Parabolic Boundary Value Problems Research Articles

Application of AGE Method to High Accuracy Variable Mesh Arithmetic Average Type Discretization for 1D Non-linear Parabolic Initial Boundary Value Problems

In this article, we discuss the application of arithmetic average discretization alternating group explicit (A-AGE) method to a new two-level implicit variable mesh formula of O(k2hl − 1+ khl + hl 3) for the solution of non-linear parabolic equation uxx = F(x,t,u,ux, ut) subject to appropriate initial and Dirichlet boundary conditions, where k > 0 and hl > 0 are the variable step lengths in time and space directions, respectively. The proposed technique is also useful to solve parabolic singular problems. Our discretization procedure requires only 3-spatial grid points. A-AGE and Newton-A-AGE algorithms are discussed in detail. Comparison of numerical results of these methods with the corresponding successive over relaxation (SOR) and Newton-SOR methods is also included.

Stability of the coincidence set of a solution to a parabolic variational inequality with an obstacle

In this paper we propose a new technique for the stability analysis of the coincidence set of a solution to a parabolic variational inequality with an obstacle inside the domain. It is based on the reformulation of the initial inequality in the form of a parabolic initial boundary value problem with an exact penalty operator.

Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems

We prove local existence, uniqueness, Holder regularity in space and time, and smooth dependence in Holder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to abstract formulations of the initial boundary value problems, has been closed. The theory works for any space dimension, and the nonlinearities in the equations as well as in the boundary conditions are allowed to be nonlocal and to have any growth. The main tools are new maximal regularity results (Griepentrog in Adv Differ Equ 12:781–840, 1031–1078, 2007) in Sobolev–Morrey spaces for linear parabolic initial boundary value problems with nonsmooth data, linearization techniques and the Implicit Function Theorem.

On the use of AGE algorithm with a high accuracy Numerov type variable mesh discretization for 1D non-linear parabolic equations

In this paper, the use of N-AGE and Newton-N-AGE iterative methods on a variable mesh for the solution of one dimensional parabolic initial boundary value problems is considered. Using three spatial grid points, a two level implicit formula based on Numerov type discretization is discussed. The local truncation error of the method is of \(O({k^2h_l^{-1} +kh_l +h_l^3})\), where h l > 0 and k > 0 are the step lengths in space and time directions, respectively. We use a special technique to handle singular parabolic equations. The advantage of using these algorithms is highlighted computationally.

Asymptotic approximations of solutions to parabolic boundary value problems in thin perforated domains of rapidly varying thickness

We consider initial boundary value problems for parabolic differential equations with rapidly oscillating coefficients in thin perforated domains of rapidly varying thickness. Under certain symmetry conditions on the domain and coefficients, we construct an asymptotic expansion of a solution to the problem with homogeneous third kind conditions on the exterior boundary and the boundary of cavities. In the case of inhomogeneous Neumann conditions, we construct an asymptotic solution without symmetry assumptions and prove an asymptotic estimate in the corresponding Sobolev space. Bibliography: 27 titles. Illustrations: 1 figure.

EXISTENCE OF MULTIPLE PERIODIC SOLUTIONS FOR SEMILINEAR PARABOLIC EQUATIONS WITH SUBLINEAR GROWTH NONLINEARITIES

In this paper, we establish a multiple existence result of T-periodic solutions for the semilinear parabolic boundary value problem with sublinear growth nonlinearities. We adapt sub-supersolution scheme and topological argument based on variational structure of functionals.

On the numerical solution of the heat conduction equations subject to nonlocal conditions

Many physical phenomena are modelled by nonclassical parabolic boundary value problems with nonlocal boundary conditions. Many different papers studied the second-order parabolic equation, particularly the heat equation subject to the specifications of mass. In this paper, we provide a whole family of new algorithms that improve the CPU time and accuracy of Crandall's formula shown in [J. Martin-Vaquero, J. Vigo-Aguiar, A note on efficient techniques for the second-order parabolic equation subject to non-local conditions, Appl. Numer. Math. 59 (6) (2009) 1258–1264] (and this algorithm improved the results obtained with BTCS, FTCS or Dufort–Frankel three-level techniques previously used in other works, see [M. Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math. 52 (2005) 39–62]) with this kind of problems. Other methods got second or fourth order only when k = s h 2 , while the new codes got nth order for k = h ; therefore, the new schemes require a smaller storage and CPU time employed than other algorithms. We will study the convergence of the new algorithms and finally we will compare the efficiency of the new methods with some well-known numerical examples.

Parabolic equations related to Boltzmann measure

In this paper we study a class of parabolic initial boundary value problems relative to an operator whose the prototype is \({{u}_{t}{-Z{\rm e}}^{{W(x)}}{\rm div}({\nabla uZ^{-1} {\rm e}}^{{-W(x)}})=g,}\) where W(x) is a smooth function and Z is a constant. We obtain an estimate of the solution comparing it with the solution to a problem relative to the operator \({{{u}}_{t}{{-}}\frac{1}{{{\varphi(x)}}}{(u}_{{x}_{{1}}}{\varphi(x))}_{{x}_{{1}}}{{=}g}^{\blacklozenge },}\) where \({{{\varphi(x)}}}\) is the density of Gauss measure, \({{g}^{\blacklozenge}}\) is a function related to g and the data depend only on the time variable and the first space variable.

Two-level fourth-order explicit schemes for diffusion equations subject to boundary integral specifications

Several processes in the natural sciences and engineering lead to nonclassical parabolic initial boundary value problems with nonlocal boundary conditions. Many authors have studied second-order parabolic equations, particularly the heat equation with this kind of condition. In [19], several methods were compared to approach the numerical solution of the one-dimensional heat equation subject to the specifications of mass. Here, two two-level fourth-order explicit algorithms are derived. They improve the CPU time and accuracy of other explicit and implicit schemes seen with this kind of problems. The convergence of the algorithms is studied and finally, numerical examples are given to compare the efficiency of the new methods with others used for this kind of problem.

