Sextic spline method for the solution of a system of obstacle problems

We use sextic spline function to develop numerical method for the solution of system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the approximate solutions obtained by the present method are better than those produced by other collocation, finite difference and spline methods. A numerical example is given to illustrate practical usefulness of our method.     

Parametric Cubic Spline Approach to the Solution of a System of Second-Order Boundary-Value Problems

We use parametric cubic spline functions to develop a numerical method for computing approximations to the solution of a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite-difference, and spline methods. A numerical example is given to illustrate the applicability and efficiency of the new method.     

Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problems

In this letter, a new numerical method is proposed for solving second order linear singularly perturbed boundary value problems with left layers. Firstly a piecewise reproducing kernel method is proposed for second order linear singularly perturbed initial value problems. By combining the method and the shooting method, an effective numerical method is then proposed for solving second order linear singularly perturbed boundary value problems. Two numerical examples are used to show the effectiveness of the present method.     

Numerical methods for scattering problems from multi-layers with different periodicities

In this paper, we consider a numerical method to solve scattering problems with multi-periodic layers with different periodicities. The main tool applied in this paper is the Bloch transform. With this method, the problem is written into an equivalent coupled family of quasi-periodic problems. As the Bloch transform is only defined for one fixed period, the inhomogeneous layer with another period is simply treated as a non-periodic one. First, we approximate the refractive index by a periodic one where its period is an integer multiple of the fixed period, and it is decomposed by finite number of quasi-periodic functions. Then the coupled system is reduced into a simplified formulation. A convergent finite element method is proposed for the numerical solution, and the numerical method has been applied to several numerical experiments. At the end of this paper, relative errors of the numerical solutions will be shown to illustrate the convergence of the numerical algorithm.     

Piecewise reproducing kernel method for singularly perturbed delay initial value problems

A direct application of the reproducing kernel method presented in the previous works cannot yield accurate approximate solutions for singularly perturbed delay differential equations. In this letter, we construct a new numerical method called piecewise reproducing kernel method for singularly perturbed delay initial value problems. Numerical results show that the present method does not share the drawback of standard reproducing kernel method and is an effective method for the considered singularly perturbed delay initial value problems.     

Preconditioning by Projectors in the Solution of Contact Problems: A Parallel Implementation

A non-overlapping domain decomposition algorithm of the Neumann–Neumann type for solving contact problems of elasticity is presented. Using the duality theory of convex programming, the discretized problem turns into a quadratic one with equality and bound constraints. The dual problem is modified by orthogonal projectors to the natural coarse space. The resulting problem is solved by an augmented Lagrangian algorithm. The projectors ensure an optimal convergence rate for the solution of the auxiliary linear problems by the preconditioned conjugate gradient method. Relevant aspects on the numerical linear algebra of these problems are presented, together with an efficient parallel implementation of the method.     

Parametric Quintic-Spline Approach to the Solution of a System of Fourth-Order Boundary-Value Problems

In this paper, we use parametric quintic splines to derive some consistency relations which are then used to develop a numerical method for computing the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations, which are solved using some numerical method. Numerical evidence is presented to show the applicability and superiority of the new method over other collocation, finite difference, and spline methods.     

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill-posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill-posed problem into well-conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill-posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.     

Coupling finite difference methods and integral formulas for elliptic problems arising in fluid mechanics

This article is devoted to the numerical analysis of two classes of iterative methods that combine integral formulas with finite‐difference Poisson solvers for the solution of elliptic problems. The first method is in the spirit of the Schwarz domain decomposition method for exterior domains. The second one is motivated by potential calculations in free boundary problems and can be viewed as a numerical analytic continuation algorithm. Numerical tests are presented that confirm the convergence properties predicted by numerical analysis. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 199–229, 2004     

Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers

A numerical method is proposed for solving singularly perturbed turning point problems exhibiting twin boundary layers based on the reproducing kernel method (RKM). The original problem is reduced to two boundary layers problems and a regular domain problem. The regular domain problem is solved by using the RKM. Two boundary layers problems are treated by combining the method of stretching variable and the RKM. The boundary conditions at transition points are obtained by using the continuity of the approximate solution and its first derivatives at these points. Two numerical examples are provided to illustrate the effectiveness of the present method. The results compared with other methods show that the present method can provide very accurate approximate solutions.     

Fast solution of elliptic control problems

Elliptic control problems with a quadratic cost functional require the solution of a system of two elliptic boundary-value problems. We propose a fast iterative process for the numerical solution of this problem. The method can be applied to very special problems (for example, Poisson equation for a rectangle) as well as to general equations (arbitrary dimensions, general region). Also, nonlinear problems can be treated. The work required is proportional to the work taken by the numerical solution of a single elliptic equation.     

