Nonlinear PDE Model Research Articles

Numerical computation of Cross nanofluid model using neural network and Adaptive Neuro-Fuzzy Inference system with statistical insights for enhanced flow optimization

In this study, we present a novel integration of numerical methodologies and advanced computational intelligence to elucidate the dynamics of cross nanofluid flow over a Riga plate through a Darcy-Forchheimer porous medium. Brownian motion and thermophoretic phenomena are considered, along with the impacts of activation energy. Slip velocities, convective, and zero-flux boundary conditions are taken into account in the 3D cross nanofluid flow model in the presence of gyrotactic microorganisms. The non-linear PDE model is transformed into a highly non-linear ODE system using von Kármán similarity variables. Taking advantage of Python-derived numerical data as a foundational dataset from the system of non-linear ODEs, we employ neural network algorithms to refine and predict flow behaviors under varying conditions. The research progresses by contrasting these predictions with empirical observations, providing a rigorous validation framework. Furthermore, we incorporate the Adaptive Neuro-Fuzzy Inference System (ANFIS) alongside statistical analyses to examine the impacts of physical parameters, offering unparalleled insight into nanofluid mechanics. This multifaceted approach not only bridges theoretical and practical aspects of fluid dynamics but also proposes a robust model for predicting nanofluid behavior, poised to catalyze advancements in thermal engineering and nanotechnology applications. The precision and adaptability of our methodology underscore its potential as a cornerstone in future fluid dynamics research, inviting scrutiny and discussion from esteemed peers in the field. We have validated our model by finding various sets of error estimations. Furthermore, the velocity profile increases with the enhancement of the Hartmann number or magnetic parameter while decreasing with higher values of the Weissenberg number. The temperature profile decreases with increasing estimates of thermal stratification and the Biot number. The concentration profile amplifies with the Brownian motion parameter while decreasing against the thermophoresis parameter.

Effect of Al2O3-Cu/Water Hybrid Nanofluid on Natural Convection in an Inverted T-Shaped Enclosure

In the present study, the effect of Al2O3-Cu/water hybrid nanofluid (HNF) flow, governed by modified Navier-Stokes equations coupled with energy conservation on heat and mass transfer due to the natural convection in an inverted T shape domain is investigated numerically by solving coupled non-linear PDE model using finite element method. The effect of the governing parameters like source length, Rayleigh number (Ra), 1 ≤ Ra ≤ 104, breadth of the vertical tube, volume fraction of nanoparticle (Φ), 0 ≤ Φ ≤ 0.01, 1 ≤ Pr ≤ 50, and aspect ratio (AR), 0.5 ≤ AR ≤ 2, etc. on Nusselt number has been analyzed and presented through tabulated values and Matlab plotting. Also, the flow and heat transfer sensitivity to the the governing parameters are depicted through the streamline and isotherm contours.

Energy stable and structure-preserving schemes for the stochastic Galerkin shallow water equations

The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated nonlinear PDE models presents theoretical and numerical challenges that motivate the central advances of this paper. Starting with a spatially one-dimensional hyperbolicity-preserving, positivity-preserving stochastic Galerkin formulation of the parametric/uncertain shallow water equations, we derive an entropy-entropy flux pair for the system. We exploit this entropy-entropy flux pair to construct structure-preserving second-order energy conservative, and first- and second-order energy stable finite volume schemes for the stochastic Galerkin shallow water system. The performance of the methods is illustrated on several numerical experiments.

Local existence of solutions to a nonlinear autonomous PDE model for population dynamics with nonlocal transport and competition

In this paper, we prove that a particular nondegenerate, nonlinear, autonomous parabolic partial differential equation with nonlocal mass transfer admits the local existence of classical solutions. The equation was developed to qualitatively describe temporal changes in population densities over space through accounting for location desirability and fast, long-range travel. Beginning with sufficiently regular initial conditions, through smoothing the PDE and employing energy arguments, we obtain a sequence of approximators converging to a classical solution.

Nonlinear PDE Models in Semi-relativistic Quantum Physics

Abstract We present the self-consistent Pauli equation, a semi-relativistic model for charged spin- 1 / 2 1/2 particles with self-interaction with the electromagnetic field. The Pauli equation arises as the O ⁢ ( 1 / c ) O(1/c) approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac–Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting, the correct self-consistent equation is the Schrödinger–Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schrödinger–Poisson equation also arises as the mean field limit of the 𝑁-body Schrödinger equation with Coulomb interaction. We propose that the Pauli–Poisson equation arises as the mean field limit N → ∞ N\to\infty of the linear 𝑁-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli–Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in 1 / c 1/c of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions & Paul and Markowich & Mauser, where we need methods like magnetic Lieb–Thirring estimates.

