AEM_3e_Chapter_16

# AEM_3e_Chapter_16 - 16 16 Numerical Solutions of Partial...

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16 16 Numerical Solutions of Partial Differential Equations EXERCISES 16.1 Laplace’s Equation 3. The fgure shows the values oF u ( x, y ) along the boundary. We need to determine u 11 , u 21 , u 12 , and u 22 . By symmetry u 11 = u 21 and u 12 = u 22 . The system is u 21 + u 12 +0+0 4 u 11 =0 0+ u 22 + u 11 +0 4 u 21 u 22 + 3 / 2+0+ u 11 4 u 12 3 / 2+ u 12 + u 21 4 u 22 or 3 u 11 + u 12 u 11 3 u 12 = 3 2 . Solving we obtain u 11 = u 21 = 3 / 16 and u 12 = u 22 =3 3 / 16. 6. ±or Gauss-Seidel the coeﬃcients oF the unknowns u 11 , u 21 , u 31 , u 12 , u 22 , u 32 , u 13 , u 23 , u 33 are shown in the matrix 0 . 25 0 . 2 5 00000 . 25 0 . 25 0 . 2 5 0000 0 . 25 0 0 0 . 2 5 000 . 2 5 . 25 0 . 25 0 0 0 . 25 0 . 25 0 . 25 0 . 25 0 00 . 25 0 . 2 5 . 25 . 2 5 . 25 0 . 25 0 . 25 0 . 25 . 25 0 . 25 0 The constant terms are 7 . 5, 5, 20, 10, 0, 15, 17 . 5, 5, 27 . 5. We use 32 . 5 as the initial guess For each variable. Then u 11 =21 . 92, u 21 =28 . 30, u 31 =38 . 17, u 12 =29 . 38, u 22 =33 . 13, u 32 =44 . 38, u 13 =22 . 46, u 23 =30 . 45, and u 33 =46 . 21. 268

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TIME X=0.25 X=0.50 X=0.75 X=1.00 X=1.25 X=1.50 X=1.75 0.000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.025 0.7074 0.9520 0.9566 0.7444 0.2545 0.0371 0.0053 0.050 0.5606 0.8499 0.8685 0.6633 0.3303 0.1034 0.0223 0.075 0.4684 0.7473 0.7836 0.6191 0.3614 0.1529 0.0462 0.100 0.4015 0.6577 0.7084 0.5837 0.3753 0.1871 0.0684 0.125 0.3492 0.5821 0.6428 0.5510 0.3797 0.2101 0.0861 0.150 0.3069 0.5187 0.5857 0.5199 0.3778 0.2247 0.0990 0.175 0.2721 0.4652 0.5359 0.4901 0.3716 0.2329 0.1078 0.200 0.2430 0.4198 0.4921 0.4617 0.3622 0.2362 0.1132 0.225 0.2186 0.3809 0.4533 0.4348 0.3507 0.2358 0.1160 0.250 0.1977 0.3473 0.4189 0.4093 0.3378 0.2327 0.1166 0.275 0.1798 0.3181 0.3881 0.3853 0.3240 0.2275 0.1157 0.300 0.1643 0.2924 0.3604 0.3626 0.3097 0.2208 0.1136 0.325 0.1507 0.2697 0.3353 0.3412 0.2953 0.2131 0.1107 0.350 0.1387 0.2495 0.3125 0.3211 0.2808 0.2047 0.1071 0.375 0.1281 0.2313 0.2916 0.3021 0.2666 0.1960 0.1032 0.400 0.1187 0.2150 0.2725 0.2843 0.2528 0.1871 0.0989 0.425 0.1102 0.2002 0.2549 0.2675 0.2393 0.1781 0.0946 0.450 0.1025 0.1867 0.2387 0.2517 0.2263 0.1692 0.0902 0.475 0.0955 0.1743 0.2236 0.2368 0.2139 0.1606 0.0858 0.500 0.0891 0.1630 0.2097 0.2228 0.2020 0.1521 0.0814 0.525 0.0833 0.1525 0.1967 0.2096 0.1906 0.1439 0.0772 0.550 0.0779 0.1429 0.1846 0.1973 0.1798 0.1361 0.0731 0.575 0.0729 0.1339 0.1734 0.1856 0.1696 0.1285 0.0691 0.600 0.0683 0.1256 0.1628 0.1746 0.1598 0.1214 0.0653 0.625 0.0641 0.1179 0.1530 0.1643 0.1506 0.1145 0.0617 0.650 0.0601 0.1106 0.1438 0.1546 0.1419 0.1080 0.0582 0.675 0.0564 0.1039 0.1351 0.1455 0.1336 0.1018 0.0549 0.700 0.0530 0.0976 0.1270 0.1369 0.1259 0.0959 0.0518 0.725 0.0497 0.0917 0.1194 0.1288 0.1185 0.0904 0.0488 0.750 0.0467 0.0862 0.1123 0.1212 0.1116 0.0852 0.0460 0.775 0.0439 0.0810 0.1056 0.1140 0.1050 0.0802 0.0433 0.800 0.0413 0.0762 0.0993 0.1073 0.0989 0.0755 0.0408 0.825 0.0388 0.0716 0.0934 0.1009 0.0931 0.0711 0.0384 0.850 0.0365 0.0674 0.0879 0.0950 0.0876 0.0669 0.0362 0.875 0.0343 0.0633 0.0827 0.0894 0.0824 0.0630 0.0341 0.900 0.0323 0.0596 0.0778 0.0841 0.0776 0.0593 0.0321 0.925 0.0303 0.0560 0.0732 0.0791 0.0730 0.0558 0.0302 0.950 0.0285 0.0527 0.0688 0.0744 0.0687 0.0526 0.0284 0.975 0.0268 0.0496 0.0647 0.0700 0.0647 0.0495 0.0268 1.000 0.0253 0.0466 0.0609 0.0659 0.0608 0.0465 0.0252 16.2 The Heat Equation EXERCISES 16.2 The Heat Equation 3. We identify c =1 , a =2 , T , n = 8, and m = 40. Then h / 8=0 . 25, k / 40=0 . 025, and λ / 5=0 . 4.
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AEM_3e_Chapter_16 - 16 16 Numerical Solutions of Partial...

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