Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

In this work, we explore a methodology to compute the empirical eigenfunctions for the order-reduction of nonlinear parabolic partial differential equations (PDEs) system with time-varying domain. The idea behind this method is to obtain the mapping functional, which relates the time-evolution scalar physical property solution ensemble of the nonlinear parabolic PDE with the time-varying domain to a fixed reference domain, while preserving space invariant properties of the raw solution ensemble. Subsequently, the Karhunen-Lo'eve decomposition is applied to the solution ensemble with fixed spatial domain resulting in a set of optimal eigenfunctions that capture the most energy of data. Further, the low dimensional set of empirical eigenfunctions is mapped (“pushed-back”) on the time-varying domain by an appropriate mapping resulting in the basis for the construction of the reduced-order model of the parabolic PDEs with time-varying domain. Finally, this methodology is applied in the representative cases of calculation of empirical eigenfunctions in the case of one and two dimensional model of nonlinear reaction-diffusion parabolic PDE systems with analytically defined domain evolutions. In particular, the design of both mappings which relate the raw data and function spaces transformations from the time-varying to time-invariant domain are designed to preserve dynamic features of the scalar physical property and we provide comparisons among reduced and high order fidelity models.

We present the mathematical equations that govern heat transfer in a polymer melt flowing in a circular tube with constant ambient temperature, taking into account the viscous dissipation effects. This leads to a nonlinear parabolic partial differential equation. It is shown that the exact solution of a linearized version of the governing equation can be presented in terms of the Whittaker function. A finite difference scheme is used to produce an approximate solution of the linearized problem. This numerical solution is shown to be a good approximation to the exact solution found in terms of the Whittaker function. The finite difference scheme is then modified to approximate the nonlinear parabolic partial differential equation and is compared with the results found using the finite element method. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 102: 289–294, 2006

In this paper, an approximate solution of non-linear parabolic partial differential equation is obtained for a non-uniform mesh. The scheme for partial differential equation subject to Neumann boundary is based on cubic B-spline collocation method. Modified cubic B-splines are proposed over non-uniform mesh to deal with the Dirichlet boundary conditions. This scheme produces a system of first order ordinary differential equations. This system is solved by Crank Nicholson method. The stability is also discussed using Von Neumann stability analysis. The accuracy and efficiency of the scheme is shown by numerical experiments. We have compared the approximate solutions with that in the literature.

This article tries to use of cubic B-spline collocation method in the form of a difference method to solving system of one dimension nonlinear parabolic partial differential equations. This method is characterized by the simplicity of the finite difference method through discussion of various parabolic pde problems, but does not have complex computation of traditional cubic B-spline collocation method. The use of this method leads to a system of algebraic equations which is suitable for computer programming. The stability and convergence of this method is discussed. The obtained numerical solutions indicate that the method is reliable and yields results compatible with the exact solutions. The numerical approximate solutions to the nonlinear parabolic partial differential equations have been computed without transforming the equation and without using the linearization.

The Hardy inequality and nonlinear parabolic equations on Carnot groups

Coefficient identification in parabolic equations

In this paper, we derive a comparison principle for non-negative weak sub- and super-solutions to doubly nonlinear parabolic partial differential equations whose prototype is ∂tuq-div(|∇u|p-2∇u)=0inΩT,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u^q - {{\\,\ extrm{div}\\,}}{\\big (|\ abla u|^{p-2}\ abla u \\big )}=0 \\qquad \ ext{ in } \\Omega _T, \\end{aligned}$$\\end{document}with q>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q>0$$\\end{document} and p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document} and ΩT:=Ω×(0,T)⊂Rn+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _T:=\\Omega \ imes (0,T)\\subset \\mathbb {R}^{n+1}$$\\end{document}. Instead of requiring a lower bound for the sub- or super-solutions in the whole domain ΩT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _T$$\\end{document}, we only assume the lateral boundary data to be strictly positive. The main results yield some applications. Firstly, we obtain uniqueness of non-negative weak solutions to the associated Cauchy–Dirichlet problem. Secondly, we prove that any weak solution is also a viscosity solution.

In the optimal control theory, the Hamiltonian formalism is a famous one to find an optimal solution. However, when the performance index is complicated or for a degenerate case with a non-convexity of the Hamiltonian function with respect to the control force the Hamiltonian method does not work to find the solution. In this paper we will address this important issue via a quite different approach, which uses the optimal control problem of nonlinear Duffing oscillator as a demonstrative example. The optimally controlled vibration problem of nonlinear oscillator is recast into a nonlinear inverse problem by identifying the unknown heat source in a nonlinear parabolic partial differential equation (PDE). Then through a semi-discretization of the resultant PDE, the inverse problem is further reformulated to be a system of n-dimensional ODEs with n unknown pointwise sources, which allows a Lie-group adaptive method (LGAM) to recover the point-wise sources. The present method has three-fold advantages: it can easily minimize a complicated performance index to find an optimal control force of the nonlinear vibration system, it is effective for highly nonlinear optimal control problem, and it does not resort on the classical Hamiltonian formulation, which provides only a necessary condition, but not a sufficient condition, for the optimality of the control law. Numerical examples show that the LGAM may find a better performance than the classical one.

