THE SUPERCLOSENESS OF THE FINITE ELEMENT METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM ON A BAKHVALOV-TYPE MESH IN 2D

For singularly perturbed convection-diffusion problems,supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes,especially in the case of 2D.The difficulties arise from the width of the mesh in the layer adjacent to the transition point,resulting in a suboptimal estimate for convergence.Existing analysis techniques cannot handle these difficulties well.To fill this gap,here a novel interpolation is designed delicately for the smooth part of the solution,bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method.Our theoretical result is uniform in the singular perturbation parameterεand is supported by the numerical experiments.

Adaptive Optimal Output Regulation of Interconnected Singularly Perturbed Systems With Application to Power Systems

This article studies the adaptive optimal output regulation problem for a class of interconnected singularly perturbed systems(SPSs) with unknown dynamics based on reinforcement learning(RL).Taking into account the slow and fast characteristics among system states,the interconnected SPS is decomposed into the slow time-scale dynamics and the fast timescale dynamics through singular perturbation theory.For the fast time-scale dynamics with interconnections,we devise a decentralized optimal control strategy by selecting appropriate weight matrices in the cost function.For the slow time-scale dynamics with unknown system parameters,an off-policy RL algorithm with convergence guarantee is given to learn the optimal control strategy in terms of measurement data.By combining the slow and fast controllers,we establish the composite decentralized adaptive optimal output regulator,and rigorously analyze the stability and optimality of the closed-loop system.The proposed decomposition design not only bypasses the numerical stiffness but also alleviates the high-dimensionality.The efficacy of the proposed methodology is validated by a load-frequency control application of a two-area power system.

Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

Chien-physics-informed neural networks for solving singularly perturbed boundary-layer problems

A physics-informed neural network(PINN)is a powerful tool for solving differential equations in solid and fluid mechanics.However,it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives.In this paper,we introduce Chien's composite expansion method into PINNs,and propose a novel architecture for the PINNs,namely,the Chien-PINN(C-PINN)method.This novel PINN method is validated by singularly perturbed differential equations,and successfully solves the wellknown thin plate bending problems.In particular,no cumbersome matching conditions are needed for the C-PINN method,compared with the previous studies based on matched asymptotic expansions.

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.

THE UNIFORM CONVERGENCE OF A DG METHOD FOR A SINGULARLY PERTURBED VOLTERRA INTEGRO-DIFFERENTIAL EQUATION

The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided.Then the existence and uniqueness of the DG solution are proven.Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established.Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants,the DG method achieves the uniform convergence in the L2 norm with respect to the singular perturbation parameter e when the space of polynomials with degree p is used.A numerical experiment validates the theoretical results.Furthermore,an ultra-convergence order 2p+1 at the nodes for the one-sided flux,uniform with respect to the singular perturbation parameter e,is observed numerically.

A Class of the Singularly Perturbed Boundary Value Problems for Elliptic Equation with a Super Surface of Turning Point in n-dimensional Space

In this paper, we study a class singular perturbed elliptic equation boundary value problem with a super surface of turning point in n-dimensional space by using the method of multiple scales and the comparison theorem. The uniformly valid asymptotic approxmations of solutions for the boundary value problem is constructed.

The Singularly Perturbed Boundary Value Problems for Elliptic Equation with Turning Point

The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.

Uniform Convergence Analysis for Singularly Perturbed Elliptic Problems with Parabolic Layers

In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.

On the model-based networked control for singularly perturbed systems

In this paper, the control of a two-time-scale plant, where the sensor is connected to a linear controller/ actuator via a network is addressed. The slow and fast systems of singularly perturbed systems are used to produce an estimate of the plant state behavior between transmission times, by which one can reduce the usage of the network. The approximate solutions of the whole systems are derived and it is shown that the whole systems via the network control are generally asymptotically stable as long as their slow and fast systems are both stable. These results are also extended to the case of network delay.

State feedback design for singularly perturbed system using unified approach

The state feedback design for singularly perturbed systems described in Delta operator is considered. The composite state feedback controller for slow and fast subsystems is designed by using the direct method. The obtained results can bring previous conclusions of continuous and discrete time systems into the unified Delta framework. A simulation example is presented to demonstrate the validity and efficiency of the design.

Numerical Solution of Quasilinear Singularly Perturbed Problems by the Principle of Equidistribution

In this paper, the numerical solution and its error analysis of quasilinear singular perturbation two-point boundary value problems based on the principle of equidistribution are given. On the non-uniform grid of the uniformly distributed arc-length monitor function, the solution of the simple upwind scheme is obtained. It is proved that the adaptive simple upwind scheme based on the principle of equidistribution has uniform convergence for small perturbation parameters. Numerical experiments are carried out and the error analysis are confirmed.

Optimal deterministic disturbances rejection for singularly perturbed linear systems

Optimal deterministic disturbances rejection control problem for singularly perturbed linear systems is considered. By using the slow-fast decomposition theory of singular perturbation, the existent and unique conditions of the feedforward and feedback composite control （FFCC） laws for both infinite-time and finite-time are proposed, and the design approaches are given. A disturbance observer is introduced to make the FFCC laws realizable physically. Simulation results indicate that the FFCC laws are robust with respect to external disturbances.

A SECOND ORDER UNIFORM DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED TURNING POINT PROBLEM

We construct a positive type difference scheme for a singularly perturbed boundary value problem with a turning point. It's proved that this scheme is the second order convergence, uniformly in ? , to the solution of the singularly perturbed B. V.P. Numerical examples are provided.

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.

Ill-conditioned stable inversion arising from singularly perturbed zero dynamics

The ill-conditioned stable inversion is studied for slightly nonminimum phase systems whose zero dynam- ics is singularly perturbed, that is, the relative degree is ill-defined. For these systems, we show that there exists an inherent limitation in the bandwidth of a reference trajectory to be tracked when a well-conditioned feedforward input via stable inversion is sought. We assert that, when the violation of this limitation occurs, the so-called reference trajectory redesign is called for. Our analysis results can provide an explicit assessment as well as useful guidance for the reference trajectory redesign if needed.

Optimal disturbances rejection control for singularly perturbed systems with time-delay

The optimal control design for singularly perturbed time-delay systems affected by external distur-bances is considered.Based on the decomposition theory of singular perturbation,the system is decom-posed into a fast subsystem without time-delay and a slow time-delay subsystem with disturbances.Theoptimal disturbances rejection control law of the slow subsystem is obtained by using the successive ap-proximation approach(SAA)and feedforward compensation method.Further,the feedforward and feed-back composite control(FFCC)law for the original problem is developed.The FFCC law consists of lin-ear analytic terms and a time-delay compensation term which is the limit of the solution sequence of theadjoint vector equations.A disturbance observer is introduced to make the FFCC law physically realiz-able.Numerical examples show that the proposed algorithm is effective.

BOUNDARY AND ANGULAR LAYER BEHAVIOR IN SINGULARLY PERTURBED SEMILINEAR SYSTEMS

Some authors employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ?-n+ , of the solutions of scalar boundary value problemsIn this paper, we extend these results to vector boundary value problems, under analogous stability conditions on the solution u = u(t) of the reduced equation 0 = h(t, u) Two types of asymptotic behavior are studied, depending on whether the reduced solution u(f) has or does not have a con tinuous first derivative in (a, b) leading to the phenomena of boundary and angular layers.

UNIQUENESS OF SOLUTIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR THIRD QRDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].

A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.