Research and teaching interests

Wieslaw Marszalek

    Research

    My recent research interests concern the analysis of differential algebraic equations (DAEs) which appear in infinite dimensional systems, such as systems of conservation laws (magnetohydrodynamics (MHD), gas dynamics, elasticity).

    DAEs are implicitly defined systems of differential equations

    F(y',y,t)=0
    with Jacobian of F w.r.t. y' identically singular. DAEs naturally arise in many areas (electrical circuits, constrained mechanical systems, chemical engineering, etc.). Another source of DAEs is the method of lines solution of partial differential equations and traveling wave solutions of dissipative systems of conservation laws.

    The structure of the DAEs depends on the manner in which the dissipative mechanism is present. For example, the dissipative mechanism influences the existence of the traveling wave solutions, the index of the DAE, and the behavior at singularities. The new type of bifurcation in MHD (the so-called Singularity Induced Bifurcation) is of great interest, since it indirectly imposes challenging demands on numerical integrators of DAEs and may eventually lead to the proof of the existence of a new type of solutions in dissipative MHD equations. This in turn may lead to a new type of shock wave in non-dissipative systems of conservation laws. Key issues in this approach are the use of the Singularity Induced Bifurcation Theorem and the ability to integrate DAEs through certain singularities.

    I am interested in further analysis of these issues, including the numerical treatment of DAEs with singularities, the canonical forms of DAEs near singularities, and the analysis of what impact the novel DAE approach may have on explaining some of the existing results in the theory of systems of conservation laws. These investigations could include examining the relationship between singular DAEs and catastrophy theory.

    Simultaneously, I am interested in theoretical issues in DAEs occuring in infinite dimensional systems and their comparisons with DAEs in finite dimensional systems. In particular, such issues include properly defined indices of DAEs, consistency in the data (initial/boundary conditions, forcing functions), and numerical algorithms for solving DAEs.

    TEACHING

    I have taught and am interested in teaching a variety of courses including Calculus, Ordinary Differential Equations, Partial Differential Equations, Digital Signal Processing and Control Theory (state space and frequency domain methods).