Parabolic Differential Equations Research Articles

Spectral-based optimal output-feedback boundary control of a cracking catalytic reactor PDE model

This research paper focuses on the design of a boundary optimal controller with output feedback for a tubular catalytic cracking reactor. The process is represented by a set of non-linear parabolic partial differential equations (PDEs). The approach involves utilising the linearised infinite-dimensional version of the model, taking advantage of the system generator being a Riesz spectral operator. A stabilising compensator is designed by leveraging the eigenvalues and eigenvectors of the system generator. This paper also includes numerical simulations to demonstrate the performance of the developed algorithm.

A robust collocation method for singularly perturbed discontinuous coefficients parabolic differential difference equations

A robust collocation method for singularly perturbed discontinuous coefficients parabolic differential difference equations

Parabolic PDEs on low-dimensional structures

The paper concerns the theory of parabolic equations on a broad class of closed subsets of Euclidean space possessing a kind of tangent structure. A necessary framework for considering evolutionary problems is developed, and fundamental results about the existence and uniqueness of solutions are established. The presented approach is based on applying the semigroup theory of linear operators combined with special geometric constructions related to the shape of the examined structure. The proposed setting is consistent with the theory of elliptic equations on lower dimensional structures and with the second-order calculus of variations developed by Bouchitté and Fragalà in [5].

Determining a parabolic system by boundary observation of its non-negative solutions with biological applications

In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in partial differential equation (PDE) analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities.

Exponentially Fitted Robust Scheme for the Solution of Singularly Perturbed Delay Parabolic Differential Equations with Integral Boundary Condition

Exponentially Fitted Robust Scheme for the Solution of Singularly Perturbed Delay Parabolic Differential Equations with Integral Boundary Condition

Dynamic Event-triggered Fuzzy Filtering for Semi-linear Parabolic PDE Systems: A Reduced-order Approach

Dynamic Event-triggered Fuzzy Filtering for Semi-linear Parabolic PDE Systems: A Reduced-order Approach

Locally risk minimizing pricing of Asian option in a semi-Markov modulated market

We consider a regime-switching model where the stock volatility dynamics is a semi-Markov process. Under this model assumption, we find the locally risk-minimizing price of some Asian options with European-style exercise. The price function is shown to satisfy a non-local degenerate system of parabolic PDEs in dimension two with a terminal condition. We show this by deriving the F-S decomposition of the discounted contingent claim. The related Cauchy problem involving the PDE is shown to be equivalent to an integral equation (IE). The existence and uniqueness of the classical solution to the PDE are determined by studying the IE and using semigroup theory. To be more precise, we first obtain the mild solution of the PDE and then we show that the mild solution has sufficient regularity. The locally risk-minimizing hedging for the option has also been identified in this work. Finally, the computational aspects of Asian option prices have been discussed by solving the equation numerically.

A modified domain decomposition spectral collocation method for parabolic partial differential equations

<p>In this paper, utilizing Legendre polynomials as the basis functions in both space and time, we present a modified domain decomposition spectral method for 2-dimensional parabolic partial differential equations. For solving the obtained linear/nonlinear algebraic equations, a dimension expanding preconditioner is applied employing the obtained saddle construction of the coefficient matrix. Numerical examples are given to show the performance of the presented method and the efficiency of the preconditioner.</p>

Dynamic dissipative control for fuzzy distributed parameter cyber physical system under input quantization and DoS attack.

This article explores the dissipative control for a class of nonlinear DP-CPS (distributed parameter cyber physical system) within a finite-time interval. By utilizing a Takagi-Sugeno (T-S) fuzzy model to represent the system's nonlinear aspects, the studied system is formulated as a class of fuzzy parabolic partial differential equation (PDE). In order to optimize network resources, both the system state and input signal are subjected to quantization using dynamic quantizers. Subsequently, a dynamic state control strategy is proposed, taking into account potential DoS attack. The finite-time boundedness of the fuzzy parabolic PDE is analyzed, with respect to the influence of quantization, through the construction of an appropriate Lyapunov functional. The article then presents the conditions for finite-time dissipative control design, alongside the adjustment parameters for the dynamic quantizers within the fuzzy closed-loop system. Furthermore, the decoupling of interlinked nonlinear terms in the control design conditions is achieved by using an arbitrary matrix. Finally, an example is provided and the simulation results indicate the effectiveness of the dissipative control method proposed.

