Parabolic Partial Differential Systems Research Articles

A new analysis of Galerkin Legendre spectral methods for coupled hyperbolic/parabolic system arising in unsteady MHD flow of Maxwell fluid and numerical simulation

In this paper, a high-order accurate numerical method for two-dimensional coupled hyperbolic and parabolic partial differential system arising in unsteady magnetohydrodynamics (MHD) flow of upper-convected Maxwell fluid is presented. We apply Galerkin-Legendre spectral method for discretizing spatial derivatives for MHD-M (MHD flow of Maxwell fluid) equations and then Crank-Nicolson scheme is used for time derivatives. Optimal a priori error bound is derived in the L2 and H1 norm for the semidiscrete formulation. We then proceed to discuss an instability mechanism in polynomial spectral methods and show that filtering suffices to ensure stability. Optimal error bound is derived between the semi-analytical solutions and filtered semi-analytical solutions. The results are illustrated by computational experiments. Furthermore, these results indicate that the proposed method is simple, accurate, efficient, and stable for solving various MHD-M and MHD flow problems or in general coupled hyperbolic-parabolic partial differential equations.

On a parabolic partial differential equation and system modeling a production planning problem

<abstract><p>We consider a parabolic partial differential equation and system derived from a production planning problem dependent on time. Our goal is to find a closed-form solution for the problem considered in our model. Our new theoretical results can be applied in the real world.</p></abstract>

Optimal control of an evolution hemivariational inequality involving history-dependent operators

In this paper we consider a class of feedback control systems described by an evolution hemivariational inequality involving history-dependent operators. Under the mild conditions, first, we prove a priori estimates of the solutions to the feedback control system. Then, an existence theorem for the feedback control system is obtained by using the well-known Bohnenblust-Karlin fixed point theorem. Moreover, we consider an optimal control problem driven by the feedback control system, and establish its solvability. Finally, a parabolic partial differential system with Clarke subgradient term is considered to illustrate the applicability of the theoretical results.

Optimal control of a class of semi‐linear stochastic evolution equations with applications

In this study, the authors derive a new type of maximum principle for both stochastic evolution and parabolic type stochastic partial differential systems. The systems only require the Hamiltonian functional to be concave in the state variable rather than in both state and control variables. They also show a connection between these two types of systems. Finally, examples are given to illustrate the authors' theoretical results.

Numerical simulation of porous silicon morphology using a monotone iterative method

In this work, a convergent monotone iterative system based on finite difference approximations is proposed to solve numerically a novel porous silicon formation model. This model includes chemical, physical and electrical processes involved in the porous growth and is formulated as a weakly coupled parabolic partial differential system. Important properties like nonnegativity, boundedness, stability and convergence of the numerical solution are demonstrated by means of the upper and lower solutions technique. The computational porous silicon formation model is validated by experimental evidence. Numerical simulations of the porous silicon morphology in two dimensions are shown along with the effects of parameters variation.

Robust Fuzzy Filter Design for Nonlinear Parabolic Partial Differential Systems with Continuous Wiener Noise and Discontinuous Poisson Noise

This paper addresses a robust \(H_{\infty }\) fuzzy filter design problem for nonlinear stochastic partial differential systems (NSPDSs) with continuous random fluctuation, discontinuous Poisson jumping noise, random external disturbance and measurement noise in the spatiotemporal domain. For NSPDSs, the robust \(H_{\infty }\) filter design problem through a measurement output needs to solve a complex second-order Hamilton Jacobi integral inequality. In order to simplify the design procedure, a fuzzy stochastic partial differential system based on a fuzzy interpolation approach is proposed to approximate the NSPDS. Then, the robust \(H_{\infty }\) fuzzy filter design problem can be reformulated as a diffusion matrix inequality (DMI) problem. Since the DMI problem is difficult to be solved via traditional algebraic techniques, we utilize the divergence theorem and Poincare inequality to transform the DMIs to a set of linear matrix inequalities (LMIs) which could be easily solved with the help of MATLAB LMI Toolbox. Finally, a robust state estimation problem of an ecology system with intrinsic spatiotemporal continuous Wiener noise and discontinuous Poisson jump fluctuation is provided as an example to illustrate the design procedure and to confirm the \(H_{\infty }\) filtering performance of the proposed \(H_{\infty }\) fuzzy filter design method.

