Nonlinear Hyperbolic Partial Differential Equations Research Articles

Periodic Optimal Control of a Plug Flow Reactor Model with an Isoperimetric Constraint

AbstractWe study a class of nonlinear hyperbolic partial differential equations with boundary control. This class describes chemical reactions of the type “$$A \rightarrow $$ 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.

Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems.

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.

Coupling of sediment transport phenomena with turbulent surface flows: mathematical modeling, finite volume approximation and test simulations

Several sediment transport models coupled with shallow water equations have been developed in the literature over the past few decades. The water flow in the presence of abrupt bottom slopes distorts and cause turbulence. This phenomenon is well-known in the sediment transport theory. Sediment transport models based on shallow water equations do not include turbulence effects or the phase-lag between water velocity and bottom velocity. In this study, we consider a one-dimensional free surface turbulent water flow with sediment and morphodynamic. We propose a new one-dimensional acceleration-dissipation mathematical sediment transport model that considers the distortion of horizontal profile velocity in transport. The model is an improved version of existing averaged sediment transport models. In the new model, there is a distinction between fluid velocity and bottom velocity through a new bedload equation. The mathematical model is a system of nonlinear hyperbolic partial differential equations that are approximated by a stable and robust finite volume scheme presented in the path-conservative framework. Numerical tests are performed to demonstrate the efficiency of the developed theory.

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Modeling and Quantifying Parameter Uncertainty of Co‐Seismic Non‐Classical Nonlinearity in Rocks

AbstractDynamic perturbations reveal unconventional nonlinear behavior in rocks, as evidenced by field and laboratory studies. During the passage of seismic waves, rocks exhibit a decrease in elastic moduli, slowly recovering after. Yet, comprehensive physical models describing these moduli alterations remain sparse and insufficiently validated against observations. Here, we demonstrate the applicability of two physical damage models—the internal variable model (IVM) and the continuum damage model (CDM)—to provide quantitative descriptions of nonlinear co‐seismic elastic wave propagation observations. While the IVM uses one internal variable to describe the evolution of elastic material moduli, the CDM damage variable is a mathematical representation of microscopic defects. We recast the IVM and CDM models as nonlinear hyperbolic partial differential equations and implement 1D and 2D numerical simulations using an arbitrary high‐order discontinuous Galerkin method. We verify the modeling results with co‐propagating acousto‐elastic experimental measurements. Subsequently, we infer the parameters for these nonlinear models from laboratory experiments using probabilistic Bayesian inversion and 2D simulations. By adopting the Adaptive Metropolis Markov chain Monte Carlo method, we quantify the uncertainties of inferred parameters for both physical models, investigating their interplay in 70,000 simulations. We find that the damage variables can trade off with the stress‐strain nonlinearity in discernible ways. We discuss physical interpretations of both damage models and that our CDM quantitatively captures an observed damage increase with perturbation frequency. Our results contribute to a more holistic understanding of co‐seismic damage and post‐seismic recovery after earthquakes bridging the worlds of theoretical analysis and laboratory findings.

Tracking model predictive control and moving horizon estimation design of distributed parameter pipeline systems

This manuscript proposes moving horizon control and state/parameter estimation designs for pipeline networks modelled by partial differential equations (PDEs) with boundary actuation. The spatial–temporal pressure and velocity dynamics within the pipelines are described by a system of six coupled one-dimensional first-order nonlinear hyperbolic PDEs. To address the discrete-time modelling challenge and preserve the infinite-dimensional nature of the pipeline system, the Cayley–Tustin transformation is deployed for model time discretization without any spatial discretization or model reduction. Considering the lack of full state information across the entire pipeline manifold, unknown states and uncertain parameters are estimated using moving horizon estimation (MHE). Based on the estimated states and parameters, a tracking model predictive control (MPC) strategy for the discrete-time infinite-dimensional pipeline system is proposed, which enables specific operation while ensuring physical constraint satisfaction. The effectiveness of the proposed controller and estimator designs is demonstrated via numerical examples.

