Chem Differential Eq HW Solutions Fall 2011 121

Chem Differential Eq HW Solutions Fall 2011 121 - Section...

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Section 7.4 The Heat Equation and Gauss’s Kernel 121 You can easily check that this solution verifes the heat equation and u ( x, 0) = x 3 . IF n =4 , u ( x, t )= 2 ± j =0 ² 4 2 j ³ t j (2 j )! j ! · x 4 - 2 j = x 4 +12 tx 2 t 2 . Here too, you can check that this solution verifes the heat equation and u ( x, 0) = x 4 . We now derive a recurrence relation that relates the solutions corresponding to n - 1, n , and n + 1. Let u n = u n ( x, t ) denote the solution with initial temperature distribution u n ( x, 0) = x n . We have the Following recurrence relation u n +1 = xu n +2 ntu n - 1 . The prooF oF this Formula is very much like the prooF oF Bonnet’s recurrence Formula For the Legendre polynomials (Section 5.6). BeFore we give the prooF, let us veriFy the Formula with n = 3. The Formula states that u 4 u 3 +6 tu 2 . Sine u 4 = x 4 tx 2 t 2 , u 3 = x 3 tx , and u 2 = x 2 t , we see that the Formula is true For n = 3. We now prove the Formula using Leibniz rule oF differentiation. As in
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