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Unformatted text preview: Chapter 10 A General Framework for Linear Partial Differential Equations Before pressing on to the higher dimensional forms of the heat, wave, and Laplace/ Poisson equations, it is worth taking some time to develop a general, abstract, linear algebraic framework that underlies many of the linear partial differential equations aris- ing in applications. The power of mathematical abstraction is that, by concentrating on the essential features and not being distracted by the at times messy particular details, it enables one to establish, relatively painlessly, very general results that can be applied throughout the subject and beyond. Each abstract concept has, as its source, an ele- mentary finite-dimensional version valid for linear algebraic systems and matrices, which is then generalized and extended to include linear boundary value problems governed by differential equations. All of the abstract definitions and results contained here will be im- mediately applicable to the boundary and initial value problems of physical interest, and serve to deepen our understanding of the underlying commonalities among systems and solution techniques. Nevertheless, a more applications-oriented reader may prefer to skip ahead to the more concrete material in the following chapters, referring to the background material developed here as necessary. Most equilibrium systems are modeled as boundary value problems involving a linear differential operator that satisfies the two key conditions of being “self-adjoint” and either “positive definite” or, more generally, “positive semi-definite”. So, our first task is to introduce the adjoint of a linear function in general, and, for our specific purposes, a linear differential operator. The adjoint is a far-reaching generalization of the elementary matrix transpose. Its formulation relies on the specification of inner products on both the domain and target spaces of the operator, and, when dealing with linear differential operators, the imposition of suitable homogeneous boundary conditions on the spaces of allowable functions. In applications, the relevant inner products are typically dictated by the underlying physics. One immediate application of the adjoint is the Fredholm Alternative, which provides the constraints required for the existence of solutions to linear systems, including linear boundary value problems. A linear operator that equals its own adjoint is called self-adjoint. The simplest example is the linear function defined by a symmetric matrix. The most important sub- classes are the positive definite and positive semi-definite operators, which are the natural analogues of positive (semi-)definite matrices. We will see how to construct self-adjoint 1/19/12 345 c circlecopyrt 2012 Peter J. Olver positive (semi-)definite operators in a canonical manner. Almost all of the linear differen- tial operators studied in this text, including the Laplacian, are, when subject to suitable boundary conditions, self-adjoint and either positive definite or positive semi-definite. Theboundary conditions, self-adjoint and either positive definite or positive semi-definite....
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