Wavelet Collocation Method for Optimal Control Problems

A Haar wavelet technique is discussed as a method for discretizing the nonlinear system equations for optimal control problems. The technique is used to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. A nonlinear programming solver can then be used to solve optimal control problems that are rather general in form. Here, general Bolza optimal control problems with state and control constraints are considered. Examples of two kinds of optimal control problems, continuous and discrete, are solved. The results are compared to those obtained by using other collocation methods.     

一类二维对偶积分方程的解及其应用

二维对偶积分方程的理论与方法,在数学上尚未建立,因而完全的分析解不可能得到,从而使一些力学、物理与工程问题无法求解．利用双重展开和边界配置方法,得到了在数学和物理学上有着广泛应用的一类二维对偶积分方程的解答．把二维对偶积分方程化简成无限代数方程组,此方法的精确度取决于计算点的配置（即所谓边界配置）．通过对固体力学中某些复杂的初值-边值问题的应用说明此是方法有效的．     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

On the determination of boundary collocation points for solving some problems for the Laplace operator

Boundary collocation is a method for obtaining approximate solutions of boundary problems for linear partial differential equations, for which complete families of particular solutions are explicitly known. The method contains various decisions which are important for its performance, such as choice of solution subspace, choice of basis for the subspace, and choice of collocation points. Using a model problem, some particular strategies for the determination of collocation points are investigated.     

Numerical solution of differential-algebraic equations using the spline collocation-variation method

Numerical methods for solving initial value problems for differential-algebraic equations are proposed. The approximate solution is represented as a continuous vector spline whose coefficients are found using the collocation conditions stated for a subgrid with the number of collocation points less than the degree of the spline and the minimality condition for the norm of this spline in the corresponding spaces. Numerical results for some model problems are presented.     

On the numerical treatment of the eigenparameter dependent boundary conditions

In this paper, we consider the numerical treatment of singular eigenvalue problems supplied with eigenparameter dependent boundary conditions using spectral methods. On the one hand, such boundary conditions hinder the construction of test and trial space functions which could incorporate them and thus providing well-conditioned Galerkin discretization matrices. On the other hand, they can generate surprising behavior of the eigenvectors hardly detected by analytic methods. These singular problems are often indirectly approximated by regular ones. We argue that spectral collocation as well as tau method offer remedies for the first two issues and provide direct and efficient treatment to such problems. On a finite domain, we consider the so-called Petterson-König’s rod eigenvalue problem and on the half line, we take into account the Charney’s baroclinic stability problem and the Fourier eigenvalue problem. One boundary condition in these problems depends on the eigenparameter and additionally, this also could depend on some physical parameters. The Chebyshev collocation based on both, square and rectangular differentiation and a Chebyshev tau method are used to discretize the first problem. All these schemes cast the problems into singular algebraic generalized eigenvalue ones which are solved by the QZ and/or Arnoldi algorithms as well as by some target oriented Jacobi-Davidson methods. Thus, the spurious eigenvalues are completely eliminated. The accuracy of square Chebyshev collocation is roughly estimated and its order of approximation with respect to the eigenvalue of interest is determined. For the problems defined on the half line, we make use of the Laguerre-Gauss-Radau collocation. The method proved to be reliable, accurate, and stable with respect to the order of approximation and the scaling parameter.     

Alternating direction implicit orthogonal spline collocation on some non-rectangular regions with inconsistent partitions

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional dependent initial-boundary value problems on rectangles. Earlier we have used the ADI technique in conjunction with orthogonal spline collocation (OSC) for discretization in space to solve parabolic problems on rectangles and rectangular polygons. Recently, we extended applications of ADI OSC schemes to the solution of parabolic problems on some non-rectangular regions that allow for consistent nonuniform partitions. However, for many regions, it is impossible to construct such partitions. Therefore, in this paper, we show how to extend our approach further to solve parabolic problems on some non-rectangular regions using inconsistent uniform partitions. Numerical results are presented using piecewise Hermite cubic polynomials for spatial discretizations and our ADI OSC scheme for parabolic problems to demonstrate its performance on several regions.     

A new approach to solve a set of nonlinear split boundary value problems

In this paper, we propose a new approach based on conjunction of the orthogonal collocation on finite elements method with decoupling and quasi-linearization technique to approximate solutions of a set of nonlinear split boundary value problems. The numerical stability, the convergence and the accuracy of the results are checked by this algorithm. The approach developed in this study is illustrated by some numerical examples. These examples are solved using a special software package which implements the proposed algorithms.     

Operator splitting for numerical solutions of the RLW Equation

In this study, the numerical behavior of the one-dimensional Regularized Long Wave (RLW) equation has been sought by the Strang splitting technique with respect to time. For this purpose, cubic B-spline functions are used with the finite element collocation method. Then, single solitary wave motion, the interaction of two solitary waves and undular bore problems have been studied and the effectiveness of the method has been investigated. The new results have been compared with those of some of the previous studies available in the literature. The stability analysis has also been taken into account by the von Neumann method.     

Trigonometric Tension B-Spline Method for the Solution of Problems in Calculus of Variations

In this paper, the tension B-spline collocation method is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. The problems are reduced to an explicit system of algebraic equations by this approximation. We apply some numerical examples to illustrate the accuracy and implementation of the method.     

