Nonlinear Parabolic Partial Differential Equations Research Articles

Protected Area Biodiversity Conservation: Population Dynamics

One of the methods for protecting the environment and conserving species biodiversity is the creation of protected areas, the task of which is to conserve rare species of animals and plants. The dynamics of populations can be studied quite effectively with the mathematical models. Modeling allows to study all possible scenarios of interaction and development of populations, to evaluate the influence of the environment on the spatial-temporal dynamics of biological communities. The paper considers a non-linear model of a protected population, the feature of which is the presence of a bilocal range and the exchange of species between its parts. It is a system of nonlinear parabolic partial differential equations that describes the dependence of migration flows on the uneven distribution of species in an area with a generalized resource. A complex of programs was coded in Python programming language for numerical study of diffusion models and visualization of the results obtained. Numerical simulation results of cross-diffusion effects and influence of generalized resource are analyzed in detail. The obtained results of research can be used for studying the processes in the protected population, for assessing the size of the protected areas being created, determining the conditions of population degeneration and recommendations for their stable existence.

Anisotropic and p, q-nonlinear partial differential equations

Anisotropic partial differential equations recently received a large interest in the mathematical literature, due to their applications to double and multiphase variational energies, as well as to anisotropic energies in integral form. More specifically, from a mathematical point of view, we need to consider a generalization of the classical Laplacian elliptic and parabolic partial differential equations, as well as the nonlinear p-Laplacian equations, which naturally arises new and interesting mathematical questions, nowadays only partially solved in the context of anisotropic p, q-growth nonlinear elliptic and parabolic partial differential equations. In this context, we describe some “mathematical pathologies”; more precisely, some singularities in the potential generated by some anisotropic energies. The singularities appear if the anisotropy of the energy is too large in some directions, while these singularities do not appear, not only if the energy is isotropic with respect to all directions, but also even if we allow an energy integral with a mild anisotropy.

The Link between Stochastic Differential Equations with Non-Markovian Coefficients and Backward Stochastic Partial Differential Equations

In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework. Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.

Analysis of a plasma-assisted reactive evaporation process for preparation of ZnO thin films: Modeling and experimentation

This paper focuses on modeling and simulation of a plasma-assisted reactive evaporation process. A dimensional unsteady-state model consisting of six nonlinear parabolic partial differential equations which account for the diffusive and convective mass transfer, bulk and surface reactions, non-uniform fluid flow, and plasma electron density profiles is developed. The equations are spatially discretized using finite difference methods and then numerically solved. The simulation results are validated by comparison with a commercial simulation software that computes similar equation systems. To validate the model, experimental measurements of film thickness are carried out on ZnO films deposited by plasma-assisted reactive evaporation. After 130 min of deposition time and deposition rate around 6 nm/min, the simulated results and the experimental measurements exhibit a good agreement (770 nm and 750 nm respectively), just around 2.59% discrepancy in the final thin film thickness, hence showing the high accuracy of the developed model.

Input-to-State Stability for a Class of One-Dimensional Nonlinear Parabolic PDEs with Nonlinear Boundary Conditions

The aim of this paper is to introduce a weak maximum principle--based approach for studying the input-to-state stability (ISS) with respect to boundary disturbances and states in certain classes for a class of one-dimensional nonlinear parabolic partial differential equations (PDEs) with nonlinear boundary conditions. To tackle the difficulties in ISS analysis due to, in particular, the nonlinear terms on the boundary, we establish first several maximum estimates for the solutions of linear parabolic PDEs with different nonlinear boundary conditions by means of the weak maximum principle. Then, using the technique of splitting and combining maximum estimates for the solutions of linear parabolic PDEs and the Lyapunov method, we establish ISS estimates for nonlinear parabolic PDEs with nonlinear boundary conditions. Two examples of specific parabolic equations with nonlinear boundary conditions are provided to illustrate the developed approach.

Observer Design for Lipschitz Nonlinear Parabolic PDE Systems With Unknown Input

In this article, a novel method to design the observer for a class of uncertain Lipschitz nonlinear parabolic partial differential equations (PDE) systems is investigated. First, the observer and the dynamic errors with undetermined parameters for the parabolic PDE systems subject to appropriate boundary conditions are presented. The conditions of the designed observer are involved. Then the analysis of asymptotic stability and $\mathcal {H}_\infty $ performance conditions for the observer design of uncertain nonlinear parabolic PDE systems are studied and presented in terms of matrix inequalities based on the Lyapunov stability theory. Finally, the effectiveness of the proposed method is validated by a numerical parabolic PDE system.

IMAGE-DRIVEN BIOPHYSICAL TUMOR GROWTH MODEL CALIBRATION.

