Analytical and numerical solutions to higher index linear variable coefficient DAE systems

摘要

Analytical and numerical solutions to the singular linear system of differential equations A(t)x′(t) + B(t)x(t) = f(t) are investigated. Closed-form analytical solutions of systems transformable to standard canonical form, SCF (see Campbell (1983)) are derived. Equivalent ways to compute the analytic solution are presented which are of interest in a symbolic/numeric computing environment. Many numerical methods for solving DAE systems are based on Gear's backward difference formula (BDF) method. It is proved that the BDF method and the modified BDF method, MBDF (see Clark (1986)) are stable and O(hk) accurate after a maximum of (s − 1)k + 1 steps for linear variable coefficient systems in SCF, where s is the dimension of the algebraic part. These new results are valid for arbitrarily higher index systems and generalize well-known results for linear constant coefficient singular systems to the variable coefficient case. Numerical experiments confirm the order and stability results.