Hyperbolic Equations Research Articles

A comparison of hyperbolic sine creep life equations and data correlation methods for these equations

ABSTRACT Hyperbolic sine (HS) minimum creep rate equations combined with the Monkman-Grant equation are attractive for correlating time to rupture data because of their physical basis in dislocation creep theory, numerical stability, and conformance to physical boundary conditions at zero and infinite stress. In this study, uniaxial creep life data of Sanicro 25 and HR6W alloys were correlated to two recently published HS creep life equations with three commonly used objective functions. For each alloy, the correlation results (fit to the data and predicted strength at 100,000 h time to rupture) were essentially the same using either HS equation. Multiple equivalent solutions (same goodness of fit but different values of the fitting constants) were obtained with the HR6W data set. A discussion on obtaining initial estimates of fitting constants is provided as an Appendix.

The problem with the missing Goursat condition at the boundary of the domain for a degenerate hyperbolic equation with a singular coefficient

The problem with the missing Goursat condition at the boundary of the domain for a degenerate hyperbolic equation with a singular coefficient

Problem of Linear Conjugation with Multipoint Conditions in the Case of Multiple Nodes for Higher-Order Strictly Hyperbolic Homogeneous Equations

Problem of Linear Conjugation with Multipoint Conditions in the Case of Multiple Nodes for Higher-Order Strictly Hyperbolic Homogeneous Equations

Observers for hyperbolic systems with multiple delays in the nonlocal boundary condition and its application to secure image encryption

In this article, we propose a step-by-step observer design for a first-order linear hyperbolic system with multiple delays in the nonlocal boundary condition. In the proposed approach inspired by the backstepping methodology, the observer gains are obtained following seven successive steps. Next, we present the application of this observer design in the context of image encryption. We consider a system where a local nonlinear term causing a chaotic vibration is added to the nonlocal boundary condition with time lags. A synchronization system for this chaotic system can be constructed using the same observer design method. The chaotic states generated by both systems can be used to encrypt and decrypt image data. The time lags, and weight functions included in the nonlocal boundary condition play the role of common encryption keys. By considering a system that includes arbitrarily many delays, this approach allows for a significant increase in the number of common encryption keys compared to conventional methods, resulting in greatly improved communication security.

An absolutely convergent fixed-point fast sweeping WENO method on triangular meshes for steady state of hyperbolic conservation laws

High order accuracy fast sweeping methods have been developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs) on rectangular meshes, while they were not available yet on unstructured meshes. In this paper, we extend high order accuracy fast sweeping methods to unstructured triangular meshes by applying fixed-point iterative sweeping techniques to a fifth-order finite volume unstructured WENO scheme, for solving steady state solutions of hyperbolic conservation laws. Similar as other fast sweeping methods, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady state solutions. An advantage of fixed-point fast sweeping methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. This also provides certain convenience in designing high order fast sweeping methods on unstructured meshes. As in the first order fast sweeping methods on triangular meshes, we introduce multiple reference points to determine alternating sweeping directions on unstructured meshes. All the cells on the mesh are ordered according to their centroids' distances to those reference points, and the resulted orderings provide sweeping directions for iterations. To make the residue of the fast sweeping iterations converge to machine zero / round off errors, we follow the approach in our early work of developing the absolutely convergent fixed-point fast sweeping WENO methods on rectangular meshes, and adopt high order WENO scheme with unequal-sized sub-stencils, specifically here a fifth-order finite volume unstructured WENO scheme for spatial discretization. Extensive numerical experiments, including problems with complex domain geometries, are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth-order fast sweeping scheme on triangular meshes. Furthermore, the proposed method is compared with the forward Euler time marching method and the popular third order total variation diminishing Rung-Kutta (TVD-RK3) time-marching method for steady state computations. Numerical examples show that the developed fixed-point fast sweeping WENO method is the most efficient scheme among them, and especially it can save up to 70% CPU time costs than TVD-RK3 in order to converge to steady state solutions.

On Lattice Boltzmann Methods based on vector-kinetic models for hyperbolic partial differential equations

In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit discretisation of vector-kinetic equation (VKE), we also consider LBE derived by semi-implicit discretisation of VKE and compare the relaxation factors of both. We study the properties such as H-inequality, total variation boundedness and positivity of both the LBEs, and infer that the LBE due to semi-implicit discretisation naturally satisfies all the properties while the LBE due to explicit discretisation requires more restrictive condition on relaxation factor compared to the usual condition obtained from Chapman-Enskog expansion. We also derive the macroscopic finite difference form of the LBEs, and utilise it to establish the consistency of LBEs with the hyperbolic system. Further, we extend this LBM framework to hyperbolic conservation laws with source terms, such that there is no spurious numerical convection due to imbalance between convection and source terms. We also present a D2Q9 model that allows upwinding even along diagonal directions in addition to the usual upwinding along coordinate directions. The different aspects of the results are validated numerically on standard benchmark problems.

