An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.     

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.     

Spline collocation methods for nonlinear Volterra integral equations with unknown delay

In this paper we introduce and study polynomial spline collocation methods for systems of Volterra integral equations with unknown lower integral limit arising in mathematical economics. Their discretization leads to implicit Runge-Kutta-type methods. The global convergence and local superconvergence properties of these methods are proved, and the theory is illustrated by a numerical example arising in the application of such equations in certain mathematical models of liquidation.     

High order implicit collocation method for the solution of two‐dimensional linear hyperbolic equation

In this article, we introduce a high‐order accurate method for solving the two dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives of linear hyperbolic equation and collocation method for the time component. The resulted method is unconditionally stable and solves the two‐dimensional linear hyperbolic equation with high accuracy. In this technique, the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite difference approximation of fourth order and collocation method give a very efficient approach for solving the two dimensional linear hyperbolic equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A discrete collocation method for Fredholm integro-differential equations with weakly singular kernels

A fully discrete version of a piecewise polynomial collocation method is constructed to solve initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Some of our theoretical results are illustrated by numerical experiments.     

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

     

Convergence of the collocation method and the mechanical quadrature method for systems of singular integro-differential equations in Lebesgue spaces

Computational schemes for the collocation method and the mechanical quadrature method for the approximate solution of systems of singular integro-differential equations with a Cauchy kernel are elaborated. The case where the systems of equations are defined on an arbitrary smooth closed contour of a complex plane is examined. The methods researched are based on Fejér points. Estimates of the rate of convergence in Lebesgue space are obtained.     

Superconvergence of the iterated collocation methods for Hammerstein equations

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [16] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also discuss the discrete collocation method for weakly singular Hammerstein equations. Some discrete collocation methods for Hammerstein equations with smooth kernels were given previously in [3, 18].     

A numerical method to solve a class of linear integro‐differential equations with weakly singular kernel

In this paper, a collocation method based on the Bessel polynomials is introduced for the approximate solution of a class of linear integro‐differential equations with weakly singular kernel under the mixed conditions. The exact solution can be obtained if the exact solution is polynomial. In other cases, increasing number of nodes, a good approximation can be obtained with applicable errors. In addition, the method is presented with error and stability analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

A taylor collocation method for solving high‐order linear pantograph equations with linear functional argument

A numerical method based on the Taylor polynomials is introduced in this article for the approximate solution of the pantograph equations with constant and variable coefficients. Some numerical examples, which consist of the initial conditions, are given to show the properties of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:1628–1638, 2011     

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     

A collocation method to solve higher order linear complex differential equations in rectangular domains

In this article, a collocation method is developed to find an approximate solution of higher order linear complex differential equations with variable coefficients in rectangular domains. This method is essentially based on the matrix representations of the truncated Taylor series of the expressions in equation and their derivates, which consist of collocation points defined in the given domain. Some numerical examples with initial and boundary conditions are given to show the properties of the method. All results were computed using a program written in scientific WorkPlace v5.5 and Maple v12. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A weak‐derivative form for linear hyperbolic systems

Difference schemes for linear hyperbolic systems are considered. As a main result, a weak derivative form (WDF) of the governing equations is derived, which is also valid near flow discontinuities. The occurrence of one‐sided derivatives in the WDF structure indicated how to difference near discontinuities. When first‐order differencing is applied to the WDF result, the (linearly identical) schemes by Godunov, Roe, and Steger‐Warming are reproduced. The extension to nonlinear systems is via a local linearization. Choosing Roe's averaging reduces the WDF algorithm to Roe's scheme, whereas other nonlinear WDF schemes are possible. The suitability of various kinds of averaging is numerically investigated. For weak shocks a surprising lack of sensitivity of the method to a particular averaging is exhibited. However, for strong shocks and where the ordinary arithmetic average is used, a slightly more pronounced difference in performance exists between Roe's scheme and WDF. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Stability of periodically switched discrete-time linear singular systems

In this paper we present some necessary and sufficient conditions for the stability of periodically switched discrete-time linear index-1 singular system, (PSSS). In particular, it is proved that, if at least one subsystem of a PSSS is asymptotically stable, then there is a switching rule, so that the whole system is also uniformly exponentially stable. Furthermore, for a periodically switched control system with no stable subsystems, there exist a switching rule and feedback matrices, such that the obtained PSSS is uniformly exponentially stable.     

