Solving differential-algebraic equation systems with Pade approximation of matrix exponent

A set of new numerical methods for solving linear differential-algebraic equation systems is developed. Homogenous systems can be solved, and nonhomogeneous systems with piecewise-polynomial right-hand side function. Calculation of system state at each integration step requires solving one or several (depending on the method order) systems of linear algebraic equations. The methods are based on decomposition of Pade approximation of the matrix exponent to simplest fractions. The proposed formulas make possible to avoid conversion of the differential-algebraic equation system to the ordinary differential equation system at the stage of system solving. The new methods are equivalent to some well-known Runge-Kutta type methods like Radau and Lobatto methods in terms of accuracy and steady areas. However, new methods are much more simple in theory and practical implementation, and they require several times less computational work. Methods with diagonal Pade approximations are A-stable, and methods with subdiagonal Pade approximations are L-stable. New methods can be used for solving stiff, oscillative and stiff-oscillative systems.