convergence theory to show that, for certain semi‐Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse‐grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re‐purposed in the MGRIT context to develop more robust parallel‐in‐time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

This study aims to establish the convolutional-type variational and reciprocity theorems within the framework of the hyperbolic two-temperature generalized thermoelasticity theory for an isotropic thermoelastic material, with the help of alternate formulation of the mixed boundary initial value problem, in which initial conditions are combined with field equations (using the Laplace transform). The convolutional-type variational principle adapts readily to numerical solutions based on the Ritz method and is useful in the finite element method. The reciprocity theorem is helpful in the theoretical development of boundary and finite element methods. The current effort can be valuable for the problem of coupling effects of thermal and mechanical fields, especially in geophysics and mining. [ABSTRACT FROM AUTHOR]

Changes in flow rates in pressurized pipeline systems, due to power loss in a pump, opening or closing of a valve, load rejection by a turbine, etc., may lead to unsteady conditions called fluid transients or water hammer in the pipeline, which, in turn, can lead to dangerous consequences. This potential phenomenon must be checked during the design phase of pipeline systems in order to prevent such dangerous situations. For this purpose, many software programs have been developed worldwide to analyze fluid transients. In this study, a computer software was developed that analyzes this type of flow in pipeline systems using C# programming language. In addition to simulating and solving problems in transient flows in pipeline systems, this will help the purpose of having a free domestic transient flow software for education and research purposes, avoiding expensive alternatives. The method of characteristics is used to solve the nonlinear, hyperbolic partial differential equations of unsteady pipe flow. The developed software was tested using various well-known benchmark cases for validation purposes. It was shown that the computed results of the new software are very similar to those obtained in the literature, proving the accuracy and reliability of the new program. In future studies, it is hoped that the program will be more comprehensive with more boundary conditions added to the program. In this paper, the developed program and some of its important features will be introduced, and validation of the program with certain benchmark studies will be provided. [ABSTRACT FROM AUTHOR]

In order to study the supercritical flow in a curved channel of a rectangular cross section, the classical shallow water equations in a cylindrical coordinate system based on the mass and momentum laws that take into account the friction and bottom slope are used. The obtained mathematical model forms nonlinear partial differential equations of first-order. For simplification, a linearization of the partial differential equations (PDEs) set is performed using small perturbation approach valid for weak bends (axial curvature radius extremely larger than the channel width). The governing equations with well-posed initial and boundary conditions were solved for a rectangular bend channel flow by applying the method of characteristics that is capable of transforming the hyperbolic partial differential equations to a system of ordinary differential equations (ODEs). The proposed model is tested and validated by comparing the results with broad available experimental data reported in the literature, and particular attention was paid to the wave maximum and its location. Comparisons indicate a reasonable agreement between the results obtained for the maximum flow depth along the outer channel wall. However, the model prediction is only reliable for a small relative curvature. Despite the model limitations, the results show the reliability and accuracy of the proposed approach for practical design purposes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following mixed local and nonlocal hyperbolic equation: u t t − Δ u + μ ( − Δ ) s u = | u | p − 2 u , in  Ω × R + , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , in  Ω , u ( x , t ) = 0 , in  ( R N ∖ Ω ) × R 0 + , where s ∈ ( 0 , 1 ), N > 2, p ∈ ( 2 , 2 s ∗ ], μ is a nonnegative real parameter, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω, Δ is the Laplace operator, ( − Δ ) s is the fractional Laplace operator. By combining the Galerkin approach with the modified potential well method, we obtain the global existence, vacuum isolating, and blow-up of solutions for the aforementioned problem, provided certain assumptions are fulfilled. Specifically, we study the existence of global solutions for the above problem in the cases of subcritical and critical initial energy levels, as well as the finite time blow-up of solutions. Then, we investigate the blow-up of solutions for the above problem in the case of supercritical initial energy level, as well as upper and lower bounds of blow-up time of solutions. [ABSTRACT FROM AUTHOR]

In this paper, two problems involving nonlinear time fractional hyperbolic partial differential equations (PDEs) and time fractional pseudo hyperbolic PDEs with nonlocal conditions are presented. Collocation technique for shifted Chebyshev of the second kind with residual power series algorithm (CTSCSK-RPSA) is the main method for solving these problems. Moreover, error analysis theory is provided in detail. Numerical solutions provided using CTSCSK-RPSA are compared with existing techniques in literature. CTSCSK-RPSA is accurate, simple and convenient method for obtaining solutions of linear and nonlinear physical and engineering problems. [ABSTRACT FROM AUTHOR]

