Parabolic Differential Equations Research Articles

Conformable evolution inclusions with causal operators

In this paper, we study local and nonlocal problems of conformable semilinear evolution inclusions with causal operators, which include time lag system and a large classes of parabolic multivalued differential equations. We first investigate the problems with the help of measure of noncompactness and afterwards when the right-hand side is Lipschitz.

Calibration of the Ueno’s Shadow Rate Model of Interest Rates

Shadow rate models of interest rates are based on the assumption that the interest rates are determined by an unobservable shadow rate. This idea dates back to Fischer Black, who understood the interest rate as an option that cannot become negative. Its possible zero values are consequences of negative values of the shadow rate. In recent years, however, the negative interest rates have become a reality. To capture this behavior, shadow rate models need to be adjusted. In this paper, we study Ueno’s model, which uses the Vasicek process for the shadow rate and adjusts its negative values when constructing the short rate. We derive the probability properties of the short rate in this model and apply the maximum likelihood estimation method to obtain the parameters from the real data. The other interest rates are—after a specification of the market price of risk—solutions to a parabolic partial differential equation. We solve the equation numerically and use the long-term rates to fit the market price of risk.

Backstepping Control of a Class of Space-Time-Varying Linear Parabolic PDEs via Time Invariant Kernel Functions

Backstepping Control of a Class of Space-Time-Varying Linear Parabolic PDEs via Time Invariant Kernel Functions

Physics-informed and graph neural networks for enhanced inverse analysis

PurposeThis paper presents an original approach for learning models, partially known, of particular interest when performing source identification or structural health monitoring. The proposed procedures employ some amount of knowledge on the system under scrutiny as well as a limited amount of data efficiently assimilated.Design/methodology/approachTwo different formulations are explored. The first, based on the use of informed neural networks, leverages data collected at specific locations and times to determine the unknown source term of a parabolic partial differential equation. The second procedure, more challenging, involves learning the unknown model from a single measured field history, enabling the localization of a region where material properties differ.FindingsBoth procedures assume some kind of sparsity, either in the source distribution or in the region where physical properties differ. This paper proposed two different neural approaches able to learn models in order to perform efficient inverse analyses.Originality/valueTwo original methodologies are explored to identify hidden property that can be recovered with the right usage of data. Both methodologies are based on neural network architecture.

Discontinuous galerkin method of the first order parabolic differential equation

The objective of this paper is to propose a modest numerical error analysis by applying DG finite element method for the parabolic differential equation. The DG method is an imperative numerical method with much mass compensation and more flexible meshing than other numerical methods. The DG method starts by discretizing the domain into a set of non-overlapping elements. This study gives a general introduction and discuss about the discontinuous Galerkin Method of first order parabolic problem. The parabolic problem satisfies the condition of the existence and uniqueness of DG solution. The main goal of this study is to theoretically explore the convergence of the solution of the above methods and show the validity of the results.

Regularizations of forward‐backward parabolic PDEs

AbstractForward‐backward parabolic equations have been studied since the 1980s, but a mathematically rigorous picture is still far from being established. As quite a number of new papers have appeared recently, we review in this work the current state of the art. We focus our analysis on the status quo regarding the three most common types of regularizations, namely semidiscretization, the viscous approximation, and regularization with higher order spatial derivatives.

WELL-POSEDNESS RESULTS FOR GENERAL REACTION–DIFFUSION TRANSPORT OF OXYGEN IN ENCAPSULATED CELLS

Abstract We provide well-posedness results for nonlinear parabolic partial differential equations (PDEs) given by reaction–diffusion equations describing the concentration of oxygen in encapsulated cells. The cells are described in terms of a core and a shell, which introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. In addition, the cells are subject to general nonlinear consumption of oxygen. As no monotonicity condition is imposed on the consumption, monotone operator theory cannot be used. Moreover, the discontinuity in the diffusion coefficient bars us from applying classical results on strong solutions. However, by directly applying a Galerkin method, we obtain uniqueness and existence of the strong form solution. These results provide the basis to study the dynamics of cells in critical states.

