In this paper, the motion planning problem is studied for nonlinear differentially flat systems using B-splines parametrization of the flat output history. In order to satisfy the constraints continuously in time, the motion planning problem is transformed into a B-splines positivity problem. The latter problem is formulated as a convex semidefinite programming problem by means of a non-negativ...

Piecewise Hermite cubics have been widely used in a variety of ways for solving partial differential equations. For a number of these techniques, knowledge about the Hermite cubic collocation approximations to the spectrum of the Laplace operator is often very useful, for error analysis and, a fortiori, possible iteration parameters. To this end, we givc here explicit closed-form expressions fo...

The aim of this paper is to present the application of a particular collocation method (recently developed by the authors) to numerically solve some differential and Volterra integral equations with constant delay. The unknown function is approximated by using deficient spline functions. The existence and uniqueness of the numerical solution are studied; some aspects of the problem related to t...

Some collocation schemes used to solve mth order ordinary differential equations are known to display superconvergence at the mesh points. Here we show that some such schemes have additional superconvergence points for the approximate solution and all of its derivatives. Using such points, we argue that a mesh selection scheme introduced by Dodson can be expected to perform well under general c...

A complete stability and convergence analysis is given for twoand three-level, piecewise Hermite bicubic orthogonal spline collocation, Laplacemodified and alternating-direction schemes for the approximate solution of linear parabolic problems on rectangles. It is shown that the schemes are unconditionally stable and of optimal-order accuracy in space and time.

In this paper the quadratic spline difference scheme for a convection-diffusion problem is derived. With the suitable choice of collocation points we provide the discrete minimum principle. The numerical results implies the uniform convergence of order O(n−2 ln n).

A diierential equation of the form Dp(x)Du(x) = f(x; u(x)) is solved by collocation, using a class of spline functions of order 3. A restriction of each spline function to any subinterval of the partition is piecewisely in the kernel of the operator DDp(x)D. A coeecient p(x) may be singular. An algorithm is described by means of which a B-spline basis could be eeciently constructed. Some numeri...

In this paper, a numerical method is proposed to approximate the numeric solutions of nonlinear Fisher's reaction– diffusion equation with modified cubic B-spline collocation method. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply modified cu...

Bordered almost block diagonal systems arise from discretizing a linearized rst-order system of n ordinary diierential equations in a two-point boundary value problem with non-separated boundary conditions. The discretization may use spline collocation, nite diierences, or multiple shooting. After internal condensation, if necessary, the bordered almost block diagonal system reduces to a standa...

The Cahn–Hilliard equation plays an important role in the phase separation in a binary mixture. This is a fourth order nonlinear partial differential equation. In this paper, we study the behaviour of the solution by using orthogonal cubic spline collocation method and derive optimal order error estimates. We discuss some computational experiments by using monomial basis functions in the spatia...