Rational Chebyshev collocation method for the similarity solution of two dimensional stagnation point flow

In this study, we propose an efficient and accurate numerical technique that is called the rational Chebyshev collocation (RCC) method to solve the two dimensional flow of a viscous fluid in the vicinity of a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow, are reduced to a third-order ordinary differential equation of a boundary value problem with a semi-infinite domain by using similarity transformation. The rational Chebyshev method reduces this nonlinear ordinary differential equation to a system of algebraic equations. This technique is a powerful type of the collocation methods for solving the boundary value problems over a semi-infinite interval without truncating it to a finite domain. We also present the comparison of this work with others and show that the present method is more accurate and efficient.     

A numerical study of the accuracy and stability of symmetric and asymmetric RBF collocation methods for hyperbolic PDEs

Differentiation matrices associated with radial basis function (RBF) collocation methods often have eigenvalues with positive real parts of significant magnitude. This prevents the use of the methods for time‐dependent problems, particulary if explicit time integration schemes are employed. In this work, accuracy and eigenvalue stability of symmetric and asymmetric RBF collocation methods are numerically explored for some model hyperbolic initial boundary value problems in one and two dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On the convergence of spline collocation methods for solving fractional differential equations

In the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi-term fractional differential equations. We then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically. Using an integral equation reformulation and special non-uniform grids, global convergence estimates are derived. From these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite Gaussian quadrature formulas as collocation points. Theoretical results are verified by some numerical examples.     

A Haar wavelet collocation approach for solving one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations

We have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one- and two-dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one- and two-dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal derivatives by the truncated Haar series. The algorithm of the method is simple and we can implement easily in any other programming language. The technique is tested on some linear and nonlinear examples from literature. The maximum absolute errors (MAEs), root mean square errors (RMSEs), and computational convergence rate are calculated for different number of collocation points (CPs) and also some 3D graphs are also drawn. The results show that the proposed technique is simply applicable and accurate.     

On spline collocation methods for boundary integral equations in the plane

Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods. In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8]. The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.     

An adaptive spectral least-squares scheme for the Burgers equation

A least-squares spectral collocation method for the one-dimensional inviscid Burgers equation is proposed. This model problem shows the stability and high accuracy of these schemes for nonlinear hyperbolic scalar equations. Here we make use of a least-squares spectral approach which was already used in an earlier paper for discontinuous and singular perturbation problems (Heinrichs, J. Comput. Appl. Math. 157:329–345, 2003). The domain is decomposed in subintervals where continuity is enforced at the interfaces. Equal order polynomials are used on all subdomains. For the spectral collocation scheme Chebyshev polynomials are employed which allow the efficient implementation with Fast Fourier Transforms (FFTs). The collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. The scheme exhibits exponential convergence where the exact solution is smooth. In parts of the domain where the solution contains discontinuities (shocks) the spectral solution displays a Gibbs-like behavior. Here this is overcome by some suitable exponential filtering at each time level. Here we observe that by over-collocation the results remain stable also for increasing filter parameters and also without filtering. Furthermore by an adaptive grid refinement we were able to locate the precise position of the discontinuity. Numerical simulations confirm the high accuracy of our spectral least-squares scheme.      

Corrected Collocation Approximations for the Harmonic Dirichlet Problem

Collocation approximations with harmonic basis functions tothe solution of the harmonic Dirichlet problem are investigated.The choice of collocation points for a best local approximationis discussed, and a result is given in terms of the abscissaeof some best quadrature formulae. A global near-best approximationis obtained by adding a correction term to the collocation approximation,utilizing basic properties of the Green's function. Numericalexamples are given, demonstrating the great improvement achieved.The same correction term can also improve on least-squares approximationsand Galerkin approximations, and the results can easily be adaptedto deal with mixed harmonic boundary value problems.     

Superconvergence and stability for boundary penalty techniques of finite difference methods

The finite difference method (FDM) is used for Dirichlet problems of Poisson's equation, and the Dirichlet boundary condition is dealt with by boundary penalty techniques. Two penalty techniques, penalty‐integrals and penalty‐collocations (i.e., fixing), are proposed in this paper. The error bounds in the discrete H¹ norm and the infinite norms are derived. The stability analysis is based on the new effective condition number (Cond_eff) but not on the traditional condition number (Cond). The bounds of Cond_eff are explored to display that both the penalty‐integral and the penalty‐collocation techniques have good stability; the huge Cond is misleading. Since the penalty‐collocation technique (i.e., the fixing technique) is simpler, it has been applied in engineering problem for a long time. It is worthy to point out that this paper is the first time to provide a theoretical justification for such a popular penalty‐collocation (fixing) technique. Hence the penalty‐collocation is recommended for dealing with the complicated constraint conditions such as the clamped and the simply support boundary conditions of biharmonic equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous Volterra integral equations

We analyse collocation methods for nonlinear homogeneous Volterra—Hammerstein integral equations with non-Lipschitz nonlinearity. We present different kinds of existence and uniqueness of nontrivial collocation solutions and give conditions for such existence and uniqueness in some cases. Finally, we illustrate these methods with an example of a collocation problem and give some examples of collocation problems that do not fit in the cases studied previously.     

A stochastic collocation method for the second order wave equation with a discontinuous random speed

In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.