Furtivity and Masking Problems in Time-Dependent Electromagnetic Obstacle Scattering

In this paper, we consider furtivity and masking problems in time-dependent three-dimensional electromagnetic obstacle scattering. That is, we propose a criterion based on a merit function to minimize or to mask the electromagnetic field scattered by a bounded obstacle when hit by an incoming electromagnetic field and, with respect to this criterion, we drive the optimal strategy. These problems are natural generalizations to the context of electromagnetic scattering of the furtivity problem in time-dependent acoustic obstacle scattering presented in Ref. 1. We propose mathematical models of the furtivity and masking time-dependent three-dimensional electromagnetic scattering problems that consist in optimal control problems for systems of partial differential equations derived from the Maxwell equations. These control problems are approached using the Pontryagin maximum principle. We formulate the first-order optimality conditions for the control problems considered as exterior problems defined outside the obstacle for systems of partial differential equations. Moreover, the first-order optimality conditions derived are solved numerically with a highly parallelizable numerical method based on a perturbative series of the type considered in Refs. 2–3. Finally, we assess and validate the mathematical models and the numerical method proposed analyzing the numerical results obtained with a parallel implementation of the numerical method in several experiments on test problems. Impressive speedup factors are obtained executing the algorithms on a parallel machine when the number of processors used in the computation ranges between 1 and 100. Some virtual reality applications and some animations relative to the numerical experiments can be found in the website http://www.econ.unian.it/recchioni/w10/.     

Numerical solution of singular perturbation problems by a terminal boundary-value technique

In this paper, we present a new approach for numerically solving linear singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in implicit form is introduced. Then, the outer region problem is solved as a two-point boundary-value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, the inner region problem is modified and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Three numerical examples have been solved to demonstrate the applicability of the method.     

A reliable algorithm for solving tenth-order boundary value problems

In this paper we present an efficient numerical algorithm for solving linear and nonlinear boundary value problems with two-point boundary conditions of tenth-order. The differential transform method is applied to construct the numerical solutions. The proposed algorithm avoids the complexity provided by other numerical approaches. Several illustrative examples are given to demonstrate the effectiveness of the present algorithm.      

Global convergence analysis of the aggregate constraint homotopy method for nonlinear programming problems with both inequality and equality constraints

In this paper, we construct appropriate aggregate mappings and a new aggregate constraint homotopy (ACH) equation by converting equality constraints to inequality constraints and introducing two variable parameters. Then, we propose an ACH method for nonlinear programming problems with inequality and equality constraints. Under suitable conditions, we obtain the global convergence of this ACH method, which makes us prove the existence of a bounded smooth path that connects a given point to a Karush–Kuhn–Tucker point of nonlinear programming problems. The numerical tracking of this path can lead to an implementable globally convergent algorithm. A numerical procedure is given to implement the proposed ACH method, and the computational results are reported.     

Splitting-based conjugate gradient method for a multi-dimensional linear inverse heat conduction problem

In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, by a variational method. The gradient of the functional to be minimized is obtained by the aid of an adjoint problem, and the conjugate gradient method with a stopping rule is then applied to this ill-posed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem, we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.     

非线性特征值问题的二次近似方法

本文研究求解非线性特征值问题的数值方法.基于矩阵值函数的二次近似,将非线性特征值问题转化为二次特征值问题,提出了求解非线性特征值问题的逐次二次近似方法,分析了该方法的收敛性.结合求解二次特征值问题的Arnoldi方法和Jacobi-Davidson方法,给出求解非线性特征值问题的一些二次近似方法.数值结果表明本文所给算法是有效的.     

Helly’s Principle and Its Application to an Infinite-Horizon Optimal Control Problem

In this paper, a numerical method is presented to solve singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a discontinuous source term. First, an asymptotic expansion approximation of the solution of the boundary-value problem is constructed using the basic ideas of the well-known WKB perturbation method. Then, some initial-value problems and terminal-value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial-value problems and terminal-value problems are singularly-perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples are provided to illustrate the method.     

A new boundary integral equation formulation for plane orthotropic elastic media

A new boundary integral equation formulation for solving plane elasticity problems involving orthotropic media is presented in this paper. Based on the real variable fundamental solutions of the considered problems, a limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) and a novel decomposition technique to the fundamental solutions, the regularized BIEs with indirect unknowns, which do not involve the direct calculation of CPV and HFP integrals, are established. The limiting process is done in global coordinates and no separate numerical treatment for strong and weak singular integrals was necessary. The current method does not need to transform the considered problems into isotropic ones as is normally done in the existing literature, so no inverse transform is required. The numerical implementation is carried out using both discontinuous quadratic elements and exact elements, which is developed to model its boundary with negligible error. The validity of the proposed scheme is demonstrated by three numerical examples. Excellent agreement between the numerical results and exact solutions was obtained even with using small amounts of element.     

On the uniform convergence of a finite difference scheme for time dependent singularly perturbed reaction-diffusion problems

In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction-diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a special mesh in space, condensing the mesh points in the boundary layer regions. In this paper three different meshes of Shishkin, Bahkvalov and Vulanovic type are used, proving the uniform convergence with respect to the diffusion parameter. The analysis of the uniform convergence is based on a new study of the asymptotic behavior of the solution of the semidiscrete problems, which are obtained after the time discretization by the Euler method. Some numerical results are showed corroborating in practice the theoretical results on the uniform convergence and the order of the method.