ABSTRACT
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.1 Introducing the Homogenous Cauchy Problem (HCP) . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Lessons Learned from a Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3 Defining the Matrix Exponential eAt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.1 One Approach-Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3.2 Another Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Putzer’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.5 Properties of eAt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.6 The Homogenous Cauchy Problem: Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.7 Higher-Order Linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.8 A Perturbed (HCP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.9 What Happens to Solutions of (HCP) as Time Goes On and On and On...? . 123 4.10 Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
The question of how phenomena evolve over time is central to a broad range of fields within the social, natural, and physical sciences. The behavior of such phenomena is governed by established laws in the underlying field that typically describe the rates at which it and related quantities evolve over time. A precise mathematical description involves the formulation of so-called evolution equations whose complexity depends largely on the realism of the model. We focus in this chapter on models in which the evolution equation is generated by a system of linear homogeneous ordinary differential equations. We are in search of an abstract paradigm into which all of these models are subsumed as special cases. Once established, we can study the rudimentary properties of the abstract paradigm and subsequently apply the results to each model.
