Nonlinear Partial Differential Equations (nonlinear + partial_differential_equation)

Distribution by Scientific Domains

Selected Abstracts

An interpolation-based local differential quadrature method to solve partial differential equations using irregularly distributed nodes

Hang Ma
Abstract To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation. Copyright 2007 John Wiley & Sons, Ltd. [source]

Robust detection and accommodation of incipient component and actuator faults in nonlinear distributed processes

AICHE JOURNAL, Issue 10 2008
Antonios Armaou
Abstract A class of nonlinear distributed processes with component and actuator faults is presented. An adaptive detection observer with a time varying threshold is proposed that provides additional robustness with respect to false declarations of faults and minimizes the fault detection time. Additionally, an adaptive diagnostic observer is proposed that is subsequently utilized in an automated control reconfiguration scheme that accommodates the component and actuator faults. An integrated optimal actuator location and fault accommodation scheme is provided in which the actuator locations are chosen in order to provide additional robustness with respect to actuator and component faults. Simulation studies of the Kuramoto-Sivashinsky nonlinear partial differential equation are included to demonstrate the proposed fault detection and accommodation scheme. 2008 American Institute of Chemical Engineers AIChE J, 2008 [source]


EVOLUTION, Issue 3 2004
Garrick T. Skalski
Abstract The spread of genes and individuals through space in populations is relevant in many biological contexts. I study, via systems of reaction-diffusion equations, the spatial spread of advantageous alleles through structured populations. The results show that the temporally asymptotic rate of spread of an advantageous allele, a kind of invasion speed, can be approximated for a class of linear partial differential equations via a relatively simple formula, c= 2,rD, that is reminiscent of a classic formula attributed to R. A. Fisher. The parameters r and D, represent an asymptotic growth rate and an average diffusion rate, respectively, and can be interpreted in terms of eigenvalues and eigenvectors that depend on the population's demographic structure. The results can be applied, under certain conditions, to a wide class of nonlinear partial differential equations that are relevant to a variety of ecological and evolutionary scenarios in population biology. I illustrate the approach for computing invasion speed with three examples that allow for heterogeneous dispersal rates among different classes of individuals within model populations. [source]

Numerical simulation of one-dimensional flows through porous media with shock waves

Maria Laura Martins-Costa
Abstract This work studies an unsaturated flow of a Newtonian fluid through a rigid porous matrix, using a mixture theory approach in its modelling. The mixture consists of three overlapping continuous constituents: a solid (porous medium), a liquid (Newtonian fluid) and an inert gas (to account for the mixture compressibility). A set of two nonlinear partial differential equations describes the problem, which is approximated by means of a Glimm's scheme, combined with an operator splitting technique. Copyright 2001 John Wiley & Sons, Ltd. [source]

Analysis of parameter sensitivity and experimental design for a class of nonlinear partial differential equations

Michael L. Anderson
Abstract The purpose of this work is to analyse the parameter sensitivity problem for a class of nonlinear elliptic partial differential equations, and to show how numerical simulations can help to optimize experiments for the estimation of parameters in such equations. As a representative example we consider the Laplace,Young problem describing the free surface between two fluids in contact with the walls of a bounded domain, with the parameters being those associated with surface tension and contact. We investigate the sensitivity of the solution and associated functionals to the parameters, examining in particular under what conditions the solution is sensitive to parameter choice. From this, the important practical question of how to optimally design experiments is discussed; i.e. how to choose the shape of the domain and the type of measurements to be performed, such that a subsequent inversion of the measured data for the model parameters yields maximal accuracy in the parameters. We investigate this through numerical studies of the behaviour of the eigenvalues of the sensitivity matrix and their relation to experimental design. These studies show that the accuracy with which parameters can be identified from given measurements can be improved significantly by numerical experiments. Copyright 2005 John Wiley & Sons, Ltd. [source]


