Autonomous Linear Functional Differential Equations: A Fundamental Solution Approach

This Book Chapter focuses attention on the fundamental solution of a first order n-vector autonomous linear retarded functional differential equation (RFDE) with finite delay. The document highlights the supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix, and the Laplace transform. Over the last ten years estimation and learning methods utilizing positive definite kernels have become rather popular, particularly in machine learning. The fundamental solution for a RFDE is presented in a form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The Book Chapter treats fundamental solutions of other functional differential equations, including: (i) pth order RFDE, (ii) RFDE with infinite delay, (iii) Neutral Functional Differential Equation (NFDE) as a composite of integral (zero order) and integro-differential (first order) functional differential equations, and (iv) Nonautonomous RFDE. Examples are provided to elucidate the behavior of fundamental solutions.