Collocation‐variation difference schemes for differential‐algebraic equations

Abstract

We consider the initial value problem in linear differential-algebraic equations and highlight the difficulties of the construction of numerical methods for these problems. We propose one-step difference schemes based on the collocation-variational approach and show the principal difference between these algorithms and the known difference schemes. The analysis of the particular cases of such schemes are given. KEYWORDS differential-algebraic equations, difference schemes | INTRODUCTIONA number of applied problems can be described by interrelated ODEs and algebraic equations. These systems are called differential-algebraic equations. Examples of the problems can be found in previous works. [1][2][3][4][5] It is assumed that the initial condition is given in such a way that the problem under consideration has a solution.Both the qualitative theory and the theory of numerical methods of DAEs have been developed since the 1970s to 1980s. This is due to the fact that the construction and the justification of numerical methods for solving DAEs is a more complex task than the development of methods for initial value problem in ODEs solved with respect to derivatives. At present, there are several approaches to the numerical solution of the problem under consideration.1. Application of various projectors, the generalized inverse of a matrix, the operation of differentiation for reduction of the original problem to a simpler problem, for example, to an ODE solved with respect to derivatives. A bibliography on this topic is presented in other studies. 3,6 2. The method of normal splines, 7 which is similar to the least squares method. 8 3. The method of solution continuation with respect to the parameter. 9 4. Using the so-called extended systems that are based on the following idea: The original DAE and several its total derivatives are combined in a single system. By applying nonsingular transformations to this system, we obtain a first-order ODE. Further research and an extensive bibliography on this topic are presented in previous works. 2,4 This approach can be used for the construction of special difference schemes for DAEs.

Integral Equations Related to Volterra Series and Inverse Problems: Elements of Theory and Applications in Heat Power Engineering

“…In the future, it is planned to construct and justify numerical methods for solving such systems. These algorithms are supposed to be based on the extrapolation formulas [67], which have performed well when solving IAEs Equation (30), on the special quadrature formula of the midpoint type [58,59], and on the collocation-variational approach that was proposed for the numerical solution to differential-algebraic equations [72][73][74][75]. Institutional Review Board Statement: Not applicable.…”