Adaptive Neural Consensus Observer Networks Design for a Class of Semilinear Parabolic PDE Systems.

This article concerns the investigation on the consensus problem for the joint state-uncertainty estimation of a class of parabolic partial differential equation (PDE) systems with parametric and nonparametric uncertainties. We propose a two-layer network consisting of informed and uninformed boundary observers where novel adaptation laws are developed for the identification of uncertainties. Particularly, all observer agents in the network transmit their information with each other across the entire network. The proposed adaptation laws include a penalty term of the mismatch between the parameter estimates generated by the other observer agents. Moreover, for the nonparametric uncertainties, radial basis function (RBF) neural networks are employed for the universal approximation of unknown nonlinear functions. Given the persistently exciting condition, it is shown that the proposed network of adaptive observers can achieve exponential joint state-uncertainty estimation in the presence of parametric uncertainties and ultimate bounded estimation in the presence of nonparametric uncertainties based on the Lyapunov stability theory. The effects of the proposed consensus method are demonstrated through a typical reaction-diffusion system example, which implies convincing numerical findings.

Single-Particle Model of Li-ion Battery – Model Calibration and Validation against Real Data in an Electric Vehicular Application

This paper investigates the problem of calibration and validation of a battery electrochemical model as a mandatory step towards accurate estimation of battery important variables, like state of charge (SoC) and state of health (SoH). Here, the Single Particle Model (SPM) is considered, which mathematically describes the battery internal governing phenomena by means of parabolic partial differential equations (PDEs), but whose parameters are notoriously difficult to measure or estimate. After suitable approximation of this model through a linear finite-dimensional model, a systematic procedure of SPM calibration is here proposed and validated against real data issued from battery cycling in an electric vehicular application, i.e., under standard driving cycle scenarios. In a novel approach of SoC estimation, the suitably calibrated SPM, together with measures of voltage and current, allow to analytically connect the internal spatially distributed ions’ concentrations to the equilibrium concentration, which, at its turn, is an image of battery SoC. Results suggest that SPM can reliably predict the battery internal ions’ concentrations and be further used for SoC accurate estimation.

LANDAU LEGENDRE WAVELET GALERKIN METHOD APPLIED TO STUDY TWO-PHASE MOVING BOUNDARY PROBLEM OF HEAT TRANSFER IN FINITE REGION

In this paper, we developed a mathematical model of solidification where specific heat and thermal conductivity are temperature dependent. This model is a two-phase moving boundary problem (MBP) of heat transfer in finite region and represents as MBP of system of parabolic nonlinear second-order partial differential equations (PDEs). We developed a Landau Legendre wavelet Galerkin method for finding the solution of the problem. The MBP of a system of PDEs is transformed into a variable boundary value problem of nonlinear ordinary defferential equations (ODEs) by the use of dimensionless variables and the Landau transform. The problem is converted into a system of algebraic equations with the application of Legendre wavelet Galerkin method. In particular case, we compared present solution with Laplace transform solution and found approximately the same. The whole investigation has been done in dimensionless form. When the specific heat and thermal conductivity exponentially vary in temperatures, it is the effect of dimensionless parameters: thermal diffusivity (α<sub>12</sub>), ratio of thermal conductivity (<i>k</i><sub>12</sub>), dimensionless temperature (θ<sub><i> f</i></sub>), Fourier number (F<sub>0</sub>), Stefan number (Ste), and ratio of densities (ρ<sub>1</sub> / ρ<sub>2</sub>) are discussed in detail.