The influence of cleat permeability and porosity on the cumulative gas produced is studied. The cleat parameters are weighed for producing reserve in the least time and the minimum breakdown pressure requirement to create fractures. The reservoir is stimulated. The breakdown pressure required is more to create fractures at closer spacing than needed to increase the width of new fractures. The non-linear hyperbolic partial differential equations (PDE) are solved by the implicit finite difference method. Cleat permeability has a more significant influence on the gas produced than the porosity. Optimized values of 300 mD and 0.03 porosity sweep more area in a short time. Due to the faster propagation of drainage, the adsorbed gas is not recovered to nearly complete before the entire reservoir/pay zone length is drained. Fracturing a formation can be successful if the same adsorbed gas is spread over a larger area, and length is insufficient to accommodate the gas in place. The pay zone length needs to be larger than the present value. The novel approach of checking both the economic feasibility of fracturing a formation and its relation with the length of the reservoir, is discovered and solved here. [ABSTRACT FROM AUTHOR]

In this article, we propose a new path-conservative discontinuous Galerkin (DG) method to solve non-conservative hyperbolic partial differential equations (PDEs). In particular, the method here applies the one-stage ADER (Arbitrary DERivatives in space and time) approach to fulfill the temporal discretization. In addition, this method uses the differential transformation (DT) procedure rather than the traditional Cauchy–Kowalewski (CK) procedure to achieve the local temporal evolution. Compared with the classical ADER methods, the current method is free of solving generalized Riemann problems at inter-cells. In comparison with the Runge–Kutta DG (RKDG) methods, the proposed method needs less computer storage, thanks to the absence of intermediate stages. In brief, this current method is one-step, one-stage, and fully-discrete. Moreover, this method can easily obtain arbitrary high-order accuracy both in space and in time. Numerical results for one- and two-dimensional shallow water equations (SWEs) show that the method enjoys high-order accuracy and keeps good resolution for discontinuous solutions. [ABSTRACT FROM AUTHOR]

We study a class of nonlinear hyperbolic partial differential equations with boundary control. This class describes chemical reactions of the type " A → product" carried out in a plug flow reactor (PFR) in the presence of an inert component. An isoperimetric optimal control problem with periodic boundary conditions and input constraints is formulated for the considered mathematical model in order to maximize the mean amount of product over the period. For the single-input system, the optimality of a bang-bang control strategy is proved in the class of bounded measurable inputs. The case of controlled flow rate input is also analyzed by exploiting the method of characteristics. A case study is performed to illustrate the performance of the reaction model under different control strategies. [ABSTRACT FROM AUTHOR]

AbstractThe study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ2 + h2) in L∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ4 + h4) in L∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings. [ABSTRACT FROM AUTHOR]

This paper is dedicated to investigating the chaos of a initial-boundary value (IBV) problem of a multi-dimensional weakly hyperbolic equation subject to two general nonlinear boundary conditions (NBCs). The existence and uniqueness of solution for the IBV problem are established. By employing the snap-back repeller and heteroclinic cycle theories, it has been proven that the IBV problem with a linear and a general NBCs exhibits chaos in the sense of both Devaney and Li–Yorke. Furthermore, these chaotic results are extended to the IBV problem with two general NBCs. Two stability criteria of the IBV problem are established, respectively, for the corresponding two cases of boundary conditions. Finally, numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre's hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function. [ABSTRACT FROM AUTHOR]

The shear shallow water (SSW) model introduces an approximation for shallow water flows by including the effect of vertical shear in the system. Six non-linear hyperbolic partial differential equations with non-conservative laws make up this system. Shear, contact, rarefaction, and shock waves are all admissible in this model. We developed the finite-volume two-step scheme, the so-called generalized Rusanov (G. Rusanov) scheme, for solving the SSW model. This method is split into two stages. The first one relies on a local parameter that permits control over the diffusion. In stage two, the conservation equation is recovered. Numerous numerical instances were taken into consideration. We clarified that the G. Rusanov scheme satisfied the C-property. We also compared the numerical solutions with those obtained from the Rusanov, Lax-Friedrichs, and reference solutions. Finally, the G. Rusanov technique may be applied for solving a wide range of additional models in developed physics and applied science. [ABSTRACT FROM AUTHOR]

