Read e-book online Separation of variables for partial differential equations PDF

By Gunter H. Meyer George Cain

ISBN-10: 020349878X

ISBN-13: 9780203498781

ISBN-10: 0203621034

ISBN-13: 9780203621035

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Note that M is a subspace of X because it is closed under vector addition and scalar multiplication. Definition A collection of vectors in a linear space is linearly independent if A sequence of vectors is linearly independent if any finite collection C drawn from the sequence is linearly independent. The definition implies that in a finite set of linearly independent vectors no one element can be expressed as a linear combination of the remaining elements. , Then the set is a basis of X, and X has dimension N.

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 .

If then the proposition is obviously true, so assume number. Then Let α be a complex Now Next, let where t is any real number. Then This expression is quadratic in t and so the fact that it is never negative means that In other words, which completes the proof. The inequality is known as Schwarz’s inequality. 5 Suppose X is an inner product space. Then the function F defined by norm on X. Proof. The proofs that F(f)≥0 and F(αf)=|α|F(f) are simple and omitted. We prove the triangle inequality. html[22/02/2009 23:51:22] is a next page > page_29 < previous page Page 29 Hence F(f+g)≤F(f)+F(g), and we see that In an inner product space, the norm page_29 next page > is indeed a norm on X .

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Separation of variables for partial differential equations by Gunter H. Meyer George Cain


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