On best possible order of convergence estimates in the collocation method and Galerkin's method for singularly perturbed boundary value problems for systems of first-order ordinary differential equations

The collocation method and Galerkin method using parabolic splines are considered. Special adaptive meshes whose number of knots is independent of the small parameter of the problem are used. Unimprovable estimates in the -norm are obtained. For the Galerkin method these estimates are quasioptimal, while for the collocation method they are suboptimal.

     

A collocation approach for solving linear complex differential equations in rectangular domains

In this paper, a collocation method is presented to find the approximate solution of high‐order linear complex differential equations in rectangular domain. By using collocation points defined in a rectangular domain and the Bessel polynomials, this method transforms the linear complex differential equations into a matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. The proposed method gives the analytic solution when the exact solutions are polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and the comparisons are made with existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on a computer using a program written in MATLAB v7.6.0 (R2008a). Copyright © 2012 John Wiley & Sons, Ltd.     

Inexact GMRES for singular linear systems

Inexact Krylov subspace methods have been shown to be practical alternatives for the solution of certain linear systems of equations. In this paper, the solution of singular systems with inexact matrix-vector products is explored. Criteria are developed to prescribe how inexact the matrix-vector products can be, so that the computed residual remains close to the true residual, thus making the inexact method of practical applicability. Cases are identified for which the methods work well, and this is the case in particular for systems representing certain Markov chains. Numerical experiments illustrate the effectiveness of the inexact approach. AMS subject classification (2000)  65F10     

Riccati equations for generalized singular estimate control systems

We consider the Bolza problem associated with boundary/point control systems governed by strongly continuous semigroups. In continuation of our work in Lasiecka and Tuffaha [I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by C ₀–semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008), pp. 229–246; I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control systems, Control Cybernet 38(4B) (2009), pp. 1429–1460], we yet extend the theory to a more general class of control problems that are not analytic providing sharp blow-up rates for the regularity. Solvability of the associated Riccati equations and an optimal feedback synthesis are established. The presence of unbounded control actions, such as boundary/point controls, naturally lead to a singularity at the terminal point t?=?T of the optimal control and of the corresponding feedback operator as before. The class of control systems considered in this article is a generalization to the class usually referred to in the literature as ‘Singular Estimate Control Systems’. The prototype is still that of a PDE system consisting of coupled hyperbolic parabolic dynamics interacting on an interface with point/boundary control. The distinct feature of the class considered in this article is that the degree of unboundedness in the control is stronger than that allowed in the usual singular estimate control system configuration, giving rise to less regular optimal state trajectories.     

Spline on a generalized hyperbolic paraboloid

In this paper, we present an approach to produce a kind of spline, which is very close to G²-continuity. For a control polygon, we can construct a polyhedron. A generalized hyperbolic paraboloid with a Bernstein-Bézier algebraic form is obtained by the barycentric coordinate system, in which parametrical forms can be represented with two parameters. Having constrained the two parameters with a functional relation for the generalized hyperbolic paraboloid, a variety of arcs could be constructed with the nature of fitting the tangent direction at the endpoints and a little curvature for the whole arc, which can be attached into a spline curve of G²-continuity. Further, using the method of simple averages, we present a new symmetry spline to a control polygon, which can improve the approximating effect for a control polygon.     

带非局部边界条件的热方程的LEGENDRE配置法

对带非局部边界条件的热方程的初边值问题提出了LEGENDRE配置法,并给出其半离散逼近和全离散逼近的稳定性和收敛性分析.数值试验验证了方法的有效性.