This note is concerned with the provision of an interpolant for o.d.e. initial value codes based upon backward differentiation formulae (b.d.f.) in which both the solution and its first time derivative are continuous over the range of integrationa C’ interpolant. The construction and implementation of the interpolant is described and the continuity achieved in practice is illustrated by two exa...

This paper studied the 2–point improved block backward differentiation formula for solving stiff initial value problems proposed by Musa et al (2013) and further established the necessary conditions for the convergence of the method. It is shown that the method is both zero stable and consistent. The order of the method is also derived.

In this paper we investigate the behavior of numerical ODE methods for the solution of systems of differential equations coupled with algebraic constraints. Systems of this form arise frequently in the modelling of problems from physics and engineering; we study some particular examples from electrical networks, fluid dynamics and constrained mechanical systems. We show that backward differenti...

A new block extended backward differentiation formula suitable for the integration of stiff initial value problems is derived. The procedure used involves the use of an extra future point which helps in improving the performance of an existing block backward differentiation formula. The method approximates the solution at 3 points simultaneously at each step. Accuracy and stability properties o...

In this paper, we consider an implicit block backward differentiation formula (BBDF) for solving Volterra Integro-Differential Equations (VIDEs). The approach given in this paper leads to numerical methods for solving VIDEs which avoid the need for special starting procedures. Convergence order and linear stability properties of the methods are analyzed. Also, methods with extensive stability r...

We consider the numerical solution of differential Riccati equations. We review the existing methods and investigate whether they are suitable for large-scale problems arising in LQR and LQG design for semi-discretized partial differential equations. Based on this review, we suggest an efficient matrix-valued implementation of the BDF for differential Riccati equations.

Statutory records of underground utility apparatus (such as pipes and cables) are notoriously inaccurate, so street surveys are usually undertaken before road excavation takes place to minimize the extent and duration of excavation and for health and safety reasons. This involves the use of sensors such as Ground Penetrating Radar (GPR). The GPR scans are then manually interpreted and combined ...

The present paper describes implementation details of a new implementation of a BDF solver within MATLAB R . It is tuned to be most efficient when applied as a solver within FEMLAB R . The present code aims at a predictable and efficient performance for index-2 differential-algebraic equations arising within the method of lines. The main novelty is the consequent implementation of a predictive ...

Quasi-linear parabolic equations are discretised in time by fully implicit backward difference formulae (BDF) as well as by implicit–explicit and linearly implicit BDF methods up to order 5. Under appropriate stability conditions for the various methods considered, we establish optimal order a priori error bounds by energy estimates, which become applicable via the Nevanlinna-Odeh multiplier te...

Abstract: The main purpose of this paper is to present some convergence results of a class of Parallel Multivalue Hybrid Methods (PMHMs) for semi-explicit index-2 differential-algebraic equations. Some numerical examples confirm our theoretical results and show that the computed orders of the 2-step 2-order PMHM (PMHM2) and the 3-step 3-order PMHM (PMHM3) are higher than the corresponding theor...