A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations

A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition

This note considers discrete-time systems with several small singular perturbation parameters. The aim is to explore analogies with the (continuous-time) multiparameter singular perturbation theory. The models considered are obtained first as discretizations of continuous-time multiparameter singularly perturbed systems, and then noted to be natural extensions of standard discrete-time single parameter singular perturbation problems. Particular attention is paid to the stability characteristics of the new models. The conclusions reached on this issue lead to the suggestion of more exotic multiparameter discrete-time models due to the richer stability properties which can be achieved.

The slow and fast reduced‐order observers and reduced‐order observer‐based controllers are designed by using the two‐stage feedback design technique for slow and fast subsystems. The new designs produce an arbitrary order of accuracy, while the previously known designs produce the accuracy of O(ϵ) only where ϵ is a small singular perturbation parameter. Several cases of reduced‐order observer designs are considered depending on the measured state space variables: only all slow variables are measured, only all fast variables are measured, and some combinations of the slow and fast variables are measured. Since the two‐stage methods have been used to overcome the numerical ill‐conditioning problem for Cases (III)–(V), they have similar procedures. The numerical ill‐conditioning problem is avoided so that independent feedback controllers can be applied to each subsystem. The design allows complete time‐scale separation for both the reduced‐order observer and controller through the complete and exact decomposition into slow and fast time scales. This method reduces both offline and online computations.

In this article, we present a parameters‐uniform numerical method for a time‐dependent weakly coupled system of two convection‐diffusion equations that has a discontinuity, along the line in the source term. The second order term of each equation is multiplied by a small singular perturbation parameter and these parameters are assumed to be different in magnitude. On an appropriate Shishkin mesh, the considered problem is discretized using an upwind central difference scheme away from the line of discontinuity, and a particular upwind central difference scheme along the line of discontinuity. The numerical approximations produced by this scheme are uniformly convergent of first‐order in time and almost first‐order in space with respect to both perturbation parameters. Numerical results are presented to validate the theoretical results.

A parameter-uniform Schwarz method for a coupled system of reaction–diffusion equations

Singular perturbation of boundary value problem for quasilinear third-order ordinary differential equations involving two small parameters

In this study, a singularly perturbed linear impulsive system with singularly perturbed impulses is considered. Many books discuss different types of singular perturbation problems. In the present work, an impulse system is considered in which a small parameter is introduced into the impulse equation. This is the main novelty of our study, since other works [25] have only considered a small parameter in the differential equation. A necessary condition is also established to prevent the impulse function from bloating as the parameter approaches zero. As a result, the notion of singularity for discontinuous dynamics is greatly extended. An asymptotic expansion of the solution of a singularly perturbed initial problem with an arbitrary degree of accuracy for a small parameter is constructed. A theorem for estimating the residual term of the asymptotic expansion is formulated, which estimates the difference between the exact solution and its approximation. The results extend those of [32], which formulates an analogue of Tikhonov’s limit transition theorem. The theoretical results are confirmed by a modelling example.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Allaberen Ashyralyev, Deniz Ağirseven; Finite Difference Method for Delay Parabolic Equations. AIP Conf. Proc. 22 September 2011; 1389 (1): 573–576. https://doi.org/10.1063/1.3636795 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

The initial-boundary value problem for the delay parabolic partial differential equation with nonlocal conditions is studied. The convergence estimates for solutions of first and second order of accuracy difference schemes in Holder norms are obtained. The theoretical statements are supported by a numerical example.

Two point Singularly Perturbed Weakly Coupled System of Convection-Reaction-Diffusion Problems(WCSCRDPs), having small parameters (ε1, ε2) multiplying the highest derivative (diffusion) and the next highest derivative (convection) term with discontinuity over the source term is studied. Generally, for adequately small values of the perturbation parameters, the exact solution involves boundary and interior layers. Hence, the analysis of the problem splits into two cases, in accordance with the ratio of the convection and diffusion parameters. The continuous bounds of the solution and its derivatives are derived. A numerical method which is parameter uniform is constructed, on a suitably fitted piecewise uniform Shishkin mesh. The error estimation analysis is discussed and the numerical method applied here exhibits almost first order convergence. Numerical examples and the corresponding tables and figures demonstrate the efficiency of the numerical scheme.

In this paper, a numerical method for a weakly coupled system of singularly perturbed convection–diffusion second order ordinary differential equations with discontinuous source term subject to Robin type boundary conditions is presented. A fitted mesh method has been used to obtain the difference scheme to solve the system of equations on a piecewise uniform Shishkin mesh. An error estimate is derived to show that the method is uniformly convergent with respect to singular perturbation parameter. Numerical results are provided to illustrate the theoretical results.

In this paper, parameter - uniform numerical methods for singularly perturbed ordinary differential equations containing two small parameters are studied.Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. A numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are given to illustrate the parameter-uniform convergence of numerical approximations.

In this paper, we tackle the problem of stabilizing singularly perturbed switched affine systems. Even though efficient control strategies based on the solution of Linear Matrix Inequalities (LMIs) have been presented in the switched affine systems literature, the presence of the small singular perturbation parameter may introduce numerical difficulties in the solution of these LMIs due to ill conditioning. Moreover, this parameter may be uncertain or its use in the control law may not be practical. Here, we propose an LMI-based control design strategy that avoids the conditioning issues and that also provides an estimation of the domain of attraction. Furthermore, the resulting controller does not depend on the singular perturbation parameter. The proposed method is illustrated on a numerical example.

This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting is used to establish the approximating equation of the system for a sufficiently small singular perturbation parameter. The approximating equation is a stochastic parabolic equation when the power exponent of the singular perturbation parameter is in $[1/2, 1)$ but is a deterministic wave equation when the power exponent is in $(1, +\infty)$. Moreover, if the power exponent of a singular perturbation parameter is bigger than or equal to $1/2$, the same limiting equation of the system is derived in the sense of distribution, as the perturbation parameter tends to zero. This limiting equation is a deterministic parabolic equation.

The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed modelordinary different alequations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounded with respect to the perturbat on parameter. For the same classcal finite difference operator on an appropriate piecewise -uniform mesh, it is shown that the numerical solutions converge, uniformly with respect to the perturbation parameter, to the exact solution of any problem from this class.KeywordsNeumann Boundary ConditionPerturbation ParameterUniform MeshDiscrete SolutionMesh RegionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.