We present a novel formulation for the calibration of a biophysical tumor growth model from a single-time snapshot, multiparametric magnetic resonance imaging (MRI) scan of a glioblastoma patient. Tumor growth models are typically nonlinear parabolic partial differential equations (PDEs). Thus, we have to generate a second snapshot to be able to extract significant information from a single patient snapshot. We create this two-snapshot scenario as follows. We use an atlas (an average of several scans of healthy individuals) as a substitute for an earlier, pretumor, MRI scan of the patient. Then, using the patient scan and the atlas, we combine image-registration algorithms and parameter estimation algorithms to achieve a better estimate of the healthy patient scan and the tumor growth parameters that are consistent with the data. Our scheme is based on our recent work (Scheufele et al., Comput. Methods Appl. Mech. Engrg., to appear), but we apply a different and novel scheme where the tumor growth simulation in contrast to the previous work is executed in the patient brain domain and not in the atlas domain yielding more meaningful patient-specific results. As a basis, we use a PDE-constrained optimization framework. We derive a modified Picard-iteration-type solution strategy in which we alternate between registration and tumor parameter estimation in a new way. In addition, we consider an ℓ 1 sparsity constraint on the initial condition for the tumor and integrate it with the new joint inversion scheme. We solve the sub-problems with a reduced space, inexact Gauss-Newton-Krylov/quasi-Newton method. We present results using real brain data with synthetic tumor data that show that the new scheme reconstructs the tumor parameters in a more accurate and reliable way compared to our earlier scheme.

Nonlinear dynamics of dewetting thin films

Fluid films spreading on hydrophobic solid surfaces exhibit complicated dynamics that describe transitions leading the films to break up into droplets. For viscous fluids coating hydrophobic solids this process is called "dewetting". These dynamics can be represented by a lubrication model consisting of a fourth-order nonlinear degenerate parabolic partial differential equation (PDE) for the evolution of the film height. Analysis of the PDE model and its regimes of dynamics have yielded rich and interesting research bringing together a wide array of different mathematical approaches. The early stages of dewetting involve stability analysis and pattern formation from small perturbations and self-similar dynamics for finite-time rupture from larger amplitude perturbations. The intermediate dynamics describes further instabilities yielding topological transitions in the solutions producing sets of slowly-evolving near-equilibrium droplets. The long-time behavior can be reduced to a finite-dimensional dynamical system for the evolution of the droplets as interacting quasi-steady localized structures. This system yields <i>coarsening</i>, the successive re-arrangement and merging of smaller drops into fewer larger drops. To describe macro-scale applications, mean-field models can be constructed for the evolution of the number of droplets and the distribution of droplet sizes. We present an overview of the mathematical challenges and open questions that arise from the stages of dewetting and how they relate to issues in multi-scale modeling and singularity formation that could be applied to other problems in PDEs and materials science.

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man made complex systems. In particular, parabolic PDEs are a fundamental tool to determine fair prices of financial derivatives in the financial industry. The PDEs appearing in financial engineering applications are often nonlinear and high dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue is that most approximation methods for nonlinear PDEs in the literature suffer under the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods have not been shown not to suffer under the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend those findings to a more general class of semilinear PDEs including as special cases semilinear Black-Scholes equations used for the pricing of financial derivatives with default risks. More specifically, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove that the computational effort of our method grows at most polynomially both in the dimension and the reciprocal of the prescribed approximation accuracy. This is, to the best of our knowledge, the first result showing that the approximation of solutions of semilinear Black-Scholes equations is a polynomially tractable approximation problem.

Steady states and dynamics of a thin-film-type equation with non-conserved mass

We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

Gradient estimates and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces

Let (Mn,g,e−ϕdv) be a smooth metric measure space. In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a nonlinear parabolic partial differential equation(Δϕ−∂t)u=pu+qua+1 in Mn×(−∞,+∞), where p,q∈C2,1(Mn×(−∞,+∞)) and a is a positive constant.

Sampled-Data Fuzzy Control With Guaranteed Cost for Nonlinear Parabolic PDE Systems via Static Output Feedback

This article introduces a sampled-data (SD) static output feedback fuzzy control (FC) with guaranteed cost for nonlinear parabolic partial differential equation (PDE) systems. First, a Takagi–Sugeno (T–S) fuzzy parabolic PDE model is employed to represent the nonlinear PDE system. Second, with the aid of the T–S fuzzy PDE model, a SD FC design with guaranteed cost under spatially averaged measurements is developed in the formulation of linear matrix inequalities by utilizing a time-dependent Lyapunov functional and inequality techniques, which can stabilize exponentially the PDE system while providing an optimized upper bound on the cost function. The membership functions of the proposed controller are determined by the measurement output and independent of the fuzzy PDE plant model. Finally, simulation results are presented to control the diffusion equation and the FitzHugh–Nagumo equation for demonstrating the effectiveness of the proposed method.

