Linear Parabolic Partial Differential Equations Research Articles

A uniform solution of a parabolic problem with angular layer behavior

For a singularly perturbed linear parabolic partial differential equation with a C1-discontinuous initial data, the solution possesses an angular layer behavior. We construct an angular layer equation subject to auxiliary conditions. The angular layer function can be written in terms of the first integral of the complementary error function after a careful analysis on the C1-jump discontinuity of the outer solution. An asymptotic expansion obtained is shown to be uniformly valid with the first order accuracy in the small parameter by employing the maximum principle and some exponentially decay estimate of the angular layer function.

Finite-dimensional approximation and error bounds for spectral systems with partially known eigenstructure

An approach is presented for directly computing bounds on the frequency-response error between finite-dimensional modal models and the full infinite-dimensional models of systems described by certain classes of linear hyperbolic and parabolic partial differential equations (PDE's). The models and bounding techniques are developed specifically to be computable when applied to hyperbolic and parabolic systems with spatially variant parameters, complicated boundary shapes, and other cases where the eigenstructure is not available in closed form and must be computed numerically. A controller design example is presented to illustrate the utility of this approach. >

Exponential split operator methods for solving coupled time-dependent Schrödinger equations

Coherent excitation of molecules with laser pulses are usually described by coupled time-dependent linear parabolic partial differential equations, i.e., Schrödinger equations. Numerical solutions of these equations based on splitting (factorization) of the exponential form of the evolution operator or time-dependent propagator are examined for accuracy of amplitude and phase as a function of various unitary splitting schemes.

Identification of transmissivity coefficients by mollification techniques. Part I: One-dimensional elliptic and parabolic problems

We discuss the numerical identification of the transmissivity coefficient in the one-dimensional linear and nonlinear elliptic and parabolic partial differential equations. This is a quite general problem arising—for example—in porous media and related to groundwater modeling and reservoir simulation. This problem belongs to the class field of inverse problems. It is well-known that it is ill-posed because small errors in the data might produce large errors in the computed solution. We combine the mollification techniques and finite differences to address the ill-posedness of this parameter estimation problem. Several numerical experiments support the stability and convergence of the algorithms.

Sinc-Galerkin estimation of diffusivity in parabolic problems

A fully Sinc-Galerkin method for the numerical recovery of spatially varying diffusion coefficients in linear parabolic partial differential equations is presented. Because the parameter recovery problems are inherently ill-posed, an output error criterion in conjunction with Tikhonov regularization is used to formulate them as infinite-dimensional minimization problems. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which displays an exponential convergence rate and is valid on the infinite time interval. The minimization problems are then solved via a quasi-Newton/trust region algorithm. The L-curve technique for determining an appropriate value of the regularization parameter is briefly discussed, and numerical examples are given which demonstrate the applicability of the method both for problems with noise-free data as well as for those whose data contain white noise.

Impulse response model for a class of distributed parameter systems

A mathematical model is developed here for the impulse response of a class of systems represented by a general linear parabolic partial differential equation. Using a generalized eigenfuction expansion and the concept of a sampled-data system, a canonical set of state equations is obtained in discrete form. The impulse inputs may be any known time-varying functions on the given boundaries of the system or may be assumed to be distributed over a finite boundary region. For this computer model, no spatial discretization is required. Each discrete-time equation represents the dynamic behavior of each eigenvalue of the system and the response variable at any discrete location is given by a spatial linear combination of these state variables. Due to the discrete formulation of the system equations, numerical stability is assured. The equations are stable for any sampling time selected and the solution at any particular time desired may be simply obtained by incorporating this value directly into the model equations. The application of this method is demonstrated by solving a two-dimensional heat transfer problem subject to a single impulse input and comparisons were made to the finite element method of solution.

On the existence and pointwise stability of periodic solutions to a linear parabolic equation in unbounded domains

It is proved that a linear parabolic partial differential equation in an unbounded domain admits a unique time periodic solution, provided the data are time-periodic. Moreover, pointwise stability of solutions are established via a suitable maximum principle.

Stability Criterion For Linear Oscillatory Parabolic Systems

This paper presents a stability criterion for a class of distributed parameter systems governed by linear oscillatory parabolic partial differential equations with Neumann boundary conditions. The results of numerical simulations that support the criterion are presented as well.

Error estimates for the semidiscrete finite element approximation of linear nonlocal parabolic equations

Existence and uniqueness are proved for nonlocal (in time) for solutions of linear parabolic partial differential equations. Instead of an initial condition, there is a relation connecting the initial value to values of the solution at other times. L2 error estimates are obtained for the semidiscrete approximation of the problem using finite elements in the space variables.

THE PARALLEL PERFORMANCE OF STANDARD PARABOLIC MARCHING SCHEMES

We compare standard parallel algorithms for solving linear parabolic partial differential equations. The comparison is based on the combined effect of their numerical properties and their parallel performance. We discuss the classical explicit methods (forward Euler, Heun and DuFort-Frankel), the standard implicit methods (BDF1, BDF2 and Crank-Nicolson), the line Hopscotch technique and the ADI formula of McKee and Mitchell. Timing results obtained on a 16-processor Intel hypercube are given. It is shown that parallelism does not alter the ranking of the methods unless the number of grid points per processor is very small.

