X2 since this formula holds for all innitesimal

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Unformatted text preview: dition u(x, 0) = h(x), where h(x) is a given function, representing the initial temperature. For any fixed choice of t0 the function u(x, t0 ) is an element of V0 , the space of piecewise smooth functions defined for 0 ≤ x ≤ 1 which vanish at the endpoints. 92 Our idea is to replace the “infinite-dimensional” space V0 by a finite-dimensional Euclidean space R n−1 and reduce the partial differential equation to a system of ordinary differential equations. This corresponds to utilizing a discrete model for heat flow rather than a continuous one. For 0 ≤ i ≤ n, let xi = i/n and ui (t) = u(xi , t) = the temperature at xi at time t. Since u0 (t) = 0 = un (t) by the boundary conditions, the temperature at time t is specified by u1 (t) u (t) , u(t) = 2 · un−1 (t) a vector-valued function of one variable. The initial condition becomes h(x1 ) h(x2 ) . u(0) = h, where h= · h(xn−1 ) We can approximate the first-order partial derivative by a difference quotient: ∂u xi + xi+1 [ui+1 (t) − ui (t)] . ui+1 (t) − ui (t) = ,t = = n[ui+1 (t) − ui (t)]. ∂x 2 xi+1 − xi 1/n Similarly, we can approximate the sec...
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This document was uploaded on 01/12/2014.

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