Robust Fuzzy $H_{\infty }$ Estimator-Based Stabilization Design for Nonlinear Parabolic Partial Differential Systems With Different Boundary Conditions

In this paper, a new robust fuzzy H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> estimator-based stabilization design for a class of N-dimensional nonlinear parabolic partial differential systems (PDSs) with either the Dirichlet or Neumann boundary conditions is proposed. First, an N-dimensional parabolic Takagi-Sugeno (T-S) fuzzy PDS is used to approximate the N-dimensional nonlinear parabolic PDS via the knowledge-based T-S fuzzy system technique. Second, based on the N-dimensional parabolic T-S fuzzy PDS, a robust fuzzy estimator-based controller is employed not only to stabilize the PDS, but also to attenuate the effects of external and measurement noises on the PDS in the spatiotemporal domain below a prescribed attenuation level. The robust fuzzy H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> estimator-based stabilization design problem can be formulated as a diffusion matrix inequality (DMI) problem. To solve DMIs via the traditional algebraic matrix techniques, we utilize the divergence theorem and the Poincare inequality to transform the DMIs into bilinear matrix inequalities with a Poincare constant defined according to the spatial domain of the corresponding PDS. Finally, the robust fuzzy H∞ estimator-based stabilization design problem can be effectively solved by a set of linear matrix inequalities instead of the BMIs with the help of the proposed decoupled method. Additionally, an optimal robust fuzzy H∞ estimator-based stabilization design can be realized via minimizing the noise attenuation level. A simulation example is provided to illustrate the design procedure and verify the performance of the proposed optimal design.

A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space

In this article we consider parabolic systems and $L_p$ regularityof the solutions. With zero boundary condition the solutionsexperience bad regularity near the boundary. This article addressesa possible way of describing the regularity nature. Our space domainis a half space and we adapt an appropriate weight into our functionspaces. In this weighted Sobolev space setting we develop aFefferman-Stein theorem, a Hardy-Littlewood theorem and sharpfunction estimations. Using these, we prove uniqueness and existenceresults for second-order elliptic and parabolic partial differentialsystems in weighed Sobolev spaces.

On Regularity of the Time Derivative for Degenerate Parabolic Systems

We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) $p$-Laplace system and models for non-Newtonian fluids of powerlaw or Carreau type. By the use of special weak different quotients adapted to the variational structure we bound fractional derivatives of $u_t$ in time and space directions. Although the estimates presented here are valid under very general assumptions they are a novelty even for the parabolic $p$-Laplace equation.

A review of mathematical analysis of nematic and smectic-A liquid crystal models

We review the mathematical analysis of some uniaxial, liquid crystal phases. Firstly, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models. It is our aim to show that, although nematic and smectic-A phases have different physical properties and are modelled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and obtain analytical results.

A note on [formula omitted]-theory of linear stochastic parabolic partial differential systems

In this article we construct a Wpγ-theory of linear stochastic parabolic partial differential systems. Here, p∈[2,∞) and γ∈(−∞,∞). We also provide an example to show that for stochastic systems we need more restriction than the algebraic condition which ensures that diffusion survives against wild convection.

A ${W}^{n}_{2}$ -Theory of Stochastic Parabolic Partial Differential Systems on C 1-domains

In this article we present a $W^n_2$ -theory of stochastic parabolic partial differential systems. In particular, we focus on non-divergent type. The space domains we consider are ℝ d , $ {\mathbb{R}}^d_+$ and eventually general bounded C 1-domains $\mathcal{O}$ . By the nature of stochastic parabolic equations we need weighted Sobolev spaces to prove the existence and the uniqueness. In our choice of spaces we allow the derivatives of the solution to blow up near the boundary and moreover the coefficients of the systems are allowed to oscillate to a great extent or blow up near the boundary.

Habitat choice of multiple pollinators in almond trees and its potential effect on pollen movement and productivity: A theoretical approach using the Shigesada–Kawasaki–Teramoto model

California's almond industry, valued at $2.3 billion per year, depends on the pollinator services of honey bees, although pollination by other insects, mainly solitary wild bees, is being investigated as an alternative because of recent declines in the number of honey bee colonies. Our objective is to model the movements of honey bees and determine the conditions under which they will forage in less favorable areas of a tree and its surroundings when other pollinators are present. We hypothesize that foraging in less favorable areas leads to increased movement between trees and increased cross pollination between varieties which is required for successful nut production. We use the Shigesada–Kawasaki–Teramoto model (1979) which describes the density of two species in a two-dimensional environment of variable favorableness with respect to intrinsic diffusions and intra and interspecific interactions of species. The model is applied to almond pollination by honey bees and other pollinators with environmental favorableness based on the distribution of flowers in trees. Using the spectral-Galerkin method in a rectangular domain, we numerically approximated the two-dimensional nonlinear parabolic partial differential system arising in the model. When cross-diffusion or interspecific effects of other pollinators was high, honey bees foraged in less favorable areas of the tree. In the model, high cross-diffusion also resulted in increased activity in honey bees which manifested itself in the field in terms of accelerations, decelerations, and changes in direction, indicating rapid redistribution of densities to an equilibrium state. Empirical analysis of the number of honey bees and other visitors in 2-min intervals to almond trees shows a negative relationship, indicating cross-diffusion effects in nature with the potential to increase movement to a different tree with a more favorable environment, potentially increasing nut production.