Determination of initial states of parabolic systems from discrete data

The authors investigate the problem of recovering the initial states of distributed parameter systems, governed by linear parabolic partial differential equations, from finite approximate data. It is shown that an approximation of the true initial state can be obtained from the solution of an appropriate finite-dimensional algebraic linear system and that this approximation depends continuously on the observed data. Error estimates are also obtained.

A fully-Galerkin method for the numerical solution of an inverse problem in a parabolic partial differential equation

A fully-Galerkin approach to the coefficient recovery (parameter identification) problem for a linear parabolic partial differential equation is introduced. The forward problem is discretised with a sinc basis in the temporal domain and a finite element basis in the spatial domain. Tikhonov regularisation is applied to deal with the ill-posedness of the inverse problem. In the solution of the resulting nonlinear optimisation problem, advantage is taken of the diagonalisation solution procedure used for the discretised forward problem. An example with noisy data is included.

A zero-dimensional shock

Abstract : In this note, we add a new wrinkle to the very old problem of determining the motion of a mass point on a spring. We adopt a general model for the spring in which the force needed to compress it to zero length is infinite. (Consequently the motion is governed by a singular nonlinear second-order ordinary differential equation.) In this setting we entertain the possibility, permitted by the governing equations, that such a total compression is actually attained. This total compression corresponds to a kind of shock. We then extract from the governing equations all the illumination they can shed on the physical behavior. Our problem, which presents novel features for ordinary differential equations, captures in microcosm deep and unresolved issues involving shocks and their suppression, which arise in the study of quasi linear hyperbolic and parabolic partial differential equations. We comment briefly on these issues in Section 8.

Optimal control computation for parabolic systems with boundary conditions involving time delays

In this paper, we consider a class of optimal control problems involving a second-order, linear parabolic partial differential equation with Neumann boundary conditions. The time-delayed arguments are assumed to appear in the boundary conditions. A necessary and sufficient condition for optimality is derived, and an iterative method for solving this optimal control problem is proposed. The convergence property of this iterative method is also investigated.

Convergence of a feasible directions algorithm for a distributed optimal control problem of parabolic type with terminal inequality constraints

In this paper, we consider a class of optimal control problems with control and terminal inequality constraints, where the system dynamics is governed by a linear second-order parabolic partial differential equation with first boundary condition. A feasible direction algorithm for solving this class of optimal control problems has already been obtained in the literature. The aim of this paper is to improve the convergence result by using a topology arising in the study of relaxed controls.

The Parameter Estimation Problem for Parabolic Equations and Discontinuous Observation Operators

Parameter estimation problems are studied for a class of linear autonomous parabolic partial differential equations with various fit-to-data criteria, which may be discontinuous with respect to the state variable. We analyze the convergence of Galerkin schemes approximating the optimization problems and generalize results to higher dimensional problems. An example is then presented for the case of point observation fit-to-data criteria in higher dimensions. Finally, discretization of coefficients is discussed for identification problems with variable coefficients.

Qualitative analysis of a coupled reaction-diffusion model in biology with time delays

Certain biochemical reaction can be modeled by a coupled system of time-delayed ordinary differential equations and linear parabolic partial differential equations. In a three-compartment model these equations are coupled through the boundary conditions. The aim of this paper is to give a qualitive analysis of this unusual coupled system. The analysis includes the existence and uniqueness of a global solution, explicit upper and lower bounds of the solution, and global stability of a steady-state solution. The global stability result is with respect to any nonnegative initial perturbation and is independent of the time delays in the process of reaction. Special attention is given to the Goodwin model for biochemical control of genes by a negative feedback mechanism with time delay and diffusion.

Asymptotic analysis of P.D.E.s with wide–band noise disturbances, and expansion of the moments

We present an asymptotic analysis -in the “ white-noise limit”- of a linear parabolic partial differential equation, whose coefficients are perturbed by a wide-band noise. After having studied some ergodic properties of a class of diffusion processes, we prove the convergence in law towards the solution of an Ito stochastic P.D.E. We then establish an expansion in powers of Δ ( 1/Δ being a measure of the bandwith of the driving noise) of the first moment of the solution

Parameter identifiability for partial differential equations

The direct method of parameter identifiability is shown to hold for linear parabolic partial differential equations. This is in contrast to other methods of parameter identifiability which, while they hold for finite dimensional lumped systems, fail to extend to partial differential equations. It is hoped that the direct method will provide some insight into the many unanswered questions concerning parameter identifiability of partial differential equations.

Time suboptimal feedback control of systems described by linear parabolic partial differential equations

AbstractFeedback control for systems described by linear parabolic partial differential equations is considered from a time suboptimal point of view. A scalar quadratic function representing the deviation from a desired distribution is first formulated, and the rate of approach to the target is maximized by minimizing this quadratic function at each time step. An orthogonal collocation technique enables an accurate low‐order lumped parameter model to be constructed. A direct search optimization can then be used to obtain the optimal quadratic function and, subsequently, the optimal feedback gain matrix for control. The resulting control applied to a diffusion process takes the system to the target rapidly and in a stable manner.