By Wolfgang Wasow
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Extra resources for Asymptotic Expansions for Ordinary Differential Equations
Proof. From ii) with α=0, we know that Thus 0= F(f+(−f)) ≤F(f)+F(−f)=2F(f). In other words, F(f)≥0. The value of the norm function F is almost always written as A vector space X together with a norm on X is called a normed linear space. The inequality iii) is commonly called the triangle inequality. 6a). The distance induced by this norm is the everyday Euclidean distance. In a linear space X, our approximation problem will be to find a member fM of a given subspace that is closest to a given vector f in the sense that || f−fM ||≤|| f−m || for all We shall be concerned only with finite-dimensional subspaces M .
This property is a consequence of the mathematical model and contradicts the observation that heat does not flow infinitely fast. 11) at remains unmeasurably small for a certain time interval before a detectable heat wave arrives so that defacto the wave speed is finite. 3 where the speed of an isotherm is found numerically. 11) to practical problems such as heat flow in a slab or bar. 11) is not as restrictive as it might appear. This is easily demonstrated if M=1. html[22/02/2009 23:51:11] next page > page_16 < previous page page_16 next page > Page 16 we see that u(x, t) is odd in x with respect to x0 for all t .
Define by and Then where A is the matrix is an inner product. html[22/02/2009 23:51:21] next page > page_28 < previous page page_28 next page > Page 28 It is clear that and if and only if To see that simply compute both inner products. The remaining two properties are evident. c) On the space of all continuous functions (real- or complex-valued) defined on the reals having period 2L, it is easy to verify that is an inner product. 4 In an inner product space, Proof. If then the proposition is obviously true, so assume number.
Asymptotic Expansions for Ordinary Differential Equations by Wolfgang Wasow