ELENA BERETTA, MICHIEL BERTSCH & ROBERTA DAL PASSO Communicated by M. GUaTIN O. Introduction In this paper we study nonnegative solutions of the fourth-order equation (0.1) ut + (u"uxxx)x = O, where n is a real positive constant. Equation (0.1) is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes. It arises in several applications (see the references in [2, 4, 5, 6]) in which u(x, t) stands for a nonnegative quantity. For example, in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant (nonnegative) quantity. The case n = 1 models the flow in a Hele-Shaw cell, while n = 3 corresponds to viscous flow on a solid surface without slip driven by surface tension (see also [6, 7, 8] for a related equation). It is well known that fourth-order equations do not satisfy a maximum principle. In particular, classical solutions of the linear equation ut + u .... ---- 0 may become negative in finite time. BERNIS 8~; FRIEDMAN [-2] have observed that if n is positive, equation (0.1) has (generalized) solutions which remain nonnegative for all times. In addition, they show that the qualitative positivity of these solutions strongly depends on the value of n. In more recent papers by KADANOFF et al., a combination of numerical calculations and matched asymptotics [4] and the analysis of travelling-wave solutions [-5] leads to results which suggest a very rich structure of the solutions of (0.1). It is the purpose of the present paper to clarify several aspects of the nonlinear equation (0.1) and its dependence on the parameter n. Our results lead to several conjectures and open problems. As was pointed out in [4], for sufficiently small values of n, solutions of (0.1) which are initially strictly positive may vanish at some point x 0 after a finite time to, but, to the best of our knowledge, there is no rigorous proof of this phenomenon in the literature. On the other hand, it is of particular interest for applications; for example, if u denotes the thickness of a liquid film, it implies that at time to the film breaks at xo.

Chapter 6 Stochastic Representations For Nonlinear Parabolic PDEs

It is known that the Swift-Hohenberg equation \documentclass[12pt]{minimal}\begin{document}$\partial u/\partial t = -(\partial _x^2 + 1)^2u + \varepsilon (u-u^3)$\end{document}∂u/∂t=−(∂x2+1)2u+ɛ(u−u3) can be reduced to the Ginzburg-Landau equation (amplitude equation) \documentclass[12pt]{minimal}\begin{document}$\partial A/\partial t\break = 4\partial _x^2 A + \varepsilon (A-3A|A|^2)$\end{document}∂A/∂t=4∂x2A+ɛ(A−3A|A|2) by means of the singular perturbation method. This means that if ɛ > 0 is sufficiently small, a solution of the latter equation provides an approximate solution of the former one. In this paper, a reduction of a certain class of a system of nonlinear parabolic equations \documentclass[12pt]{minimal}\begin{document}$\partial u/\partial t = \mathcal {P}u + \varepsilon f(u)$\end{document}∂u/∂t=Pu+ɛf(u) is proposed. An amplitude equation of the system is defined and an error estimate of solutions is given. Further, it is proved under certain assumptions that if the amplitude equation has a stable steady state, then a given equation has a stable periodic solution. In particular, near the periodic solution, the error estimate of solutions holds uniformly in t > 0.

This paper is mainly devoted to the boundary behavior of non-negative solutions to the equation $$\begin{aligned} \mathrm{H }u =\partial _tu-\nabla \cdot \mathrm{A }(x,t,\nabla u) = 0 \end{aligned}$$in domains of the form \(\Omega _T=\Omega \times (0,T)\) where \(\Omega \subset \mathbb R^n\) is a bounded non-tangentially accessible (NTA) domain and \(T>0\). The assumptions we impose on \(A\) imply that \(H\) is a non-linear parabolic operator with linear growth. Our main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary. Furthermore, to each such solution one can associate a natural Riesz measure supported on the lateral boundary and one of our main result is a proof of the doubling property for this measure. Our results generalize, to the setting of non-linear equations with linear growth, previous results concerning the boundary behaviour, in Lipschitz cylinders and time-independent NTA-cylinders, established for non-negative solutions to equations of the type \(\partial _tu-\nabla \cdot (\mathrm{A }(x,t)\nabla u)=0\), where \(\mathrm{A }\) is a measurable, bounded and uniformly positive definite matrix-valued function. In the latter case the measure referred to above is essentially the caloric or parabolic measure associated to the operator and related to Green’s function. At the end of the paper we also remark that our arguments are general enough to allow us to generalize parts of our results to general fully non-linear parabolic partial differential equations of second order.

Error estimates for discontinuous Galerkin method for nonlinear parabolic equations

The problem of calculating the evaporation from the soil surface is formulated as an optimal control problem. The controlled process of the vertical water transfer in soil is described by a onedimensional nonlinear second-order parabolic partial differential equation. The objective function is the squared Euclidean distance between the calculated values of the soil moisture at various depths and certain prescribed values. To improve the efficiency of finding a numerical solution, the sensitivity of the soil moisture at various depths to the variations of evaporation is estimated by means of fast automatic differentiation. The analysis of these estimates made it possible to determine an effective thin subsurface soil layer in which the moisture is most sensitive to the variations of evaporation; it is in this soil layer where the objective function should be calculated.

Via mobile sensing measurements, this study applies the Takagi-Sugeno (T-S) fuzzy model to deal with the mobile fuzzy control design problem for nonlinear time-delay parabolic partial differential equation (PDE) systems. Initially, we use a T-S fuzzy model to accurately represent the nonlinear time-delay parabolic PDE system. Subsequently, under the assumption that the actuators and sensors are collocated while the spatial domain is divided by several subdomains, a control scheme containing the fuzzy controllers and the guidance of mobile actuator/sensor pairs is proposed based on the obtained T-S fuzzy model, where the projection modification guidance to be designed can guarantee that each mobile actuator/sensor pair moves within the prescribed area. Then, using the Lyapunov direct method and integral inequalities, a membership-function-dependent design of fuzzy controllers plus mobile actuator/sensor guidance laws is developed to render the resulting closed-loop time-delay system exponentially stable. Moreover, the exponential decay rate can also be increased by the proposed mobile guidance laws. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method and the application of mobile actuator/sensor pairs contributes to accelerating the convergence speed of the closed-loop state.