Quantum algorithms for nonlinear partial differential equations

Computations of nonlinear partial differential equations (PDEs) are some of the most important problems in science and engineering, but come with tremendous costs. Quantum algorithms have the potential to speed up these computations, but current methods are bound to weak or polynomial nonlinearity, short time of validity, or display no quantum advantage otherwise. We construct new quantum algorithms–based on level set methods – for Hamilton-Jacobi and scalar hyperbolic PDEs that generate algorithms with advantages with respect to critical numerical parameters, even for computing the physical observables. These algorithms hold for arbitrary nonlinearity for the equations under study and are valid globally in time. In addition, we also introduce a new quantum encoding, called level set encoding, that allows the efficient computation of physical observables. These PDEs are important for many applications like geometric optics, semi-classical limit of the Schrödinger equations, and high-frequency limit of symmetric linear hyperbolic systems (for example the elastic wave and Maxwell equations). It can display up to exponential advantage –depending on the initial conditions and domain of the relevant observables– in both the dimension of the PDE and the error in computing its observables with respect to classical finite difference methods for these PDEs. Compared to classical finite difference methods, our algorithm also has an advantage with respect to the number of initial data, which has potential important applications in Monte Carlo simulations and uncertainty quantification. For more general PDEs, while an advantage with respect to a very large number of initial data is possible, this is mostly unrealistic for problems of practical interest, and new methods are required.

An Experimental Study Focusing on the Filling Process and Consolidation Characteristics of Geotextile Tubes Filled with Fine-Grained Tungsten Tailings

With advancements in mineral processing technology, the disposal of fine-grained tailings has increasingly become a significant challenge. The geotextile tube method, characterized by its use of a permeable fabric and its cost-effectiveness, has gradually been applied in dam construction and other engineering projects involving tailings. This method offers a novel approach to addressing the storage issues of fine-grained tailings and promotes sustainable utilization. In this paper, the fine tailings that remained after the cyclone classification of Ganzhou tungsten ore were taken as the research object. Specifically, this research endeavored to evaluate the effects of various filling heights and concentrations on the geotextile tube-filling and consolidation process. The results revealed that the filling concentration had a significant impact on the filling benefit of the geotextile tubes, while the filling height had a minimal effect. During the consolidation drainage stage, the dry density, internal friction angle, cohesion, and compression modulus of the tailings in the bags increased with an increasing consolidation time and filling concentration. However, the physical and mechanical properties of the tailings in the geotextile tubes decreased with an increased filling height. Ultimately, this research developed a hyperbolic equation that makes it possible to forecast the ultimate settlement value at various filling heights and concentrations, better representing how the settlement of geotextile tubes changes over the consolidation time.

Global solvability of a mixed problem for a singular semilinear hyperbolic 1d system

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 $\Pi=\{(x,t)\colon 0<x<l<\infty,\ 0<t<T<\infty\}$ for the first order hyperbolic system of semi-linear equations of the form $$ \dfrac{\partial u}{\partial t}+\Lambda(x,t) \dfrac{\partial u}{\partial x}=f(x,t,u,v,w), \dfrac{\partial v}{\partial x}=g(x,t,u,v,w), \dfrac{\partial w}{\partial t }=h(x,t,u,v,w), $$ for a singular with orthogonal (degenerate) and non-orthogonal to the coordinate axes characte\-ristics and with nonlinear boundary conditions, where $\Lambda(x,t)=\mathrm{diag}(\lambda_1(x,t),\ldots,\lambda_k(x,t)),$ $u=(u_1,\ldots,u_k),$ $v=(v_1,\ldots,v_m),$ $w=(w_1,\ldots,w_n),$ $f=(f_1,\ldots,f_k),$ $g=(g_1,\ldots,g_m),$ $h=(h_1,\ldots,h_n)$ and besides $\textrm{sign } \lambda_i(0,t)=\mathrm{const}\neq 0$, $\textrm{sign } \lambda_i(l,t)=\mathrm{const}\neq 0 $ $\text{for all} \ t \in [0, T] $ and for all $i \in \{1,\ldots,k\}$. 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.

On Hyperbolic Equations with a Translation Operator in Lowest Derivatives

In the half-plane, a solution to a two-dimensional hyperbolic equation with a translation operator in the lowest derivative with respect to a spatial variable varying along the entire real axis is constructed in an explicit form. It is proven that the solutions obtained are classical if the real part of the symbol of a differential-difference operator in the equation is positive.

Adaptive observer design for coupled ODE–hyperbolic PDE systems with application to traffic flow estimation

This paper studies the state estimation problem for a class of coupled linear ODE–hyperbolic PDE systems with unknown in-domain parameters. Based on the swapping transformation and the least squares parameter estimation method, a Luenberger-type adaptive boundary observer is designed to estimate the ODE–PDE states under the unknown parameters. The exponential convergence conditions w.r.t. the observer feedback gains are given by employing the Lyapunov technique. Finally, an application to traffic state estimation of the Aw–Rascle–Zhang (ARZ) model involving the unknown relaxation time and the boundary input speed deviation is provided to validate the theoretical result.