A [formula omitted]-theory of elliptic and parabolic partial differential systems in [formula omitted] domains

In this paper we consider the second-order parabolic partial differential systems and elliptic systems with C1 space domains. We prove the existence and uniqueness results in the Sobolev spaces with weights. In this solution spaces the derivatives of solutions are allowed to blow up near the boundary and also the coefficients of the systems may oscillate to a great extent or blow up near the boundary.

Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients

The aim of this paper is to formulate Campanato-type boundary estimates for solutions of the Rothe approximate scheme to parabolic partial differential systems with constant coefficients. The core observation is that such estimates hold <i>independently of the approximate systems</i>.

Constructing eigenfunctions of strongly coupled parabolic boundary value systems

This paper deals with the study of existence conditions as well as the construction of eigenfunctions of strongly coupled parabolic boundary value partial differential systems.

OPTIMAL CONTROL OF HARVESTING IN A PARABOLIC SYSTEM MODELING TWO SUBPOPULATIONS

An optimal harvesting problem for a parabolic partial differential system modeling two subpopulations of the same species is investigated. The two subpopulations are competing for resources. Under conditions on the smallness of the time interval and certain biological parameters, existence and uniqueness of an optimal control pair are established.

High-order adaptive methods for parabolic systems

We consider the adaptive solution of parabolic partial differential systems in one and two space dimensions by finite element procedures that automatically refine and coarsen computational meshes, vary the degree of the piecewise polynomial basis and, in one dimension, move the computational mesh. Two-dimensional meshes of triangular, quadrilateral, or a mixture of triangular and quadrilateral elements are generated using a finite quadtree procedure that is also used for data management. A posteriori estimates, used to control adaptive enrichment, are generated from the hierarchical polynomial basis. Temporal integration, within a method-of-lines framework, uses either backward difference methods or a variant of the singly implicit Runge-Kutta (SIRK) methods. A high-level user interface facilitates use of the adaptive software.

Towards the automatic numerical solution of partial differential equations

Adaptive mesh enrichment techniques are becoming recognized as a means of automating many of the decisions associated with the computer solution of partial differential equations. Towards this end, we describe an intelligent interface where partial differential equations and their data may be entered in a relatively natural mathematical language. Fortran subroutines for functions and Jacobians that are required by the numerical software are generated automatically with symbolic differentiation used to evaluate Jacobians. We further describe the use of this symbolic interface with an adaptive local mesh refinement finite element method for two-dimensional parabolic partial differential systems. Solutions are calculated using Galerkin's method with a piecewise bilinear polynomial basis in space and backward Euler integration in time. Estimates of the local discretization error, obtained by a p-refinement technique, are used to control a local h-refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded. Fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them. A version of the Schwarz alternating principle is used to coordinate solutions between overlapping fine grids.

A moving-mesh finite element method with local refinement for parabolic partial differential equations

We discuss a moving-mesh finite element method for solving initial boundary value problems for vector systems of partial differential equations in one space dimension and time. The system is discretized using piecewise linear finite element approximations in space and a backward difference code for stiff ordinary differential systems in time. A spatial-error estimation is calculated using piecewise quadratic approximations that use the superconvergence properties of parabolic systems to gain computational efficiency. The spatial-error estimate is used to move and locally refine the finite element mesh in order to equidistribute a measure of the total spatial error and to satisfy a prescribed error tolerance. Ordinary differential equations for the spatial-error estimate and the mesh motion are integrated in time using the same backward difference software that is used to determine the numerical solution of the partial differential system. We present several details of an algorithm that may be used to develop a general-purpose finite element code for one-dimensional parabolic partial differential systems. The algorithm combines mesh motion and local refinement in a relatively efficient manner and attempts to eliminate problemdependent numerical parameters. A variety of examples that motivate our mesh-moving strategy and illustrate the performance of our algorithm are presented.