I am writing a package to solve linear differential equations of the form $\frac{d\vec{x}}{dt} = A\vec{x}$ where $A$ is nil potent. The knowledge of the nil potency guarantees that the matrix exponential just produces polynomials, which can be explicitly calculated and evaluated much faster than a matrix exponential.
I am looking into ways to expand my package and one way to do so would be to exploit knowledge about mass matrix DAEs of the form: $$M \frac{d\vec{x}}{dt} = A\vec{x}$$ where $M$ is singular and $A$ is nilpotent. However since i am unfamiliar with DAEs i am unsure whether there exist any explicit knowledge about the solution structure i could exploit.
My questions are:
- Is there explicit solution for initial value problems of Linear (mass matrix) DAEs?
- If there is no global explicit solution, how could i use knowledge of the global solution of $\frac{d\vec{x}}{dt} = A\vec{x}$ to solve $M \frac{d\vec{x}}{dt} = A\vec{x}$ approximately?
- How would i calculate the local approximation error to do step size control in your proposed scheme?