An iterative representer-based scheme for data inversion in reservoir modeling

In this paper, we develop a mathematical framework for data inversion in reservoir models. A general formulation is presented for the identification of uncertain parameters in an abstract reservoir model described by a set of nonlinear equations. Given a finite number of measurements of the state and prior knowledge of the uncertain parameters, an iterative representer-based scheme (IRBS) is proposed to find improved parameters. In this approach, the representer method is used to solve a linear data assimilation problem at each iteration of the algorithm. We apply the theory of iterative regularization to establish conditions for which the IRBS will converge to a stable approximation of a solution to the parameter identification problem. These theoretical results are applied to the identification of the second-order coefficient of a forward model described by a parabolic boundary value problem. Numerical results are presented to show the capabilities of the IRBS for the reconstruction of hydraulic conductivity from the steady-state of groundwater flow, as well as the absolute permeability in the single-phase Darcy flow through porous media.

A Posteriori Error Estimates for Optimal Control Problems

We present new a posteriori error estimates for approximate solutions of optimal control problems. Our approach is based on so-called functional a posteriori error estimates that were derived in the last decade for many elliptic and parabolic boundary value problems. They provide guaranteed two-sided bounds of approximation errors for any conforming approximation of a problem considered. In the paper, we consider optimal control problems with distributed control functions that enter right-hand sides of elliptic type state equations and apply functional a posteriori estimates to approximate solutions of state equations. This procedure leads to guaranteed two-sided estimates for the cost functional and also allows to obtain guaranteed a posteriori estimates for state and control functions.

NUMERICAL QUENCHING FOR A SEMILINEAR PARABOLIC EQUATION

This paper concerns the study of the numerical approximation for the nonlinear parabolic boundary value problem with the source term leading to the quenching in finite time. We find some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also prove that the semidiscrete quenching time converges to the real one when the mesh size goes to zero. A similar study has been also investigated taking a discrete form of the above problem. Finally, we give some numerical experiments to illustrate our analysis.

Boundary-value problem for a parabolic system of integro-differential equations with integral conditions

Using operators of fractional integration and differentiation, we prove a theorem on the wellposedness of a general parabolic boundary-value problem for a system of integro-differential equations with integral operators in boundary conditions.

Space-time adaptive wavelet methods for parabolic evolution problems

With respect to space-time tensor-product wavelet bases, parabolic initial boundary value problems are equivalently formulated as bi-infinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spatial domain is of product type, the use of spatial tensor product wavelet bases is proved to overcome the so-called curse of dimensionality, i.e., the reduction of the convergence rate with increasing spatial dimension.

Estimation of derivatives with respect to parameters of a functional of a diffusion process moving in a domain with absorbing boundary

A statistical method is proposed for estimating derivatives with respect to parameters of a functional of a diffusion process moving in a domain with absorbing boundary. The functional considered defines the probability representation of the solution of a corresponding parabolic first boundary-value problem. The problem posed is tackled by numerically solving stochastic differential equations (SDE) using the Euler method. An error of the proposed method is evaluated, and estimates of the variance of the resultant parametric derivatives are given. Some numerical results are presented.

Maximal [formula omitted]-regularity of parabolic problems with boundary dynamics of relaxation type

In this paper we investigate vector-valued parabolic initial boundary value problems of relaxation type. Typical examples for such boundary conditions are dynamic boundary conditions or linearized free boundary value problems like in the Stefan problem. We present a complete L p -theory for such problems which is based on maximal regularity of certain model problems.

Parabolic boundary value problems connected with Newton’s polygon and some problems of crystallization

A new class of boundary value problems for parabolic operators is introduced which is based on the Newton polygon method. We show unique solvability and a priori estimates in corresponding L 2-Sobolev spaces. As an application, we discuss some linearized free boundary problems arising in crystallization theory which do not satisfy the classical parabolicity condition. It is shown that these belong to the new class of parabolic boundary value problems, and two-sided estimates for their solutions are obtained.

A note on efficient techniques for the second-order parabolic equation subject to non-local conditions

Many physical phenomena are modelled by non-classical parabolic boundary value problems with non-local boundary conditions. In [M. Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math. 52 (2005) 39–62], several methods were compared to approach the numerical solution of the one-dimensional heat equation subject to the specifications of mass. One of them was the ( 3 , 3 ) Crandall formula. The scheme showed in Eq. (64) in that paper is of order O ( h 2 ) , not of order O ( h 4 ) as proposed by that author. However, it is possible with several changes to derive a Crandall algorithm of order O ( h 4 ) . Here, we compare the efficiency of the new method with the previous results in the same tests, and we reach errors 10 3 to 10 5 times smaller with the new scheme.

On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems

We deal with the stability analysis of difference schemes for a one-dimensional parabolic equation subject to integral conditions. It is based on the spectral structure of the transition matrix of a difference scheme. The stability domain is defined by using the hyperbola which is the locus of points where the transition matrix has trivial eigenvalues. The stability conditions obtained are much more general compared with those known in the literature. We analyze three separate cases of nonlocal integral conditions and solve an example illustrating the efficiency of the technique.

Some convergence results for quasilinear parabolic boundary value problems in cylindrical domains of large size

The goal of this paper is to study the asymptotic behavior of the solution of the quasilinear parabolic boundary value problems defined on cylindrical domains when one or several directions go to infinity. We show that the dimension of the space can be reduced and the rate of convergence is analyzed. The evolution p -Laplacian equations and the generalized heat problems are considered.