This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coefficient. Our approach involves approximating the hyperbolic equation's solution by its truncated Fourier expansion in the time domain and using the recently developed polynomial-exponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The method not only delivers precise solutions but also showcases remarkable computational efficiency. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the International Journal of Robust & Nonlinear Control on the topic of time-delay systems. It provides a historical overview of functional differential equations and highlights the contributions of Nikolay Nikolayevich Krasovskii and Aristide Halanay to the field. The special issue includes 21 research papers covering stability, robustness, observer theory, predictors, and delay-based approaches to partial differential equation models. Each paper is briefly described, covering topics such as stabilization of uncertain systems, sampled-data control for nonlinear systems, observer theory for differential delay systems, sensor fault estimation, and control approaches for vehicular platoons. [Extracted from the article]

The article starts from the challenge of the critical cases in difference operator stability for neutral functional differential equations (NFDE). Such cases occur in the NFDE associated to 1D$$ 1D $$ hyperbolic partial differential equations (PDE) dynamics in mechanical and hydraulic engineering. For some of such applications it resulted that the aforementioned critical (nonasymptotic) stability is connected to the character and level of the energy losses. It is shown that suitable choice of the losses to be taken into account can remove the critical stability properties and give the difference operator the asymptotic stability thus ensuring asymptotic stability for the system's dynamics and also other asymptotic properties. [ABSTRACT FROM AUTHOR]

For population systems modeled by age‐structured hyperbolic partial differential equations (PDEs), we redesign the existing feedback laws, designed under the assumption that the dilution input is directly actuated, to the more realistic case where dilution is governed by actuation dynamics (modeled simply by an integrator). In addition to the standard constraint that the population density must remain positive, the dilution dynamics introduce constraints of not only positivity of dilution, but possibly of given positive lower and upper bounds on dilution. We present several designs, of varying complexity, and with various measurement requirements, which not only ensure global asymptotic (and local exponential) stabilization of a desired positive population density profile from all positive initial conditions, but do so without violating the constraints on the dilution state. To develop the results, we exploit the relation between first‐order hyperbolic PDEs and an equivalent representation in which a scalar input‐driven mode is decoupled from input‐free infinite‐dimensional internal dynamics represented by an integral delay system. [ABSTRACT FROM AUTHOR]

The dispersion error is currently the dominant error for computed solutions of wave propagation problems with high-frequency components. In this paper, we define and give explicit examples of interior \alpha -dispersion-relation-preserving schemes, of interior order of accuracy 4, 5, 6, and 7, with a complete methodology to construct them. These are dual-pair finite-difference schemes for systems of hyperbolic partial differential equations which satisfy the summation-by-parts principle and preserve the dispersion relation of the continuous problem uniformly to an \alpha \% error tolerance for their interior stencil. We give a general framework to design provably stable finite-difference operators whose interior stencil preserves the dispersion relation for hyperbolic systems such as the elastic wave equation. The operators we derive here can resolve the highest frequency (π-mode) present on any equidistant grid at a tolerance of % lessapprox 5% maximum error within the interior stencil, with minimal extra stencil points. As standard finite-difference schemes have a 100\% dispersion error for high-frequency components, fine meshes must be used to resolve these components. Our derived schemes may compute solutions with the same accuracy as traditional schemes on far coarser meshes, which in high dimensions significantly improves the computational cost. [ABSTRACT FROM AUTHOR]

Strokes are one of the leading causes of death in the United States. Stroke treatment involves removal or dissolution of the obstruction (usually a clot) in the blocked artery by catheter insertion. A computer simulation to systematically plan such patient‐specific treatments needs a network of about 105 blood vessels including collaterals. The existing computational fluid dynamic (CFD) solvers are not employed for stroke treatment planning as they are incapable of providing solutions for such big arterial trees in a reasonable amount of time. This work presents a novel one‐dimensional mathematical formulation for blood flow modeling in an elastic blood vessel with a centrally placed rigid catheter. The governing equations are first‐order hyperbolic partial differential equations, and the hypergeometric function needs to be computed to obtain the characteristic system of these hyperbolic equations. We employed the Discontinuous Galerkin method to solve the hyperbolic system and validated the implementation by comparing it against a well‐established 3D CFD solver using idealized vessels and a realistic truncated arterial network. The results showed clinically insignificant differences in steady flow cases, with overall variations between 1D and 3D models remaining below 10%. Additionally, the solver accurately captured wave reflection phenomena at domain discontinuities in unsteady cases. A primary advantage of this model over 3D solvers is its ease in obtaining a discretized geometry of complex vasculatures with multiple arterial branches. Thus, the 1D computational model offers good accuracy and applicability in simulating complex vasculatures, demonstrating promising potential for investigating patient‐specific endovascular interventions in strokes. [ABSTRACT FROM AUTHOR]