Optimal control for a nonlinear stochastic PDE model of cancer growth

ABSTRACT We study an optimal control problem for a stochastic model of tumour growth with drug application. This model consists of three stochastic hyperbolic equations describing the evolution of tumour cells. It also includes two stochastic parabolic equations describing the diffusions of nutrient and drug concentrations. Since all systems are subject to many uncertainties, we have added stochastic terms to the deterministic model to consider the random perturbations. Then, we have added control variables to the model according to the medical concepts to control the concentrations of drug and nutrient. In the optimal control problem, we have defined the stochastic and deterministic cost functions and we have proved the problems have unique optimal controls. For deriving the necessary conditions for optimal control variables, the stochastic adjoint equations are derived. We have proved the stochastic model of tumour growth and the stochastic adjoint equations have unique solutions. For proving the theoretical results, we have used a change of variable which changes the stochastic model and adjoint equations (a.s.) to deterministic equations. Then we have employed the techniques used for deterministic ones to prove the existence and uniqueness of optimal control.

Pricing renewable energy certificates with a Crank–Nicolson Lagrange–Galerkin numerical method

The valuation problem of renewable energy certificates can be formulated in terms of a nonlinear PDE model where the underlying stochastic factors are the accumulated green certificates sold by an authorized producer and the natural logarithm of the renewable generation rate. In the present paper, the nonlinear convective term is treated with the Bermúdez–Moreno duality method for maximal monotone operators as in Baamonde-Seoane et al. (2021). The main novelty of this article comes from the proposed techniques for the numerical solution of the resulting linear problem. In this case, we propose a Lagrange–Galerkin method which mainly consists of Crank–Nicolson characteristics for time discretization combined with finite elements for the discretization in the accumulated green certificates and the natural logarithm of the renewable generation rate directions. Finally, several numerical examples are presented to illustrate the good performance of the method and model, and its comparison with other numerical schemes employed to solve the same problem.

Global stability under dynamic boundary conditions of a nonlinear PDE model arising from reinforced random walks

This paper is devoted to the study of the global stability of classical solutions to an initial and boundary value problem (IBVP) of a nonlinear PDE system, converted from a model of reinforced random walks, subject to time-dependent boundary conditions. It is shown that under certain integrability conditions on the boundary data, classical solutions to the IBVP exist globally in time and the differences between the solutions and their corresponding ansatz, determined by the initial and boundary conditions, converge to zero, as time goes to infinity. Though the final states of the boundary functions are required to match, the boundary values do not necessarily equal to each other at any finite time. In addition, there is no smallness restriction on the magnitude of the initial perturbations. Furthermore, numerical simulations are performed to investigate the long-time dynamics of the IBVP when the final states of the boundary functions do not match or the boundary functions are periodic in time.

Convolutional Autoencoders and Clustering for Low-dimensional Parametrization of Incompressible Flows

The design of controllers for general nonlinear PDE models is a difficult task because of the high dimensionality of the partially discretized equations. It has been observed that the embedding of nonlinear systems into the class of linear parameter varying systems (LPV) gives way to apply linear theory and methods from numerical linear algebra for controller design. The feasibility of the LPV approach hinges on the dimension of the inherent parametrization. In this work we propose and evaluate combinations of convolutional neural networks and clustering algorithms for very low-dimensional parametrizations of incompressible Navier-Stokes equations.

Time-periodic steady-state solution of fluid-structure interaction and cardiac flow problems through multigrid-reduction-in-time

In this paper, a time-periodic MGRIT algorithm is proposed as a means to reduce the time-to-solution of numerical algorithms by exploiting the time periodicity inherent to many applications in science and engineering. The time-periodic MGRIT algorithm is applied to a variety of linear and nonlinear single- and multiphysics problems that are periodic-in-time. It is demonstrated that the proposed parallel-in-time algorithm can obtain the same time-periodic steady-state solution as sequential time-stepping. It is shown that the required number of MGRIT iterations can be estimated a priori and that the new MGRIT variant can significantly and consistently reduce the time-to-solution compared to sequential time-stepping, irrespective of the number of dimensions, linear or nonlinear PDE models, single-physics or coupled problems and the employed computing resources. The numerical experiments demonstrate that the time-periodic MGRIT algorithm enables a greater level of parallelism yielding faster turnaround, and thus, facilitating more complex and more realistic problems to be solved.