The Homotopy Perturbation Method for Solving Nonlocal Initial-Boundary Value Problems for Parabolic and Hyperbolic Partial Differential Equations

To obtain approximate-exact solutions to nonlocal initial-boundary value problems (IBVPs) of linear and nonlinear parabolic and hyperbolic partial differential equations (PDEs) subject to initial and nonlocal boundary conditions of integral type, the homotopy perturbation method (HPM) is utilized in this study. The HPM is used to solve the specified nonlocal IBVPs, which are then transformed into local Dirichlet IBVPs. Some examples demonstrate how accurate and efficient the HPM.

Physics-informed neural networks with adaptive localized artificial viscosity

Physics-informed Neural Network (PINN) is a promising tool that has been applied in a variety of physical phenomena described by partial differential equations (PDE). However, it has been observed that PINNs are difficult to train in certain “stiff” problems, which include various nonlinear hyperbolic PDEs that display shocks in their solutions. Recent studies added a diffusion term to the PDE, and an artificial viscosity (AV) value was manually tuned to allow PINNs to solve these problems. In this paper, we propose three approaches to address this problem, none of which rely on an a priori definition of the artificial viscosity value. The first method learns a global AV value, whereas the other two learn localized AV values around the shocks, by means of a parametrized AV map or a residual-based AV map. We applied the proposed methods to the inviscid Burgers equation and the Buckley-Leverett equation, the latter being a classical problem in Petroleum Engineering. The results show that the proposed methods are able to learn both a small AV value and the accurate shock location and improve the approximation error over a nonadaptive global AV alternative method.

A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation

AbstractIn this research work, we proposed a Haar wavelet collocation method (HWCM) for the numerical solution of first- and second-order nonlinear hyperbolic equations. The time derivative in the governing equations is approximated by a finite difference. The nonlinear hyperbolic equation is converted into its full algebraic form once the space derivatives are replaced by the finite Haar series. Convergence analysis is performed both in space and time, where the computational results follow the theoretical statements of convergence. Many test problems with different nonlinear terms are presented to verify the accuracy, capability, and convergence of the proposed method for the first- and second-order nonlinear hyperbolic equations.

Tracking Distributed Parameters System Dynamics with Recursive Dynamic Mode Decomposition with Control

The goal of this work is to introduce a new online data driven modeling solution for tracking the dynamics of systems, the behavior of which is governed by (nonlinear) partial differential equations (PDEs). In order to reach this goal, this paper introduces a recursive algorithm for black box time varying model identification along with an online algorithm for reducing the order of the identified model. The resulting system model is a linear time varying low order state space representation usable, e.g., for controller design or prediction. The model reduction step relies on the recursive determination of the orthonormal basis of a specific data based proper orthogonal decomposition. The system identification procedure is initialized with the dynamic mode decomposition with control and updated recursively via the introduction of a specific recursive least squares algorithm. In both steps, a forgetting factor is used to focus on the most recent variations of the system. As case studies, a linear heat diffusion equation with a time varying diffusion parameter is first considered. The Burgers equation, a nonlinear hyperbolic PDE, is then utilized to assess the introduced methods applicability with nonlinear systems. Identification results based on simulation show the high accuracy of this new approach.

CNN-Based Temporal Video Segmentation Using a Nonlinear Hyperbolic PDE-Based Multi-Scale Analysis

An automatic temporal video segmentation framework is introduced in this article. The proposed cut detection technique performs a high-level feature extraction on the video frames, by applying a multi-scale image analysis approach combining nonlinear partial differential equations (PDE) to convolutional neural networks (CNN). A nonlinear second-order hyperbolic PDE model is proposed and its well-posedness is then investigated rigorously here. Its weak and unique solution is determined numerically applying a finite difference method-based numerical approximation algorithm that quickly converges to it. A scale-space representation is then created using that iterative discretization scheme. A CNN-based feature extraction is performed at each scale and the feature vectors obtained at multiple scales are concatenated into a final frame descriptor. The feature vector distance values between any two successive frames are then determined and the video transitions are identified next, by applying an automatic clustering scheme on these values. The proposed PDE model, its mathematical investigation and discretization, and the multi-scale analysis based on it represent the major contributions of this work. Some temporal segmentation experiments and method comparisons that illustrate the effectiveness of the proposed framework are finally described in this research paper.