Event-triggered reliable [formula omitted] fuzzy filtering for nonlinear parabolic PDE systems with Markovian jumping sensor faults

This paper is devoted to even-triggered reliable H∞ filter design for a class of nonlinear partial differential equation (PDE) systems with Markovian jumping sensor faults. Initially, a Takagi-Sugeno (T-S) fuzzy model is adopted to reconstruct the nonlinear systems. Then, the time delays and signal quantization that often occur in network transmission are taken into account, and an integral-type event-triggered scheme is developed to improve the transmission channel utilization. Furthermore, non-parallel-distributed-compensation technique is introduced to increase the flexibility of filter design and the filter’s parameters can be obtained by solving several linear matrix inequalities. Finally, two numerical simulations and an application study to Catalytic Rod are provided to demonstrate the effectiveness and practicability of the proposed methodology.

Sampled-data fuzzy control with space-varying gains for nonlinear time-delay parabolic PDE systems

This paper introduces a sampled-data fuzzy control (SDFC) with space-varying gains for nonlinear time-delay parabolic partial differential equation (PDE) systems. First, a Takagi-Sugeno (T-S) fuzzy time-delay parabolic PDE model is employed to represent the nonlinear time-delay PDE system. Second, with the aid of the T-S fuzzy time-delay PDE model, a SDFC design with space-varying gains is developed in the formulation of space-dependent linear matrix inequalities (LMIs) by constructing an appropriate Lyapunov functional, which can stabilize exponentially the time-delay PDE system. These stabilization conditions can be applied to either slowing-varying time delay or fast-varying one. Finally, simulation results on two examples are presented to demonstrate the effectiveness of the proposed method.

Multi-dimensional BSDEs driven by G-Brownian motion and related system of fully nonlinear PDEs

In this paper, we study the well-posedness of multi-dimensional backward stochastic differential equations driven by G-Brownian motion (G-BSDEs). The existence and uniqueness of solutions are obtained via a contraction argument for Y component and a backward iteration of local solutions. Furthermore, we show that, the solution of multi-dimensional G-BSDE in a Markovian framework provides a probabilistic formula for the viscosity solution of a system of nonlinear parabolic partial differential equations.

Control design for non‐linear distributed parameter systems with mobile actuators and sensors

This study deals with the H ∞ control design problem for a class of non-linear distributed parameter systems described by parabolic partial differential equations (PDEs) via mobile collocated actuators and sensors. Initially, the spatial domain is decomposed into multiple subdomains according to the number of actuator/sensor pairs and the projection modification algorithm is employed to guarantee each actuator/sensor pair is only capable of moving within the respective subdomain. Subsequently, the well-posedness of the closed-loop PDE system is analysed by means of the operator semigroup theory. Then, a control-plus-guidance design method for the non-linear PDE system is developed in the form of bilinear matrix inequalities, such that the resulting closed-loop system is exponentially stable while satisfying a prescribed H ∞ performance of disturbance attenuation, and the mobile actuator/sensor guidance can enhance the transient performance of the closed-loop system. Finally, a numerical example and a practical application example are respectively given to show the effectiveness of the proposed design method.

A class of numerical algorithms based on cubic trigonometric B-spline functions for numerical simulation of nonlinear parabolic problems

In this work, the authors developed two new B-spline collocation algorithms based on cubic trigonometric B-spline functions to find approximate solutions of nonlinear parabolic partial differential equations (PDEs) with Dirichlet and Neumann boundary conditions. In the first algorithm, cubic trigonometric B-spline functions are directly used for approximate solutions of parabolic PDEs with Neumann boundary conditions. But, the Dirichlet boundary conditions cannot be handled directly by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and the second algorithm is developed with the help of modified cubic trigonometric B-spline functions. The proposed algorithms reduced the parabolic problem into a system of first-order nonlinear ordinary differential equations (ODEs) in time variable. Then, strong stability-preserving-Runge–Kutta3 (SSP-RK3) scheme is used to solve the obtained system. Some well-known parabolic problems are solved to check the accuracy and efficiency of the proposed algorithms. The algorithms can be extended to solve multidimensional problems arising as model equations in physical, chemical and biophysical phenomenon.

A non-linear parabolic PDE with a distributional coefficient and its applications to stochastic analysis

We consider a non-linear parabolic partial differential equation (PDE) on Rd with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition and blow-up times. We prove a global existence and uniqueness result assuming further properties on the non-linearity. To conclude we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.

A simple algorithm for numerical solution of nonlinear parabolic partial differential equations

In this paper, numerical solution of nonlinear two-dimensional parabolic partial differential equations with initial and Dirichlet boundary conditions is considered. The time derivative is approximated using finite difference scheme whereas space derivatives are approximated using Haar wavelet collocation method. The proposed method is developed for semilinear and quasilinear cases, however, it can easily be extended to other types of nonlinearities as well. The proposed method is also illustrated for nonlinear heat equation and Burgers’ equation. The proposed method is implemented upon five test problems and the numerical results are shown using tables and figures. The numerical results validate the accuracy and efficiency of the proposed method.

Local elliptic gradient estimates for a nonlinear parabolic equation under the Ricci flow

In this study, we derive local elliptic (Hamilton type and Souplet–Zhang type) gradient estimates for positive solutions of the nonlinear parabolic partial differential equation (Δt−∂t)u=qu+au(ln⁡u)α under the Ricci flow.