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Unformatted text preview: Chapter 6 Generalized Functions and Greens Functions Boundary value problems, involving both ordinary and partial differential equations, can be profitably viewed as the infinite-dimensional function space versions of finite dimen- sional systems of linear algebraic equations. As a result, linear algebra not only provides us with important insights into their underlying mathematical structure, but also motivates both analytical and numerical solution techniques. In the present chapter, we develop the method of Greens functions, pioneered by the early nineteenth century self-taught English mathematician (and miller!) George Green, whose famous Theorem you already encoun- tered in multi-variable calculus. We begin with the simpler case of ordinary differential equations, and then move on to solving the two-dimensional Poisson equation, where the Greens function provides an powerful alternative to the method of separation of variables. For inhomogeneous linear systems, the basic Superposition Principle says that the response to a combination of external forces is the self-same combination of responses to the individual forces. In a finite-dimensional system, any forcing function can be decomposed into a linear combination of unit impulse forces, each applied to a single component of the system, and so the full solution can be expressed as a linear combination of the solutions to the impulse problems. This simple idea will be adapted to boundary value problems governed by differential equations, where the response of the system to a concentrated impulse force is known as the Greens function. With the Greens function in hand, the solution to the inhomogeneous system with a general forcing function can be reconstructed by superimposing the effects of suitably scaled impulses. Understanding this construction will become increasingly important as we progress on to partial differential equations, where direct analytical solution techniques are far harder to come by. The obstruction blocking the direct implementation of this idea is that there is no ordinary function that represents an idealized concentrated impulse! Indeed, while this approach was pioneered by Green in the early 1800s, and then developed into an effective computational tool by the English engineer Oliver Heaviside in the early 1900s, it took another 50 years before mathematicians were able to develop a completely rigorous the- ory of generalized functions , also known as distributions . In the language of generalized functions, a unit impulse is represented by a delta function . While we do not have the analytical tools to completely develop the mathematical theory of generalized functions in its full, rigorous glory, we will spend the first section learning the basic concepts and Warning: We follow common practice and refer to the delta distribution as a function, even though, as we will see, it is most definitely not a function in the usual sense....
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