Chapter 16

# Chapter 16 - 16 Numerical Solutions of Partial Differential...

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16 Numerical Solutions of Partial Differential Equations Exercises 16.1 1. The fgure shows the values oF u ( x, y ) along the boundary. We need to determine u 11 and u 21 . The system is u 21 +2+0+0 4 u 11 =0 1+2+ u 11 +0 4 u 21 or 4 u 11 + u 21 = 2 u 11 4 u 21 = 3 . Solving we obtain u 11 =11 / 15 and u 21 =14 / 15. 2. The fgure shows the values oF u ( x, y ) along the boundary. We need to determine u 11 , u 21 , and u 31 . By symmetry u 11 = u 31 and the system is u 21 +0+0+100 4 u 11 u 31 +0+ u 11 +100 4 u 21 0+0+ u 21 4 u 31 or 4 u 11 + u 21 = 100 2 u 11 4 u 21 = 100 . Solving we obtain u 11 = u 31 = 250 / 7 and u 21 = 300 / 7. 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+ u 22 + u 11 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. 4. 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 . The system is u 21 + u 12 +8+0 4 u 11 u 22 + u 11 4 u 21 u 22 +0+16+ u 11 4 u 12 u 12 + u 21 4 u 22 or 4 u 11 + u 21 + u 12 = 8 u 11 4 u 21 + u 22 u 11 4 u 12 + u 22 = 16 u 21 + u 12 4 u 22 . Solving we obtain u 11 / 3, u 21 =4 / 3, u 12 =16 / 3, and u 22 =5 / 3. 739

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Exercises 16.1 5. The fgure shows the values oF u ( x, y ) along the boundary. ±or Gauss-Seidel the co- eﬃcients 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 . 2 5 000 . 2 5 . 2 5 . 25 0 . 25 0 0 0 . 25 0 . 25 0 . 25 0 . 25 0 00 . 25 0 . 25 0 0 0 . 25 . 25 0 0 0 . 25 0 . 25 0 . 25 0 . 25 . 25 0 . 25 0 The constant terms in the equations are 0, 0, 6 . 25, 0, 0, 12 . 5, 6 . 25, 12 . 5, 37 . 5. We use 25 as the initial guess For each variable. Then u 11 =6 . 25, u 21 = u 12 =12 . 5, u 31 = u 13 =18 . 75, u 22 = 25, u 32 = u 23 =37 . 5, and u 33 =56 . 25 6. The coeﬃcients oF the unknowns are the same as shown above in Problem 5. 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. 7. (a) Using the di²erence approximations For u xx and u yy we obtain u xx + u = 1 h 2 ( u i +1 ,j + u i,j +1 + u i 1 ,j + u i,j 1 4 u ij )= f ( x, y ) so that u i +1 ,j + u i,j +1 + u i 1 ,j + u i,j 1 4 u ij = h 2 f ( x, y ) . (b) By symmetry, as shown in the fgure, we need only solve For u 1 , u 2 , u 3 , u 4 , and u 5 . The di²erence equations are u 2 +0+0+1 4 u 1 = 1 4 ( 2) u 3 +0+ u 1 +1 4 u 2 = 1 4 ( 2) u 4 u 2 + u 5 4 u 3 = 1 4 ( 2) 0+0+ u 3 + u 3 4 u 4 = 1 4 ( 2) u 3 + u 3 +1+1 4 u 5 = 1 4 ( 2) or u 1 =0 . 25 u 2 +0 . 375 u 2 . 25 u 1 . 25 u 3 . 375 u 3 . 25 u 2 . 25 u 4 . 25 u 5 . 125 u 4 . 5 u 3 . 125 u 5 . 5 u 3 . 625 . Using Gauss-Seidel iteration we fnd u 1 . 5427, u 2 . 6707, u 3 . 6402, u 4 . 4451, and u 5 . 9451. 740
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 0.6000 1.0000 1.0000 0.6000 0.4000 0.0000 0.0000 0.050 0.5200 0.8400 0.8400 0.6800 0.3200 0.1600 0.0000 0.075 0.4400 0.7120 0.7760 0.6000 0.4000 0.1600 0.0640 0.100 0.3728 0.6288 0.6800 0.5904 0.3840 0.2176 0.0768 0.125 0.3261 0.5469 0.6237 0.5437 0.4000 0.2278 0.1024 0.150 0.2840 0.4893 0.5610 0.5182 0.3886 0.2465 0.1116 0.175 0.2525 0.4358 0.5152 0.4835 0.3836 0.2494 0.1209 0.200 0.2248 0.3942 0.4708 0.4562 0.3699 0.2517 0.1239 0.225 0.2027 0.3571 0.4343 0.4275 0.3571 0.2479 0.1255 0.250 0.1834 0.3262 0.4007 0.4021 0.3416 0.2426 0.1242 0.275 0.1672 0.2989 0.3715 0.3773 0.3262 0.2348 0.1219 0.300 0.1530 0.2752 0.3448 0.3545 0.3101 0.2262 0.1183 0.325 0.1407 0.2541 0.3209 0.3329 0.2943 0.2166 0.1141 0.350 0.1298 0.2354 0.2990 0.3126 0.2787 0.2067 0.1095 0.375 0.1201 0.2186 0.2790 0.2936 0.2635 0.1966 0.1046 0.400 0.1115 0.2034 0.2607 0.2757 0.2488 0.1865 0.0996

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Chapter 16 - 16 Numerical Solutions of Partial Differential...

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