Deep-time neural networks: An efficient approach for solving high-dimensional PDEs

This paper presents the Deep-Time Neural Network (DTNN), an efficient and novel deep-learning approach for solving partial differential equations (PDEs). DTNN leverages the power of deep neural networks to approximate the solution for a class of quasi-linear parabolic PDEs. We demonstrate that DTNN significantly reduces the computational cost and speeds up the training process compared to other models in the literature. The results of our study indicate that DTNN architecture is promising for the fast and accurate solution of time-dependent PDEs in various scientific and engineering applications. The DTNN architecture addresses the pressing need for enhanced time considerations in the deeper layers of Artificial Neural Networks (ANNs), thereby improving convergence time for high-dimensional PDE solutions. This is achieved by integrating time into the hidden layers of the DTNN, demonstrating a marked improvement over existing ANN-based solutions regarding efficiency and speed.

Adaptive fault estimation and accommodation for distributed parameter systems with coupled parabolic partial differential equations

This paper presents a novel model-based approach for fault detection, estimation and accommodation in linear distributed parameter systems (DPS) governed by parabolic partial differential equations (PDEs), prone to sensor and actuator faults. Unlike traditional methods, our approach utilises a filter-based observer directly employing the PDE representation and relies solely on boundary measurements, eliminating the need for extensive sensor arrays. By generating fault detection residuals from a comparison of measured and observer outputs, our method offers efficient fault detection. Innovative parameter update laws facilitate accurate estimation of fault parameters, enabling precise adjustment of the nominal controller to mitigate fault effects. Rigorous Lyapunov analysis ensures the robustness of our approach. Additionally, we derive a formula for estimating time-to-accommodation (TTA), providing valuable insights into fault resolution timelines. Simulations on a linearised diffusion process validate the effectiveness of our scheme, highlighting its superiority over existing approaches.

Boundary Treatment for High-Order IMEX Runge–Kutta Local Discontinuous Galerkin Schemes for Multidimensional Nonlinear Parabolic PDEs

Boundary Treatment for High-Order IMEX Runge–Kutta Local Discontinuous Galerkin Schemes for Multidimensional Nonlinear Parabolic PDEs

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs.

Modeling of adsorptive treatment of lead (II) ions contaminated effluents using cashew nut shell alkali activated carbon: Batch kinetic, isothermal and thermodynamic studies

The numerical study of the decontamination of lead(II) ions-rich effluent employing activated carbon derived from cashew nut shell was the central focus of this work. Chemical activation with zinc chloride (ZnCl2) was used to produce powdered activated carbon at optimized activation conditions of impregnation ratio (0.531), activation temperature (1083 K) and activation time (63min). The physiochemical properties of raw and activated cashew nut shell were determined using Fourier transform infrared spectroscopy, scanning, X-ray diffractor (XRD), scanning electron microscopy (SEM), and Brunauer-Emmet-Teller (BET) analyses. Batch adsorption experiments were performed at varied process conditions of temperature (303–323) K, contact time (5–80min), and initial lead(II) ion concentrations (100–400 mg.L−1). The adsorption behavior of the batch–type operation was mathematically described in the form of non-linear parabolic partial differential equations (PDE’S) incorporating film, pore and surface diffusion mechanisms. The batch mathematical model was solved by implementing a custom MATLAB program-OkiyNwabannebatch and calibrated to predict the desired response (percentage removal) using measured adsorption data. The equilibrium adsorption, kinetics and thermodynamics of lead(II) ions adsorption onto activated cashew nut shell were also assessed using nonlinear isotherm, kinetic and thermodynamic models. Sensitivity analysis unveiled that temperature with sensitivity value of 37.31 % had a predominant effect on the model performance. The simulated equilibrium data was well explicated by the Freundlich model for lead(II) ions adsorption onto activated cashew nut shell. The pseudo-second order kinetic model best reproduced the simulated adsorption data for lead(II) ions uptake by cashew nut shell adsorbent. Likewise, the kinetic data followed Boyd model, indicating that the adsorption process is controlled by external (film) diffusion for the uptake of lead(II) ions by alkali activated cashew nut shell. Thermodynamic analysis using the Gibbs and Van’t Hoff’s models indicated that lead(II) ions adsorption by modified cashew nut shell is spontaneous and exothermic. The physical nature of the adsorption process was elucidated from the low values of the isosteric heats of adsorption (ΔHX) evaluated (<80 kJ mol−1). This study showed that zinc chloride activated cashew nut shell has good potential to be used as a cost-effective and efficient adsorbent for the uptake of lead(II) ions from aqueous solution.

