Title: Solution Techniques and Error Analysis of General Classes of Partial Differential Equations
Abstract: While constructive insight for a multitude of phenomena appearing in the physical and biological sciences, medicine, engineering and economics can be gained through the analysis of mathematical models posed in terms of systems of ordinary and partial di↵erential equations, it has been observed that a better description of the behavior of the investigated phenomena can be achieved through the use of functional di↵erential equations (FDEs) or partial functional di↵erential equations (PFDEs). PFDEs or functional equations with ordinary derivatives are subclasses of FDEs. FDEs form a general class of di↵erential equations applied in a variety of disciplines and are characterized by rates of change that depend on the state of the system. As opposed to traditional partial di↵erential equations (PDEs), the formulation of PFDEs, and hence, their methods of solution, are generally significantly complicated by the functional dependence of the system. Consequently, mathematical analysis has become essential to address important questions on PFDEs, their properties and solutions. This thesis is devoted to a general class of parabolic PFDEs and works out the details of the proof techniques of a related paper that help to address these questions. In particular, we examine error bounds of approximate solutions with the aim to address whether or not they converge to the exact solutions as a result of refining the associated discretizations. vi TABLE OF CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Publication Year: 2016
Publication Date: 2016-01-01
Language: en
Type: article
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