Abstract
Finite differences are used to approximate derivatives of a function f, in order to solve differential and partial differential equations. In this way the continuous problem can be replaced by a discrete one. In the first section we consider finite differences that lead to approximations of first and second derivatives. We consider the univariate case in Exercise 13.1 , and the bivariate Laplacian in Exercise 13.2.
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Notes
- 1.
Use Taylor expansions with remainder.
- 2.
Consider row i of \(\boldsymbol{A}_{h}\boldsymbol{v}\) where \(v_{i}:=\min _{j=1,\ldots,n}v_{j}\) to show that v i ≥ 0.
- 3.
Use the results of Exercise 2.8.
- 4.
Use meshgrid .
- 5.
Use while.
- 6.
Use Theorem 2.1 that gives properties of \(\left (\boldsymbol{I} + \frac{\varDelta t} {h^{2}} \boldsymbol{A}\right )\).
- 7.
- 8.
Use meshgrid, surf, contour .
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Lyche, T., Merrien, JL. (2014). Finite Differences for Differential and Partial Differential Equations. In: Exercises in Computational Mathematics with MATLAB. Problem Books in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43511-3_13
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