Fractional controllability of semilinear hyperbolic systems

Fractional controllability of semilinear hyperbolic systems

The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy

Abstract We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time T of the solution is independent of the spatial variable.

Feedback boundary control of multi-dimensional hyperbolic systems with relaxation

This paper is concerned with boundary stabilization of multi-dimensional hyperbolic systems of partial differential equations. By adapting the Lyapunov function previously proposed by the second author for linearized hyperbolic systems with relaxation structure, we derive certain control laws so that the corresponding solutions decay exponentially in time. The result is illustrated with an application to water flows in open channels. The effectiveness of the derived control laws is confirmed by numerical simulations.

Approximate solutions to the nonlinear hyperbolic population balance equation: convergence, error estimate and numerical simulations

Approximate solutions to the nonlinear hyperbolic population balance equation: convergence, error estimate and numerical simulations

Homotopy Analysis Transform Method for a Singular Nonlinear Second-Order Hyperbolic Pseudo-Differential Equation

In this study, we employed the homotopy analysis transform method (HATM) to derive an iterative scheme to numerically solve a singular second-order hyperbolic pseudo-differential equation. We evaluated the effectiveness of the derived scheme in solving both linear and nonlinear equations of similar nature through a series of illustrative examples. The stability of this scheme in handling the approximate solutions of these examples was studied graphically and numerically. A comparative analysis with existing methodologies from the literature was conducted to assess the performance of the proposed approach. Our findings demonstrate that the HATM-based method offers notable efficiency, accuracy, and ease of implementation when compared to the alternative technique considered in this study.

Stability of hybrid time integration scheme for Lord–Shulman thermopiezoelectricity

Based on the existing initial boundary value and variational problems of Lord–Shulman thermopiezoelectricity a transient analysis of the behavior of piezoelectric materials is performed numerically. For space discretization of the variational problem the finite element method is used and for time discretization a hybrid time integration scheme is constructed on the basis of Newmark scheme for hyperbolic equation and generalized trapezoidal rule for parabolic equations. The unconditional stability of the developed time integration scheme is proved for some specific values of the scheme parameters by making use of energy balance law for the obtained time-discretized variational problem. Finally, the applicability of the constructed numerical scheme is demonstrated by the results of the numerical experiment which are compared with the ones available in literature.

Vieta–Lucas matrix approach for the numeric estimation of hyperbolic partial differential equations

This study is focused on the hyperbolic partial differential equations (HPDEs) to obtain their numerical solution. Here, the Vieta–Lucas Tau operational approach (VLTM) is proposed and utilized to solve the initial value problems related to HPDEs. In this approach, shifted Vieta–Lucas polynomials (SVLPs) are employed as the basis functions to assume the general solution. The integral and derivative operational matrices for SVLPs are built over a generalized domain. These operational matrices are used to convert this problem into a collection of algebraic equations. The examination of the convergence errors is well covered. The projected accuracy and efficacy of the method are confirmed when several cases are taken into account. Excellent agreement between exact and estimated findings is obtained, and in certain situations, estimated values are remarkably accurate. Finally, we can state that the VLTM method is dependable, efficient, precise, and easy to use.

A novel approach to the characteristic splitting scheme for mildly compressible flows based on the weighted averaged flux method

In the present work we improve the characteristic splitting approach for computing mildly compressible flows. This splitting approach is consisted of two separate steps namely an advective step and a purely hyperbolic, acoustic step. Typically, a pressure correction method is used for solving the acoustic step which may lead to overly dispersive, incorrect results. As a remedy to this issue, a novel solution approach on the basis of the weighted averaged flux method (WAF) is proposed, which solves the non-conservative, hyperbolic equations of the acoustic step directly. The stability and accuracy of this novel WAF method is demonstrated and an implicit, monotone formulation for three-dimensional unstructured grids is given. The simulation of the non-reacting flow in a swirl burner proves the applicability of the method.

On Global Solutions of Two-Dimensional Hyperbolic Equations with General-Kind Nonlocal Potentials

In the case of one spatial independent variable, we study hyperbolic differential-difference equations with potentials represented as linear combinations of translations of the desired function along the spatial variable. The qualitative novelty of this investigation is that, unlike previous research, it is not assumed that the real part of the symbol of the differential-difference operator contained in the equation has a constant sign. Previously, it was possible to remove that substantial restriction (i.e., the specified sign constancy) only for the case where the nonlocal term (i.e., the translated potential) is unique. In the present paper, we consider the case of the general-kind one-variable nonlocal potential, i.e., equations with an arbitrary amount of translated terms. No commensurability assumptions are imposed on the translation lengths. The following results are presented: We find a condition relating the coefficients at the nonlocal terms of the investigated equation and the length of the translations, providing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation.