In this paper we give necessary and sufficient conditions for convergence of continuous collocation approximations of solutions of first kind Volterra integral equations. The results close some longstanding gaps in the theory of polynomial spline collocation methods for such equations. The convergence analysis is based on a Runge-Kutta or ODE approach.

A collocation procedure is developed for the linear and nonlinear Fredholm and Volterra integral equations, using the globally defined B-spline and auxiliary basis functions. The solution is collocated by cubic B-spline and then the integral equation is approximated by the 5-points Gauss–Turán quadrature formula with respect to the Legendre weight function. Combination of these two approaches i...

The present article is concerned with the numerical solution of Benjamin-BonaMahony-Burgers (BBM-Burger) equation by quartic B-spline collocation method. The method is based on quartic B-spline basis functions for space integration, and Crank-Nicolson formulation for time integration. Numerical examples considered by different researchers are discussed to illustrate the efficiency, robustness a...

We present an exponential B-spline collocation method for solving convection-diffusion equation with Dirichlet’s type boundary conditions. The method is based on the Crank-Nicolson formulation for time integration and exponential B-spline functions for space integration. Using the Von Neumann method, the proposed method is shown to be unconditionally stable. Numerical experiments have been cond...

Based on the quadratic spline function, a quadratic spline collocation method is presented for the time fractional bioheat equation governing the process of heat transfer in tissues during the thermal therapy. The corresponding linear system is given. The stability and convergence are analyzed. Some numerical examples are given to demonstrate the efficiency of this method. c ©2016 All rights re...

In this paper, a numerical method for the solution of a strongly coupled reaction-diffusion system, with suitable initial and Neumann boundary conditions, by using cubic B-spline collocation scheme on a uniform grid is presented. The scheme is based on the usual finite difference scheme to discretize the time derivative while cubic B-spline is used as an interpolation function in the space dime...

In this paper, a collocation finite difference scheme based on new cubic trigonometric B-spline is developed and analyzed for the numerical solution of a one-dimensional hyperbolic equation (wave equation) with non-local conservation condition. The usual finite difference scheme is used to discretize the time derivative while a cubic trigonometric B-spline is utilized as an interpolation functi...

We propose and analyze a numerical method for solving fourth order differential equations modelling two point boundary value problems. The scheme is based on B-splines collocation. The error analysis is carried out and convergence rates are derived.

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...

A B-spline finite element method is used to solve the equal width equation numerically. This approach involves a collocation method using quintic B-splines at the knot points as element shape. Time integration of the resulting system of ordinary differential equations is effected using the fourth order Runge–Kutta method, instead of the finite difference method, the resulting system of ordinary...