We consider for a small parameter > 0 a parabolic convection–diffusion problem with Péclet number of order ( − 1) in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order () connected through a domain (node) of diameter (). Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as → 0 , i.e. when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems (= 0) are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed. [ABSTRACT FROM AUTHOR]

We study nonlinear inverse problems for first order hyperbolic equations. We find a solution and an unknown coefficient depending on the time variable. For additional conditions we consider integral and boundary overdetermination conditions. We establish the existence of a regular solution, i.e., the solution possesses all generalized derivatives in the sense of Sobolev entering the equation. [ABSTRACT FROM AUTHOR]

We study a hyperbolic equation with an arbitrary number of potentials undergoing translation in arbitrary directions. Differential-difference equations arise in various applications that are not covered by the classical theory of differential equations. In addition, they are of considerable interest from a theoretical point of view, since the nonlocal nature of such equations gives rise to various effects that do not arise in the classical case. We find a condition on the vector of coefficients for nonlocal terms in the equation and on the vectors of potential translations that ensures the global solvability of the equation under consideration. By imposing the specified condition on the equation and using the classical Gelfand–Shilov scheme, we explicitly construct a three-parameter family of smooth global solutions to the equation under study. [ABSTRACT FROM AUTHOR]

A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, using solution point constraints. Applying a transform method, we reduce the inverse problems to direct ones, which are studied for well-posedness in special weighted Sobolev spaces. This means that the inverse problem is said to be well posed in the sense of Tikhonov (or conditionally well posed). The main aim of this study is to develop a finite difference method for solution of the transformed hyperbolic problems with a non-local differential operator and initial conditions. Numerical test examples are also analyzed. [ABSTRACT FROM AUTHOR]

Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in the space-time domain. Considering the non-Fourier effect of heat conduction, the finite speed of thermal wave propagation has been attained. The role of the fractional order parameter has been examined scientifically. The results obtained by considering the fractional order theory and the integer order theory perfectly coincide as a limiting case of fractional order parameter approaches one. [ABSTRACT FROM AUTHOR]

This article studies the problem of prescribed‐time constrained tracking control for a class of 2×2$$ 2\times 2 $$ hyperbolic partial differential equation (PDE) systems with actuator dynamics, which are described by a set of nonlinear ordinary differential equations (ODEs). Since the control input only appears in the ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite challenging. The most important is that for the control of such ODE controlled PDE systems we mainly make the following two contributions: (1) the controlled output of the PDE system tracks the reference signal within the prescribed time; (2) the controlled output and all the actuator states are constrained. It is the first time that such a prescribed‐time constrained tracking control problem is addressed for the PDE‐ODE coupled system considered in this article. Through rigorous theoretical proof, it is demonstrated that all the system states and control signals are bounded and sufficiently continuous by configuring appropriate design parameters. Finally, the performance is investigated via numerical simulation. [ABSTRACT FROM AUTHOR]

Summary: This paper studies the adaptive synchronization of complex spatio‐temporal networks modeled by semi‐linear hyperbolic partial differential equations (CSTNSLHPDEs) as well as considering time‐invariant and time‐varying delays in a one‐dimensional space. Firstly, a distributed adaptive controller is proposed, where different nodes are with different adaptive gains. Secondly, four cases, CSTNSLHPDEs with time‐invariant delays and one single weight, with time‐invariant delays and multi‐weights, with time‐varying delays and one single weight, and with time‐varying delays and multi‐weights, are successively analyzed, and synchronization conditions of these four cases are obtained by using the proposed distributed adaptive controller. In the end, examples illustrate the effectiveness of the proposed distributed adaptive controller. [ABSTRACT FROM AUTHOR]

By using the power-exponential function method and the extended hyperbolic auxiliary equation method, we present the explicit solutions of the subsidiary elliptic-like equation. With the aid of the subsidiary elliptic-like equation and a simple transformation, we obtain the exact solutions of Hirota equation which include bright solitary wave, dark solitary wave, bell profile solitary wave solutions and Jacobian elliptic function solutions. Some of these solutions are found for the first time, which may be useful for depicting nonlinear physical phenomena. This approach can also be applied to solve the other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

