Galerkin Approximations of General Delay Differential Equations with Multiple Discrete or Distributed Delays

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Virginia Tech


Delay differential equations (DDEs) are often used to model systems with time-delayed effects, and they have found applications in fields such as climate dynamics, biosciences, engineering, and control theory. In contrast to ordinary differential equations (ODEs), the phase space associated even with a scalar DDE is infinite-dimensional. Oftentimes, it is desirable to have low-dimensional ODE systems that capture qualitative features as well as approximate certain quantitative aspects of the DDE dynamics. In this thesis, we present a Galerkin scheme for a broad class of DDEs and derive convergence results for this scheme. In contrast to other Galerkin schemes devised in the DDE literature, the main new ingredient here is the use of the so called Koornwinder polynomials, which are orthogonal polynomials under an inner product with a point mass. A main advantage of using such polynomials is that they live in the domain of the underlying linear operator, which arguably simplifies the related numerical treatments. The obtained results generalize a previous work to the case of DDEs with multiply delays in the linear terms, either discrete or distributed, or both. We also consider the more challenging case of discrete delays in the nonlinearity and obtain a convergence result by assuming additional assumptions about the Galerkin approximations of the linearized systems.



Delay Differential Equations, Galerkin Approximations