Thus the dierential operator l d2 dx2 is approximated

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Unformatted text preview: ond-order partial derivative: ∂2u . (xi , t) = ∂x2 ∂u . =n ∂x ∂u ∂x xi +xi+1 ,t 2 − ∂u ∂x xi−1 +xi ,t 2 1/n xi + xi+1 ∂u ,t − 2 ∂x xi−1 + xi ,t 2 . = n2 [ui−1 (t) − 2ui (t) + ui+1 (t)]. Thus the partial differential equation ∂u ∂2u = ∂t ∂x2 can be approximated by a system of ordinary differential equations dui = n2 (ui−1 − 2ui + ui+1 ). dt This is a first order linear system which can be presented in vector form as −2 1 0 ··· 0 1 −2 1 · · · 0 du 2 0 1 −2 · · · · , = n P u, where P = dt · · · ··· · 0 0 · · · · −2 93 the last matrix having n − 1 rows and n − 1 columns. Finally, we can rewrite this as du = Au, dt where A = n2 P, (4.16) a system exactly like the ones studied in Section 2.5. In the limit as n → ∞ one can use the Mathematica program of §2.6 to check that the eigenvalues of A approach the eigenvalues of d2 /dx2 as determined in the preceding section, and the eigenvectors approximate more and more closely the standard orthonormal basis of sine functions. One cou...
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