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,
2
Similarly, one can have formula for x y y etc. The above formulae can also be represented in the form of computational molecule as follows: u
1
+o(h2) ‐‐‐ 1
‐
0
x 2h u i‐1, j u i, j u i+1, j 1 u i, j+1 u (1 / 2k )
y 0 u i , j ‐ u i ,j‐1 +o(k2) 2u
1
2
2
x
h 1 ‐ +o(h2) 1 1 i‐1, j u
u i , j u i+1, j ‐ 1 2u
2
xy 4h 0 1 +o(h2)
1 0 ‐ u i, j 1 0 ‐ The above approximations, when substituted in the PDE reduce it to the set of difference
equation which can be solved iteratively. However due to truncation error, care has to be taken
for convergence and stability of the solution. Example: Obtain the equivalent finite difference formulation of the given PDE u
u 2 u
a t
x x 2
Write the truncation error also. Solution:
Using formula (2.3),we have u i , j 1 u i , j
t a u i 1, j u i , j
x u i 1, j 2ui , j u i 1, j
(x) 2 r
2r r
u i , j 1 u i 1, j ar ui , j 1 ar u i 1, j x x x The truncation error is o(∆t) + o(∆x)....
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This note was uploaded on 04/07/2014 for the course MATH 545 taught by Professor Prof.ramabhargava during the Spring '14 term at Indian Institute of Technology, Roorkee.
 Spring '14
 Prof.RamaBhargava
 Differential Equations, Equations, Partial Differential Equations

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