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2.1
Sept 10, 2015 17:25:35 GMT
Post by matthorr on Sept 10, 2015 17:25:35 GMT
As mentioned in class, sometimes we need to expand a derivative of a function and not just the function itself. Given one is interested in the second derivative of u at point j - 2, please write down the Taylor series expansion of the second derivative for around point j for the next 3 terms as well as the general nth extra term: \( (u_{xx})_{j−2} = (u_{xx})_j \quad +? \)
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2.1
Sept 15, 2015 13:37:59 GMT
Post by matthorr on Sept 15, 2015 13:37:59 GMT
\( (u_{xx})_{j-2} = (u_{xx})_{j} - 2\Delta x(u_{xxx})_j + 2\Delta x^2(u_{xxxx})_j - \frac{4}{3}\Delta x^3(u_{xxxxx})_j + \frac{1}{n!}(-2\Delta x)^n \frac{\partial^{(n+2)}}{\partial x^{(n+2)}} u_j \)
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