ABSTRACT This article presents a mathematical model describing the unsteady heat and mass transfer during the freeze drying of biological materials. The model was built from the mass and energy balances in the dried and frozen regions of the material undergoing freeze drying. A set of coupled nonlinear partial differential equations permitted the description of the temperature and pressure profiles, together with the position of the sublimation interface. These equations were transformed to a finite element scheme and numerically solved using the Newton-Raphson approach to represent the nonlinear problem and the interface position. Most parameters involved in the model (i.e., thermal conductivity, specific heat, density, heat and mass transfer coefficients etc.) were obtained from experimental data cited in the literature. The dehydration kinetics and the temperature profiles of potato and apple slabs were experimentally determined during freeze drying. The simulation results agreed closely with the water content experimental data. The prediction of temperature profiles within the solid was, however, less accurate. [source]

On a class of PDEs with nonlinear distributed in space and time state-dependent delay terms

Alexander V. Rezounenko
Abstract A new class of nonlinear partial differential equations with distributed in space and time state-dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state-dependent delay and discuss advantages of this class. Local and long-time asymptotic properties, including the existence of global attractor and a principle of linearized stability, are studied. Copyright 2008 John Wiley & Sons, Ltd. [source]

Radial basis collocation method and quasi-Newton iteration for nonlinear elliptic problems

H.Y. Hu
Abstract This work presents a radial basis collocation method combined with the quasi-Newton iteration method for solving semilinear elliptic partial differential equations. The main result in this study is that there exists an exponential convergence rate in the radial basis collocation discretization and a superlinear convergence rate in the quasi-Newton iteration of the nonlinear partial differential equations. In this work, the numerical error associated with the employed quadrature rule is considered. It is shown that the errors in Sobolev norms for linear elliptic partial differential equations using radial basis collocation method are bounded by the truncation error of the RBF. The combined errors due to radial basis approximation, quadrature rules, and quasi-Newton and Newton iterations are also presented. This result can be extended to finite element or finite difference method combined with any iteration methods discussed in this work. The numerical example demonstrates a good agreement between numerical results and analytical predictions. The numerical results also show that although the convergence rate of order 1.62 of the quasi-Newton iteration scheme is slightly slower than rate of order 2 in the Newton iteration scheme, the former is more stable and less sensitive to the initial guess. 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source]

State estimation of a solid-state polymerization reactor for PET based on improved SR-UKF

Ji Liu
Abstract A state estimator for the continuous solid-state polymerization (SSP) reactor of polyethylene terephthalate (PET) is designed in this study. Because of its invalidity in the application to some of the practical examples such as SSP processes, the square-root unscented Kalman filter (SR-UKF) algorithm is improved for the state estimation of arbitrary nonlinear systems with linear measurements. Discussions are given on how to avoid the filter invalidation and accumulating additional error. Orthogonal collocation method has been used to spatially discretize the reactor model described by nonlinear partial differential equations. The reactant concentrations on chosen collocation points are reconstructed from the outlet measurements corrupted with a large noise. Furthermore, the error performance of the developed ISR-UKF is investigated under the influence of various initial parameters, inaccurate measurement noise parameters and model mismatch. Simulation results show that this technique can produce fast convergence and good approximations for the state estimation of SSP reactor. Copyright 2009 Curtin University of Technology and John Wiley & Sons, Ltd. [source]

Using Image and Curve Registration for Measuring the Goodness of Fit of Spatial and Temporal Predictions

BIOMETRICS, Issue 4 2004
Cavan Reilly
Summary Conventional measures of model fit for indexed data (e.g., time series or spatial data) summarize errors in y, for instance by integrating (or summing) the squared difference between predicted and measured values over a range of x. We propose an approach which recognizes that errors can occur in the x -direction as well. Instead of just measuring the difference between the predictions and observations at each site (or time), we first "deform" the predictions, stretching or compressing along the x -direction or directions, so as to improve the agreement between the observations and the deformed predictions. Error is then summarized by (a) the amount of deformation in x, and (b) the remaining difference in y between the data and the deformed predictions (i.e., the residual error in y after the deformation). A parameter, ,, controls the tradeoff between (a) and (b), so that as ,,, no deformation is allowed, whereas for ,= 0 the deformation minimizes the errors in y. In some applications, the deformation itself is of interest because it characterizes the (temporal or spatial) structure of the errors. The optimal deformation can be computed by solving a system of nonlinear partial differential equations, or, for a unidimensional index, by using a dynamic programming algorithm. We illustrate the procedure with examples from nonlinear time series and fluid dynamics. [source]