Local wellposedness for a nonlinear parabolic-hyperbolic coupled system modeling Micro-Electro-Mechanical System (MEMS) is studied. The particular device considered is a simple capacitor with two closely separated plates, one of which has motion modeled by a semilinear hyperbolic equation. The gap between the plates contains a gas and the gas pressure is taken to obey a quasilinear parabolic Reynolds' equation. Local-in-time existence of strict solutions of the system is shown, using well-known local-in-time existence results for the hyperbolic equation, then Hölder continuous dependence of its solution on that of the parabolic equation, and finally getting local-in-time existence for a combined abstract parabolic problem. Semigroup approaches are vital for the local-in-time existence results. [ABSTRACT FROM AUTHOR]

We consider a hyperbolic equation with variable leading part and nonlinearity in the lower-order term. The coefficients of the equation are smooth functions constant beyond some compact domain in the three-dimensional space. A plane wave with direction falls to the heterogeneity from the exterior of this domain. A solution to the corresponding Cauchy problem for the original equation is measured at boundary points of the domain for a time interval including the moment of arrival of the wave at these points. The unit vector is assumed to be a parameter of the problem and can run through all possible values sequentially. We study the inverse problem of determining the coefficient of the nonlinearity on using this information about solutions. We describe the structure of a solution to the direct problem and demonstrate that the inverse problem reduces to an integral geometry problem. The latter problem consists of constructing the desired function on using given integrals of the product of this function and a weight function. The integrals are taken along the geodesic lines of the Riemannian metric associated with the leading part of the differential equation. We analyze this new problem and find some stability estimate for its solution, which yields a stability estimate for solutions to the inverse problem. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a one-dimensional hyperbolic equation whose lower-order coefficient and right-hand side oscillate in time with a high frequency and the amplitude of the lower-order coefficient is small. Under study is the reconstruction of the cofactors of these rapidly oscillating functions independent of the space variable from a partial asymptotics of a solution at some point of the space. The classical theory of inverse problems examines the numerous problems of determining unknown sources, and coefficients without rapid oscillations for various evolutionary equations, where the exact solution to the direct problem appears in the additional overdetermination condition. Equations with rapidly oscillating data are often encountered in modeling the physical, chemical, and other processes that occur in media subjected to high-frequency electromagnetic, acoustic, vibrational, and others fields, which demonstrates the topicality of perturbation theory problems on the reconstruction of unknown functions in high-frequency equations. We give some nonclassical algorithm for solving such problems that lies at the junction of asymptotic methods and inverse problems. In this case the overdetermination condition involves a partial asymptotics of solution of a certain length rather than the exact solution. [ABSTRACT FROM AUTHOR]

Integral identities that are fulfilled for measure of difference between the exact solution of a parabolic-hyperbolic equation and any functions from a corresponding energy class are proved. These identities make it possible to derive two-sided a posteriori estimates for approximate solution to the corresponding Cauchy problem. The left-hand side of such an estimate is a natural measure of deviation from the solution, and the right-hand side depends on the problem data and the approximate solution itself and, therefore, it can be explicitly calculated. These estimates can be used to control the accuracy of approximate solutions and to compare solutions to Cauchy problems with different initial conditions. These estimates also allow one to quantitatively assess the effects occurring due to inaccuracies in the initial data and in the coefficients of the equation. [ABSTRACT FROM AUTHOR]

In the paper the speed‐in‐action problem is investigated for the second‐order nonlinear hyperbolic equation. The main feature of the paper is that for the first time the problem is considered with a nonlocal condition in differ from many works devoted the similar problems. A theorem on the existence of the optimal control is proved. Freshet differentiability of the corresponding functional is shown. The necessary optimality conditions are derived in the form of variational inequalities. [ABSTRACT FROM AUTHOR]

The iterated Crank-Nicolson (ICN) method is a successful numerical algorithm for solving partial differential equations. When unequal weights are used in the ICN predictor-corrector process, the convergence rate is reduced to first order. In this paper, we propose two modified ICN algorithms that achieve second order convergence rate, based on two different ways of choosing the weights. The first approach employs geometrically averaged weights in two consecutive iterations, and the second one uses arithmetically averaged weights for two consecutive time steps. The stability and second order accuracy of our methods are verified using stability and truncation error analysis and numerically demonstrated by solving linear and semi-linear hyperbolic partial differential equations, Burgers' equation, and the Gross-Pitaevskii equation. [ABSTRACT FROM AUTHOR]