The energy-diminishing weak Galerkin finite element method for the computation of ground state and excited states in Bose-Einstein condensates

In this paper, we employ the weak Galerkin (WG) finite element method and the imaginary time method to compute both the ground state and the excited states in Bose-Einstein condensate (BEC) which is governed by the Gross-Pitaevskii equation (GPE). First, we use the imaginary time method for GPE to get the nonlinear parabolic partial differential equation. Subsequently, we apply the WG method to spatially discretize the parabolic equation. This yields a semi-discrete scheme, in which an energy function is explicitly defined. For the case β⩾0, we demonstrate that the energy is diminishing with respect to time t at each time step. Applying the backward Euler scheme for temporal discretization yields a fully discrete scheme. For the case β=0, we provide a mathematical justification, establishing the convergence analysis for the numerical solution of the ground state. Moreover, based on the theory of solving eigenvalue problems using the WG method, we present the error estimates between the ground state and its numerical solution under the H1 and L2 norms. Numerical experiments are provided to illustrate the effectiveness of the proposed schemes. Moreover, the results indicate that our method also can compute the first excited state, achieving optimal convergence orders.

Convergence Rates for Identification of Robin Coefficient from Terminal Observations

This paper deals with the problem of identification of a Robin coefficient (also known as impedance coefficient) in a parabolic PDE from terminal observations of the temperature distributions. The problem is ill-posed in the sense that small perturbation in the observation may lead to a large deviation in the solution. Thus, in order to obtain stable approximations, we employ the Tikhonov-regularization. We propose a weak source condition motivated by the work of Engl and Zou (2000) and obtain a convergence rate of O(δ12)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\delta ^\\frac{1}{2})$$\\end{document}, where δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} is the noise level of the observed data. The obtained rate is better than some of the previous known rates. Moreover, the advantage of the proposed source condition is that the above mentioned convergence rate is obtained without the need for characterizing the range space of modelling operator and its Fréchet derivative, which is in contrast to the general convergence theory of Tikhonov-regularization for non linear operators.

Existence and Regularity of Random Attractors for Stochastic Evolution Equations Driven by Rough Noise

AbstractThis work establishes the existence and regularity of random pullback attractors for parabolic partial differential equations with rough nonlinear multiplicative noise under natural assumptions on the coefficients. To this aim, we combine tools from rough path theory and random dynamical systems. An application is given by partial differential equations with rough boundary noise, for which flow transformations are not available.

Stability and regularization for ill-posed Cauchy problem of a stochastic parabolic differential equation

Abstract In this paper, we investigate an ill-posed Cauchy problem involving a stochastic parabolic equation. We first establish a Carleman estimate for this equation. Leveraging this estimate, we derive the conditional stability and convergence rate of the Tikhonov regularization method for the aforementioned ill-posed Cauchy problem. To complement our theoretical analysis, we employ kernel-based learning theory to implement the completed Tikhonov regularization method for several numerical examples.

Sliding mode control to stabilization of coupled time fractional parabolic PDEs subject to disturbances

AbstractIn this article, the stabilization of coupled time fractional parabolic partial differential equations subject to external disturbances is investigated. By using sliding mode control method and backstepping approach, a boundary state feedback controller is designed to reject the matched disturbance and achieve the Mittag‐Leffler input‐to‐state stability of closed‐loop system. The existence of the generalized solution to the closed‐loop system is proven by Galerkin approximation scheme. Simulations are presented to illustrate the validity of our theoretical results.

Stochastic Burgers type equations with two reflecting walls: Existence, uniqueness and exponential ergodicity

In this paper, we study stochastic Burgers type equations with two reflecting walls driven by multiplicative noise. We first establish the existence and uniqueness of the solution by Picard iteration technique, the main trouble is to handle the complicated nonlinear term ∂g(u(x,t))∂x which involves the derivative of the solution u(x, t) and the singularities due to reflection. Furthermore, we prove the exponential ergodicity for the equations driven by possibly degenerate, multiplicative noise through asymptotic coupling. More generally, we also give a simplified criterion for exponential ergodicity of parabolic stochastic partial differential equations with reflections driven by degenerate multiplicative noise.

A Hybrid Technique for Space-Time Fractional Parabolic Differential Equations

A Hybrid Technique for Space-Time Fractional Parabolic Differential Equations

Numerical solution of source identification multi-point problem of parabolic partial differential equation with Neumann type boundary condition

We study a source identification boundary value problem for a parabolic partial differential equation with multi-point Neumann type boundary condition. Stability estimates for the solution of the overdetermined mixed BVP for multi-dimensional parabolic equation were established. The first and second order of accuracy difference schemes for the approximate solution of this problem were proposed. Stability estimates for both difference schemes were obtained. The result of numerical illustration in test example was given.