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pdes for electrostatics & heat flow 467xti-1,j i,j i+1,ji,j+1Figure 17.13 The algorithm for the heat equation in which the temperature at the locationx = i∆x and time t =(j + 1)∆t is computed from the temperature values at three points of anearlier time. The nodes with white centers correspond to known initial and boundaryconditions. (The boundaries are placed artificially close for illustrative purposes.)Substitution of these approximations into (17.58) yields the heat difference equationT (x, t +∆t) − T (x, t)∆t= K T (x +∆x, t)+T (x − ∆x, t) − 2T (x, t)Cρ∆x 2 . (17.70)We reorder (17.70) into a form in which T can be stepped forward in t:T i,j+1 = T i,j + η [T i+1,j + T i−1,j − 2T i,j ] , η = K∆tCρ∆x 2 , (17.71)where x = i∆x and t = j∆t. This algorithm is explicit because it provides a solutionin terms of known values of the temperature. If we tried to solve for the temperatureat all lattice sites in Figure. 17.13 simultaneously, then we would have animplicit algorithm that requires us to solve equations involving unknown valuesof the temperature. We see that the temperature at space-time point (i, j +1) iscomputed from the three temperature values at an earlier time j and at adjacentspace values i ± 1,i. We start the solution at the top row, moving it forward in timefor as long as we want and keeping the temperature along the ends fixed at 0 K(Figure 17.14).−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 467