Solutions to nonlinear pseudo hyperbolic partial differential equations with nonlocal conditions by using residual power series method

Solutions to nonlinear pseudo hyperbolic partial differential equations with nonlocal conditions by using residual power series method

Second-order flows for computing the ground states of rotating Bose-Einstein condensates

Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modeled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. In comparison with the gradient flow type approaches: When explicit temporal discretization strategies are adopted, the proposed methods allow for larger stable time step sizes; While for semi-implicit discretization, using the same step size, a much smaller number of iterations are needed for the proposed methods to reach the stopping criterion, and every time step encounters almost the same computational complexity. Rich and detailed numerical examples are documented for verification and comparison.

Nonlinear fractional diffusion model for space-time neutron dynamics

In this work, a new fractional multidimensional neutron diffusion model is introduced using Cattaneo equation instead of the commonly used Fick's law for two neutron energy groups. The model represents a coupled system of nonlinear hyperbolic fractional partial differential equations considering finite fast and finite thermal neutron velocities. The model provides new quantities that relate some of the most important nuclear parameters together producing some newly physical concepts. These quantities include fast and thermal neutron relaxation times, Riemann-Liouville fractional integrals of the neutron densities and derivatives of macroscopic absorption cross sections. This model can be useful for mathematicians and physicists since the anomalous subdiffusion process describes the neutron movement closer to reality. The effect of the anomalous subdiffusion exponent (0<γ≤1) on the neutron flux is discussed. The model is successfully applied to simple homogenous and heterogonous reactor cores with step, ramp and sinusoidal perturbations for different values ofγ to estimate the average reactor powers in the fuel and moderator regions. Also, the new fractional model is applied to LRA BWR benchmark problem including adiabatic heating and Doppler feedback. On the other hand and based on the concept of prompt jump, the proposed fractional model is simplified and then analytically solved using Mittag-Liffler function. Solving fractional models numerically consumes considerable computational time costs due to the persistent memory of fractional operators, especially for smaller values of γ. For this regard, adaptively multi-step differential transform MDTM method is implemented to solve the proposed fractional model. The MDTM is developed using step size control technique for which the length of the step integration varies according to the solution behaviour.

Small Collaboration: Modeling Phenomena from Nature by Hyperbolic Partial Differential Equations

Nonlinear hyperbolic partial differential equations constitute a plethora of models from physics, biology, engineering, etc. In this workshop we cover the range from modeling, mathematical questions of well-posedness, numerical discretization and numerical simulations to compare with the phenomenon from nature that was modeled in the first place. Both kinetic and fluid models were discussed.

Numerical methods for the hyperbolic Monge-Ampère equation based on the method of characteristics

We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation (PDE) in two independent variables. The MOC gives rise to two mutually coupled systems of ordinary differential equations (ODEs). As a special case we consider the Monge–Ampère (MA) equation, for which we present a general method of determining the location and number of required boundary conditions. We solve the systems of ODEs using explicit one-step methods (Euler, Runge-Kutta) and spline interpolation. Reformulation of the Monge–Ampère equation as an integral equation yields via its residual a proxy for the error of the numerical solution. Numerical examples demonstrate the performance and convergence of the methods.

A 1D–0D–3D coupled model for simulating blood flow and transport processes in breast tissue