In the case of oscillatory potential, we present some new Lyapunov -type inequalities for linear hyperbolic and elliptic equations on a rectangular domain in R². No sign restriction is imposed on the potential function. As applications of the Lyapunov-type inequalities obtained, we give some estimations for disconjugacy of hyperbolic and elliptic Dirichlet boundary value problems. [ABSTRACT FROM AUTHOR]

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell's equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell's equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell's equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space. [ABSTRACT FROM AUTHOR]

The potential of a finite-element technique based on an egalitarian meshing of the space–time domain D of physical problems described by parabolic or hyperbolic differential equations is explored. A least-squares minimization technique is applied in the meshed domain D to obtain stiffness-like matrices associated with various linear differential operators. Applications discussed include problems of boundary growth, and diffusive coalescence, in which D cannot be regarded as the Cartesian product of two independent domains in space and time. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some of important qualitative properties of solutions of secondorder hyperbolic equations, whose coefficients of the terms involving the second-order derivatives are independent of the desired function and its derivatives. Solutions of these equations have a special property called curvilinear parallelogram identity (or mean-value property), which can be used to solve some initial-boundary value problems. [ABSTRACT FROM AUTHOR]

Using the method of characteristics and the Banach fixed point theorem (for the Bielecki metric), in the paper it is established the existence and uniqueness of a global (continuous) solution of the mixed problem in the rectangle Π = {(x, t): 0 < x < l < ∞, 0 < t < T < ∞} for the first order hyperbolic system of semi-linear equations of the form... for a singular with orthogonal (degenerate) and non-orthogonal to the coordinate axes characteristics and with nonlinear boundary conditions, where Λ(x, t) = diag(λ1(x, t),. . ., λk(x, t)), u = (u1, . . ., uk), v = (v1, . . ., vm), w = (w1, . . ., wn), f = (f1, . . ., fk), g = (g1, . . ., gm), h = (h1, . . ., hn) and besides sign λi(0, t) = const ̸= 0, sign λi(l, t) = const ̸= 0 for all t ∈ [0, T] and for all i ∈ {1, . . ., k}. The presence of non-orthogonal and degenerate characteristics of the hyperbolic system for physical reasons indicates that part of the oscillatory disturbances in the medium propagates with a finite speed, and part with an unlimited one. Such a singularity (degeneracy of characteristics) of the hyperbolic system allows mathematical interpretation of many physical processes, or act as auxiliary equations in the analysis of multidimensional problems. [ABSTRACT FROM AUTHOR]

For a second-order hyperbolic equation with inhomogeneity |u|m-1u, m > 1, a forward and an one-dimensional inverse problems are studied. The inverse problem is devoted to determining the coefficient under heterogeneity. As an additional information, the trace of the derivative with respect to x of the solution to the forward initial-boundary value problem is given at x = 0 on a finite interval. Conditions for the unique solvability of the forward problem are found. For the inverse problem a local existence and uniqueness theorems are established and a stability estimate of its solutions is found. [ABSTRACT FROM AUTHOR]

The study presents a novel approach called FEM-POD, which aims to enhance the computational efficiency of the Finite Element Method (FEM) in solving problems related to non-Fourier heat conduction. The present method employs the Proper Orthogonal Decomposition (POD) technique. Firstly, spatial discretization of the second-order hyperbolic differential equation system is achieved through the Finite Element Method (FEM), followed by the application of the Newmark method to address the resultant ordinary differential equation system over time, with the resultant numerical solutions collected in snapshot form. Next, the Singular Value Decomposition (SVD) is employed to acquire the optimal proper orthogonal decomposition basis, which is subsequently combined with the FEM utilizing the Newmark scheme to construct a reduced-order model for non-Fourier heat conduction problems. To demonstrate the effectiveness of the suggested method, a range of numerical instances, including different laser heat sources and relaxation durations, are executed. The numerical results validate its enhanced computational accuracy and highlight significant time savings over addressing non-Fourier heat conduction problems using the full order FEM with the Newmark approach. Meanwhile, the numerical results show that when the number of elements or nodes is relatively large, the CPU running time of the FEM-POD method is even hundreds of times faster than that of classical FEM with the Newmark scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O (Δ t + h ¯ 4) , where Δ t is the time step and h ¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method. [ABSTRACT FROM AUTHOR]