Numerical solution of a nonlinear PDE model for pricing Renewable Energy Certificates (RECs)

In this article we present a valuation method for Renewable Energy Certificates (RECs) or green certificates. For this purpose, we propose a non-linear PDE model with two stochastic factors: the accumulated green certificates sold by an authorized generator and the natural logarithm of the renewable electricity generation rate. One novelty of the work comes from the numerical treatment of the non-linear convective term in the PDE. Thus, we use the Bermdez-Moreno algorithm to deal with this non-linear term. This duality algorithm is based on the Yosida regularization of non-linear maximal monotone operators. In order to solve the obtained linearized problem, we use numerical methods based on semi-Lagrangian schemes in the direction without diffusion while we consider an implicit second order finite differences scheme in the direction with diffusion term. Finally, we show illustrative results of the performance of the proposed model and numerical methods that have been implemented.

On-line Identification of the Sine-Gordon Nonlinear PDE Model

Dynamic identifier is developed for a nonlinear sine-Gordon model. It is shown that the spatially-varying magnitude of the sine-wise external force comes identifiable provided that the model is persistently excited by a nonzero input. The present investigation focuses on the case where the full state information is available for the measurement. Capabilities of the theoretical results obtained are numerically illustrated in the case study of a spatially-invariant plant parameter.

A [formula omitted] PDE model for effective denoising

In this paper, we have proposed a higher order non-linear PDE model containing diffusion terms of TV−L2 and TV−H−1 model for effective image denoising. Fourier spectral method for space and convexity splitting method for time have been used to solve the proposed PDE model. Stability analysis for the time discretized equation has been derived. Numerical experiment has been done on some test images and the results have been compared with the results of some existing models.

Analysis and simulation of a PDE model for surface relaxation

We study a fourth-order, singular, nonlinear PDE model for surface relaxation. A weak solution for the model is defined using an inequality formulation. A numerical scheme based on semi-implicit time stepping, mixed finite elements and regularization is proposed to approximate the PDE model. We investigate the convergence of the scheme with respect to the discretization and the regularization parameters. Finally, formal arguments show that the model can be viewed as a gradient flow with respect to an appropriate Riemannian metric.

On the field-induced transport of magnetic nanoparticles in incompressible flow: Modeling and numerics

By methods from non-equilibrium thermodynamics, we derive a class of nonlinear pde-models to describe the motion of magnetizable nanoparticles suspended in incompressible carrier fluids under the influence of external magnetic fields. Our system of partial differential equations couples Navier–Stokes and magnetostatic equations to evolution equations for the magnetization field and the particle number density. In the second part of the paper, a fully discrete mixed finite-element scheme is introduced which is rigorously shown to be energy-stable. Finally, we present numerical simulations in the 2D-case which provide first information about the interaction of particle density, magnetization and magnetic field.

Modeling and mathematical analysis of an initial boundary value problem for hepatitis B virus infection

In this work, we investigate the hepatitis B virus infection. We first derive a nonlinear PDE model for the studied biological phenomenon. The obtained initial boundary value problem is completely analyzed. To begin with the analysis of the model, we use Lou and Zhao Lemma concerning globally attractive steady states to prove boundedness of potential solutions. Then we prove global existence, uniqueness and positivity of the solution by a variational method combined with semigroups theory and some other useful tools from functional analysis. Moreover, the basic reproduction number R0 determining the extinction or the persistence of the HBV infection is thoroughly computed via a new method based on spectral properties of differential operators, the residue Theorem and some arguments from numerical analysis. Also, the global asymptotical properties of the HBV-free equilibrium of the model are derived via a skillful construction of a suitable Lyapunov function. Local stability of the endemic equilibrium of the model is studied as well. Finally, numerical simulations are performed to support the theoretical results obtained.

Automatic Exploration Techniques of Numerical Bifurcation Diagrams Illustrated by the Ginzburg--Landau Equation

This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model that describes the states of a wide range of superconductors. For two-dimensional domains, a robust method is developed that performs a numerical continuation of the equations, automatically exploring the whole solution landscape. The strength of the applied magnetic field is used as the bifurcation parameter. Our branch switching algorithm is based on Lyapunov--Schmidt reduction, but we will show that for an important class of domains an alternative method based on the equivariant branching lemma can be applied as well. The complete algorithm has been implemented in Python and tested for multiple examples. For each example a complete solution landscape was constructed, showing the robustness of the algorithm.

