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 ...

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...

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...

A systematic way of extending a general fixed-stepsize multistep formula to a minimum storage variable-stepsize formula has been discovered that encompasses fixed-coefficient (interpolatory), variable-coefficient (variable step), and fixed leading coefficient as special cases. In particular, it is shown that the " interpolatory" stepsize changing technique of Nordsieck leads to a truly variable...

We analyze stability properties of BDF methods of order up to 5 for linear parabolic equations as well as of implicit–explicit BDF methods for nonlinear parabolic equations by energy techniques; time dependent norms play also a key role in the analysis.

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite family standard SDEs and establish its well-posedness, show that it is equivalent to equation. also representation formula in terms non-linear semi-linear partial differential equation Hamilton–Jacobi–Bellman type. As application, consider the tim...

Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity suitability high-dimensional problems: physics-informed neural networks, based on the Feynman–Kac formula of backward stochastic The article is accompanied by a suit...