Wiener Tauberian theorem and half-space problems for parabolic and elliptic equations

<abstract><p>For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem.</p></abstract>

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations.

In this computational paper, we focused on the efficient numerical implementation of semi-implicit methods for models in materials science. In particular, we were interested in a class of nonlinear higher-order parabolic partial differential equations. The Cahn-Hilliard (CH) equation was chosen as a benchmark problem for our proposed methods. We first considered the Cahn-Hilliard equation with a convexity-splitting (CS) approach coupled with a backward Euler approximation of the time derivative and tested the performance against the bi-harmonic-modified (BHM) approach in terms of accuracy, order of convergence, and computation time. Higher-order time-stepping techniques that allow for the methods to increase their accuracy and order of convergence were then introduced. The proposed schemes in this paper were found to be very efficient for 2D computations. Computed dynamics in 2D and 3D are presented to demonstrate the energy-decreasing property and overall performance of the methods for longer simulation runs with a variety of initial conditions. In addition, we also present a simple yet powerful way to accelerate the computations by using MATLAB built-in commands to perform GPU implementations of the schemes. We show that it is possible to accelerate computations for the CH equation in 3D by a factor of 80, provided the hardware is capable enough.

A New Efficient Hybrid Method Based on FEM and FDM for Solving Burgers’ Equation with Forcing Term

This paper presents a study on the numerical solutions of the Burgers’ equation with forcing effects. The article proposes three hybrid methods that combine two‐point, three‐point, and four‐point discretization in time with the Galerkin finite element method in space (TDFEM2, TDFEM3, and TDFEM4). These methods use backward finite difference in time and the finite element method in space to solve the Burgers’ equation. The resulting system of the nonlinear ordinary differential equations is then solved using MATLAB computer codes at each time step. To check the efficiency and accuracy, a comparison between the three methods is carried out by considering the three Burgers’ problems. The accuracy of the methods is expressed in terms of the error norms. The combined methods are advantageous for small viscosity and can produce highly accurate solutions in a shorter time compared to existing numerical schemes in the literature. In contrast to many existing numerical schemes in the literature developed to solve Burgers’ equation, the methods can exhibit the correct physical behavior for very small values of viscosity. It has been demonstrated that the TDFEM2, TDFEM3, and TDFEM4 can be competitive numerical methods for addressing Burgers‐type parabolic partial differential equations arising in various fields of science and engineering.

Boundary Control for Suppressing Chaotic Response to Dynamic Hydrogen Blending in a Gas Pipeline

It is known that periodic forcing of nonlinear flows can result in a chaotic response under certain conditions. Such non-periodic and chaotic solutions have been observed in simulations of heterogeneous gas flow in a pipeline with periodic, time-varying boundary conditions. In this paper, we examine a proportional feedback law for boundary control of a parabolic partial differential equation system that represents the flow of two gases through a pipe. We demonstrate that periodic variation of the mass fraction of the lighter gas at the pipe inlet can result in the chaotic propagation of gas pressure waves, and show that appropriate flow control can suppress this response. We examine phase space solutions for the single pipe system subject to boundary control, and use numerical experiments to characterize conditions for the controller gain to suppress chaos.

Global-in-time solutions and Hölder continuity for quasilinear parabolic PDEs with mixed boundary conditions in the Bessel dual scale

Global-in-time solutions and Hölder continuity for quasilinear parabolic PDEs with mixed boundary conditions in the Bessel dual scale

Observer-Based Periodic Event-Triggered and Self-Triggered Boundary Control of a Class of Parabolic PDEs

Observer-Based Periodic Event-Triggered and Self-Triggered Boundary Control of a Class of Parabolic PDEs

Event-Triggered Fuzzy Adaptive Stabilization of Parabolic PDE–ODE Systems

Event-Triggered Fuzzy Adaptive Stabilization of Parabolic PDE–ODE Systems