Special Issue on Recent Trends in Stochastic Models, Statistics and Their Applications

This special issue was organized on the occasion of the 13th Workshop on Stochastic Models, Statistics and Their Applications (SMSA 2017) held at Humboldt University of Berlin on February 20-24, 2017. Key areas of the workshop focused on models and methods for problems arising in industry, finance, natural sciences, engineering, and technology. In these areas, solutions are needed, which are able to deal with complex and possibly high-dimensional data, make use of innovative methodologies, and have proven applicability to real-world data. In this special issue, we have 13 papers. The papers consider stochastic models for diverse data sets and streams such as bridge vehicle loads, watermark identification data, text documents, images of digitized handwritten digits, financial asset returns, option prices, and equity data. The methods and algorithms required to handle come from machine learning, stochastic processes, sequential (on-line) change-point detection, regression analysis, image analysis, and robust divergence measures. The work of Barbulescu analyzes the properties of Generalized Regression Neural Network (GRNN) model applied to the prediction of the selected Romanian stock market prices. In the work of Luo et al, a nonlinear PDE model to estimate the American call options and put options under financial crisis scenario is implemented with additional penalty introduced to Black-Scholes formula. The work of Liu et al proposes a compound Poisson bivariate process model of vehicle load with Levy copula function and validates it in a reliability analysis of bridges in China. Migalska proposes in her paper a density-free symmetry verification test for arbitrary symmetries in images. In their work, Darkhovsky et al propose a novel method based on the notion of ϵ-complexity of continuous vector functions for an off-line detection of multiple change-points in multidimensional time series. The work of Rafajlowicz et al proposes a method of spike detection and suppression (SDS method), which extends the jump detector proposed by the Authors in their earlier work and applies it for a laser power control in a 3D additive manufacturing process. Sliwinski et al propose two algorithms for nonlinearity recovery in Hammerstein systems, one based on orthogonal series expansions and the other using aggregation and validate them in numerical simulations. The work of Sousa et al provides an insight into the behavior of GARCH process describing financial markets affected by rumours and other random factors through an analysis of process mean and variance by estimating run length. Kisslinger et al describe a new toolkit based on scaled Bregman divergences for detecting distributional changes in random data, including streams and clouds. The work of Sarnowski et al derives an optimal decision function for a sudden change detection in political business cycle model based on the Markovian process. The work of Walkowiak et al proposes a two-step procedure for open-set classification of text documents and validates it on simulated data and on the collection of Wikipedia documents. The work by Xie et al combines two-tier stochastic frontier model with the traditional Richardson Model and applies it to the Chinese stock market analysis. Finally, the work of Yuan et al derives the Hamilton-Jacobi-Bellman equation for an optimal consumption problem of maximizing the expected discounted value of utility in an economic growth model with random technological shocks. We thank numerous anonymous reviewers for helping in evaluating all submitted manuscripts.

Dynamics from a mathematical model of a two-state gas laser

Motivated by recent work in the area, we consider the behavior of solutions to a nonlinear PDE model of a two-state gas laser. We first review the derivation of the two-state gas laser model, before deriving a non-dimensional model given in terms of coupled nonlinear partial differential equations. We then classify the steady states of this system, in order to determine the possible long-time asymptotic solutions to this model, as well as corresponding stability results, showing that the only uniform steady state (the zero motion state) is unstable, while a linear profile in space is stable. We then provide numerical simulations for the full unsteady model. We show for a wide variety of initial conditions that the solutions tend toward the stable linear steady state profiles. We also consider traveling wave solutions, and determine the unique wave speed (in terms of the other model parameters) which allows wave-like solutions to exist. Despite some similarities between the model and the inviscid Burger’s equation, the solutions we obtain are much more regular than the solutions to the inviscid Burger’s equation, with no evidence of shock formation or loss of regularity.

On-line identification of the heat Peclet number in a gasification reactor

Motivated by the need of developing systematic safe operation as well as monitoring and control schemes, in this study the problem of on-line identifying the heat Peclet number of a bistable tubular gasification reactor is addressed. The Pe number, the ratio of dispersive-radiative to convective heat transport characteristic times, is a key parameter in the modeling of the transient and stationary complex reactor behavior. First, the nonlinear PDE model is discretized with finite differences, yielding a low-order model of global-nonlinear dynamics. Then, the combination of identification ideas from systems theory and tracer technology yields that the Pe number can be on-line identified from the kurtosis and skewness of the effluent temperature response to an induced feed temperature rectangular pulse. The proposed approach is applied to a reactor PDE model calibrated with experimental data, finding that, with respect to a previous study with response variance: (i) kurtosis halves the estimate uncertainty, and (ii) the amplitude of the temperature pulse is halved.