Research and Teaching Interests
Research: My recent research interests concern the analysis of differential algebraic equations (DAEs) which appear in infinite dimensional systems, such as magnetohydro-dynamics (MHD), gas dynamics, elasticity, constrained mechanical (multi-body) systems, power systems and circuit theory. The circuit theory includes the topics of nonlinear memristive circuits with mixed-mode oscillations
and chaos, bifurcations, links to Newton's second law and the least action principle.
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 (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, folded pseudo-equilibria, 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.
Simultaneously, I am interested in theoretical issues in DAEs occurring 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, Mathematical Modeling of Applied Problems, Linear Algebra, DC/AC Circuits, Control Theory (state space and frequency domain methods). I use Matlab & Maple in my teaching and am also quite familiar with the ZOOM, Canvas, BBB, Moody & eCollege online platforms.
and chaos, bifurcations, links to Newton's second law and the least action principle.
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 (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, folded pseudo-equilibria, 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.
Simultaneously, I am interested in theoretical issues in DAEs occurring 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, Mathematical Modeling of Applied Problems, Linear Algebra, DC/AC Circuits, Control Theory (state space and frequency domain methods). I use Matlab & Maple in my teaching and am also quite familiar with the ZOOM, Canvas, BBB, Moody & eCollege online platforms.
