Uniformly convergent second-order difference scheme for a singularly perturbed periodical boundary value problem

A periodic boundary value problem with a small parameter multiplying the first- and second-order derivatives is considered. The problem is discretized using a hybrid difference scheme on a Shishkin mesh. We show that the scheme is almost second-order convergent in the maximum norm, which is independent of a singular perturbation parameter. Numerical experiment supports these theoretical results.     

Numerical solution of a singularly perturbed three-point boundary value problem

We consider a uniform finite difference method on an S-mesh (Shishkin type mesh) for a singularly perturbed semilinear one-dimensional convection–diffusion three-point boundary value problem with zeroth-order reduced equation. We show that the method is first-order convergent in the discrete maximum norm, independently of the perturbation parameter except for a logarithmic factor. An effective iterative algorithm for solving the non-linear difference problem and some numerical results are presented.     

A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises B-spline collocation method on a non-uniform mesh of Shishkin type. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The present method is boundary layer resolving as well as second-order accurate in the maximum norm. A brief analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter ? by decomposing the solution into smooth and singular components. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.     

A fully discrete ε-uniform method for singular perturbation problems on equidistant meshes

We propose a fully discrete ε-uniform finite-difference method on an equidistant mesh for a singularly perturbed two-point boundary-value problem (BVP). We start with a fitted operator method reflecting the singular perturbation nature of the problem through a local BVP. However, to solve the local BVP, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. Thus, we show that it is possible to develop a ε-uniform method, totally in the context of finite differences, without solving any differential equation exactly. We further study the convergence properties of the numerical method proposed and prove that it nodally converges to the true solution for any ε. Finally, a set of numerical experiments is carried out to validate the theoretical results computationally.     

A second-order finite difference scheme for a class of singularly perturbed delay differential equations

In this paper a class of delay differential equations with a perturbation parameter ? is examined. A hybrid finite difference scheme on an appropriate piecewise uniform mesh of Shishkin-type is derived. We show that the scheme is almost second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.     

A high-order finite difference scheme for a singularly perturbed fourth-order ordinary differential equation

In this paper a singularly perturbed fourth-order ordinary differential equation is considered. The differential equation is transformed into a coupled system of singularly perturbed equations. A hybrid finite difference scheme on a Vulanovi?–Shishkin mesh is used to discretize the system. This hybrid difference scheme is a combination of a non-equidistant generalization of the Numerov scheme and the central difference scheme based on the relation between the local mesh widths and the perturbation parameter. We will show that the scheme is maximum-norm stable, although the difference scheme may not satisfy the maximum principle. The scheme is proved to be almost fourth-order uniformly convergent in the discrete maximum norm. Numerical results are presented for supporting the theoretical results.     

Accurate Solution of a System of Coupled Singularly Perturbed Reaction-diffusion Equations

We study a system of coupled reaction-diffusion equations. The equations have diffusion parameters of different magnitudes associated with them. Near each boundary, their solution exhibit two overlapping layers. A central difference scheme on layer-adapted piecewise uniform meshes is used to solve the system numerically. We show that the scheme is almost second-order convergent, uniformly in both perturbation parameters, thus improving previous results [5]. We present the results of numerical experiments to confirm our theoretical results.AMS Subject Classifications: 65L10, 65L11.     

Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection–diffusion equations

In this paper, we discuss the parameter-uniform finite difference method for a coupled system of singularly perturbed convection–diffusion equations. The leading term of each equation is multiplied by a small but different magnitude positive parameter, which leads to the overlap and interact boundary layer. We analyze the boundary layer and construct a piecewise-uniform mesh on the variant of the Shishkin mesh. We prove that our schemes converge almost first-order uniformly with respect to small parameters. We present some numerical experiments to support our theoretical analysis.     

Numerical Solution of a Reaction-Diffusion Elliptic Interface Problem with Strong Anisotropy

We consider a singularly perturbed reaction-diffusion elliptic problem in two dimensions (x,y), with strongly anisotropic coefficients and line interface. The second order derivative with respect to x is multiplied by a small parameter ². We construct finite volume difference schemes on condensed Shihskin meshes and prove -uniform convergence in discrete energy and maximum norms. Numerical experiments that agree with the theoretical results are given.     

The discrete minimum principle for quadratic spline discretization of a singularly perturbed problem

We consider a singularly perturbed boundary value problem with two small parameters. The problem is numerically treated by a quadratic spline collocation method. The suitable choice of collocation points provides the discrete minimum principle. Error bounds for the numerical approximations are established. Numerical results give justification of the parameter-uniform convergence of the numerical approximations.     

A numerical method for multipoint boundary value problems with application to a restricted three body problem

A type of parallel shooting method is proposed for the solution of nonlinear multipoint boundary value problems. It extends the usual quasilinearization method and a previous shooting method developed for such problems, and reduces to usual multiple shooting techniques for two point boundary value problems. The effectiveness of the method for stiff problems is illustrated by an application to the problem of finding periodic solutions of a restricted three body problem with given Jacobian constant and unknown period.     

A parameter uniform numerical method for a system of singularly perturbed convection–diffusion equations with discontinuous convection coefficients

In this paper, a weakly coupled system of two singularly perturbed convection-diffusion equations with discontinuous convection coefficients is examined. A finite difference scheme on Shishkin mesh generating the parameter uniform convergence in the global maximum norm is constructed for solving this problem. Numerical results which are in agreement with the theoretical results are presented.     

Well posedness and numerical solution for a non-local pseudohyperbolic initial boundary value problem

ABSTRACT

In this paper, we study a non-local initial boundary value problem for a one-dimensional pseudohyperbolic equation. We first establish the existence and uniqueness of strong solution, then a numerical solutions for the system will be derived by using the finite-difference method.     

A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters

In the present paper, a parameter-uniform numerical method is constructed and analysed for solving one-dimensional singularly perturbed parabolic problems with two small parameters. The solution of this class of problems may exhibit exponential (or parabolic) boundary layers at both the left and right part of the lateral surface of the domain. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution, we consider the implicit Euler method for time stepping on a uniform mesh and a special hybrid monotone difference operator for spatial discretization on a specially designed piecewise uniform Shishkin mesh. The resulting scheme is shown to be first-order convergent in temporal direction and almost second-order convergent in spatial direction. We then improve the order of convergence in time by means of the Richardson extrapolation technique used in temporal variable only. The resulting scheme is proved to be uniformly convergent of order two in both the spatial and temporal variables. Numerical experiments support the theoretically proved higher order of convergence and show that the present scheme gives better accuracy and convergence compared of other existing methods in the literature.     

Adaptive boundary control of a flexible manipulator with input saturation

In this study, we consider the anti-windup design as one of the approaches for the boundary control problem of a flexible manipulator in the presence of system parametric uncertainties, external disturbances and bounded inputs. The dynamics of the system are represented by partial differential equations (PDEs). Using the singular perturbation approach, the PDE model is divided into two simpler subsystems. With the Lyapunov's direct method, an adaptive boundary control scheme is developed to regulate the angular position and suppress the elastic vibration simultaneously and the adaptive laws are designed to compensate for the system parametric uncertainties and the disturbances. The proposed control scheme allows the application of smooth hyperbolic functions, which satisfy physical conditions and input restrictions, be easily realised. Numerical simulations demonstrate the effectiveness of the proposed scheme.