In this work, we present mixed dimensional models for simulating blood flow and transport processes in breast tissue and the vascular tree supplying it. These processes are considered, to start from the aortic inlet to the capillaries and tissue of the breast. Large variations in biophysical properties and flow conditions exist in this system necessitating the use of different flow models for different geometries and flow regimes. In total, we consider four different model types. First, a system of 1D nonlinear hyperbolic partial differential equations (PDEs) is considered to simulate blood flow in larger arteries with highly elastic vessel walls. Second, we assign 1D linearized hyperbolic PDEs to model the smaller arteries with stiffer vessel walls. The third model type consists of ODE systems (0D models). It is used to model the arterioles and peripheral circulation. Finally, homogenized 3D porous media models are considered to simulate flow and transport in capillaries and tissue within the breast volume. Sink terms are used to account for the influence of the venous and lymphatic systems. Combining the four model types, we obtain two different 1D-0D-3D coupled models for simulating blood flow and transport processes: The first model results in a fully coupled 1D-0D-3D model covering the complete path from the aorta to the breast combining a generic arterial network with a patient specific breast network and geometry. The second model is a reduced one based on the separation of the generic and patient specific parts. The information from a calibrated fully coupled model is used as inflow condition for the patient specific sub-model allowing a significant computational cost reduction. Several numerical experiments are conducted to calibrate the generic model parameters and to demonstrate realistic flow simulations compared to existing data on blood flow in the human breast and vascular system. Moreover, we use two different breast vasculature and tissue data sets to illustrate the robustness of our reduced sub-model approach.

Error estimates for DeepONets: a deep learning framework in infinite dimensions

Abstract DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite-dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepONets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ordinary differential equation, an elliptic partial differential equation (PDE) with variable coefficients and nonlinear parabolic and hyperbolic PDEs. While the approximation of arbitrary Lipschitz operators by DeepONets to accuracy $\epsilon $ is argued to suffer from a ‘curse of dimensionality’ (requiring a neural networks of exponential size in $1/\epsilon $), in contrast, for all the above concrete examples of interest, we rigorously prove that DeepONets can break this curse of dimensionality (achieving accuracy $\epsilon $ with neural networks of size that can grow algebraically in $1/\epsilon $).Thus, we demonstrate the efficient approximation of a potentially large class of operators with this machine learning framework.

Solving the eikonal equation for compressional and shear waves in anisotropic media using peridynamic differential operator

SUMMARY Traveltimes of compressional (P) and shear (S) waves have proven essential in many earthquake and exploration seismology applications. An accurate and efficient traveltime computation for P and S waves is crucial for the success of these applications. However, solving the eikonal equation with a complex phase velocity field in anisotropic media is challenging. The eikonal equation is a first-order nonlinear hyperbolic partial differential equation. It represents the high-frequency asymptotic approximation of the wave equation. The fast marching and sweeping methods are commonly used due to their efficiency in numerically solving the eikonal equation. However, these methods suffer from numerical inaccuracy in anisotropic media with sharp heterogeneity, irregular surface topography and complex phase velocity fields. This study presents a new method for the solution of the eikonal equation by employing the peridynamic differential operator (PDDO). The PDDO provides the non-local form of the eikonal equation by introducing an internal length parameter (horizon) and a weight function with directional non-locality. The operator is immune to discontinuities in the form of sharp changes in field or model variables and invokes the direction of traveltime in a consistent manner. The weight function controls the strength of association among points within the horizon. Solutions are constructed in a consistent manner without upwind assumptions through simple discretization. The robustness of this approach is established by considering different types of eikonal equations on complex velocity models in anisotropic media. The examples demonstrate its unconditional numerical stability and results compare well with the reference solutions.

On the solution of hyperbolic equations using the peridynamic differential operator

Numerical solution of hyperbolic differential equations, such as the advection equation, poses challenges. Classically, this issue has been addressed by using a scheme known as the upwind scheme. It simply invokes more points from the upwind side of the flow stream when calculating derivatives. This study presents a generalized upwind scheme, referred to as directional nonlocality, for the numerical solution of linear and nonlinear hyperbolic Partial Differential Equations (PDEs) using the peridynamic differential operator (PDDO). The PDDO provides the nonlocal form of the differential equations by introducing an internal length parameter (horizon) and a weight function. The weight function controls the degree of interaction among the points within the horizon. A modification to the weight function, i.e., upwinded-weight function, accounts for directional nonlocality along which information travels. This modification results in a stable PDDO discretization of hyperbolic PDEs. Solutions are constructed in a consistent manner without special treatments through simple discretization. The capability of this approach is demonstrated by considering time dependent linear and nonlinear hyperbolic equations as well as the time invariant Eikonal equation. Numerical stability is ensured for the linear advection equation and the PD solutions compare well with the analytical/reference solutions.