NAMA : Rahmad Hidayat NIM : 090401010 Sebuah batang AB dengan panjang 10 dijaga pada suhu 0 oC. Pada t = 0, temperatur batang naik seragam dari ujung-ujungnya sampai 2 oC pada titik tengah batang AB. Tentukan persamaan temperatur u(x,t) pada setiap titik pada batang, jarak x dari kiri pada suatu waktu,t. Penyelesaian : Tc= 2 oC Tb= 0 oC Ta= 0 oC L = 10 L/2 = 5 C B A Persamaan panas pada satu dimensi : = Dimana k = konduktivitas termal material σ = panas spesifik material ρ = massa jenis benda u(0,t) = 0 , u(10,t) = 0 untuk t u(x,0) = f(x) untuk 0 Persamaannya : u(x,t) = n = 1,2,3,….,dst x=l/2 f(l/2) = = 2 sin n = 1 Maka persamaannya menjadi : u(x,t) = n = 1,2,3,….,dst = = dan c = 3 Tentukan persamaan solusi u = f(x,t) jika kondisi batas u(0,t) = 0 dan u(2,t) = 0 untuk t u(x,0) = x(2-x) dan Penyelesaian : u(x,t) = = 0 Maka u(x,t) = u(x,0) = x(2-x) = f(x) u(x,0) = = f(x) dx dx = dx = - = ] = Maka persamaannya menjadi u(x,t) = 4. Buktikan I = Penyelesaian : Misalkan y = x = dy = n. dx dx = I = = I = I = I = I = I = I = v-1 = v = + 1 v = I = = I = = ( terbukti )  5. Selesaikan dengan metode runge kutta  y" = x-y-y' ; x = 0 ; y = 0 ; y' = 1 ; x = 0(0,2)1,0 n x K1 K2 K3 K4 y y' y" h= 0.2 p q 0 1 -0.0200000 0.0217000 0.0262870 0.0348346 0.0000000 1.0000000 -1.0000000     0.009329 0.036936 1 1.2 -0.0168373 0.0350846 0.0418345 0.0546467 0.2093290 1.1846810 -0.8418668     0.020027 0.063883 2 1.4 -0.0167290 0.0548639 0.0656028 0.0857782 0.4662925 1.5040936 -0.8364482     0.034579 0.103327 3 1.6 -0.0189959 0.0854558 0.1032126 0.1360838 0.8016904 2.0207310 -0.9497966     0.056558 0.164808 4 1.8 -0.0238160 0.1343947 0.1644547 0.2194888 1.2623941 2.8447724 -1.1908013     0.091678 0.264457 5 2 -0.0320555 0.2149192 0.2667838 0.3611999 1.9230263 4.1670582 -1.6027725     0.149883 0.43085 y" = (1-x)y' – y ; x = 0 ; y = 1 ; y' = 1 ; x = 0(0,2)1,0 n x K1 K2 K3 K4 y y' y" h= 0.2 p q 0 0 0.0000000 0.0240000 0.0242400 0.0294544 1.0000000 1.0000000 0.0000000     0.01608 0.041978 1 0.2 -0.0049633 0.0338270 0.0349907 0.0423394 1.2160800 1.2098907 -0.2481675     0.021285 0.058337 2 0.4 -0.0115679 0.0469695 0.0498964 0.0605976 1.4793429 1.5015764 -0.5783971     0.028433 0.080921 3 0.6 -0.0209124 0.0650923 0.0711126 0.0870857 1.8080909 1.9061790 -1.0456193     0.038431 0.112861 4 0.8 -0.0346732 0.0906709 0.1019518 0.1262762 2.2277575 2.4704839 -1.7336607     0.05265 0.158949 5 1 -0.0554901 0.1274743 0.1476004 0.1852926 2.7745041 3.2652312 -2.7745041     0.073195 0.226651 y" = 1+x - y2 ; x = 0 ; y = 2 ; y' = 1 ; x = 0(0,1)0,5 n x K1 K2 K3 K4 y y' y" h= 0.1 p q 0 0 -0.0150000 0.0104438 0.0105074 0.0111576 2.0000000 1.0000000 -3.0000000     0.001984 0.012687 1 0.1 -0.0165917 0.0114916 0.0117022 0.0124928 2.1019837 1.1268661 -3.3183355     0.002201 0.014096 2 0.2 -0.0185726 0.0127307 0.0131220 0.0140793 2.2168710 1.2678287 -3.7145172     0.002427 0.015737 3 0.3 -0.0210205 0.0141867 0.0148028 0.0159595 2.3460806 1.4252026 -4.2040943     0.002656 0.017639 4 0.4 -0.0240318 0.0158895 0.0167878 0.0181844 2.4912572 1.6015960 -4.8063625     0.002882 0.019836 5 0.5 -0.0277265 0.0178742 0.0191282 0.0208147 2.6542986 1.7999531 -5.5453012     0.003092 0.022364 y" = (x+2)y - 2y' ; x = 0 , y = 1, y' = 0 ; x = 0(0,2)1,0 n x K1 K2 K3 K4 y y' y" h= 0.1 p q 0 0 0.0100000 0.0050375 0.0050251 0.0050754 1.0000000 0.0000000 2.0000000     0.006688 0.011734 1 0.1 0.0093969 0.0052330 0.0052018 0.0053395 1.0066875 0.1173352 1.8793734     0.006611 0.011869 2 0.2 0.0089151 0.0056018 0.0055604 0.0057918 1.0250316 0.2360215 1.7830266     0.006692 0.012344 3 0.3 0.0085417 0.0061557 0.0061140 0.0064504 1.0553262 0.3594588 1.7083325     0.006937 0.013177 4 0.4 0.0082662 0.0069154 0.0068851 0.0073434 1.0982092 0.4912301 1.6532418     0.007356 0.014404 5 0.5 0.0080809 0.0079116 0.0079069 0.0085108 1.1546877 0.6352654 1.6161884     0.007966 0.016076 5 0.6 0.0079801 0.0091863 0.0092255 0.0100069 1.2261807 0.7960278 1.5960144 0.008797 0.01827 5 0.7 0.0079595 0.0107963 0.0109026 0.0119039 1.3145808 0.9787302 1.5919079 0.009886 0.021087 5 0.8 0.0080168 0.0128156 0.0130196 0.0142966 1.4223400 1.1896009 1.6033501 0.011284 0.024661 5 0.9 0.0081503 0.0153413 0.0156829 0.0173088 1.5525841 1.4362135 1.6300668 0.013058 0.029169 5 1 0.0083599 0.0184991 0.0190315 0.0211024 1.7092636 1.7279053 1.6719802 0.015297 0.034841 y" = (y'-xy')/x^2 ; x = 1, y = 0, y' = 1 ; x = 1,0(0,2)2,0 n x K1 K2 K3 K4 y y' y" h= 0.1 p q 0 1 -0.0050000 0.0052313 0.0057684 0.0066634 0.0000000 1.0000000 -1.0000000     0.002 0.007888 1 1.1 -0.0044825 0.0067199 0.0073640 0.0084432 0.1019999 1.0788755 -0.8964985     0.0032 0.01071 2 1.2 -0.0042017 0.0085064 0.0093006 0.0106228 0.2130879 1.1859708 -0.8403313     0.004535 0.014012 3 1.3 -0.0041056 0.0106815 0.0116796 0.0133203 0.3362201 1.3260883 -0.8211211     0.006085 0.017979 4 1.4 -0.0041666 0.0133614 0.0146321 0.0166896 0.4749141 1.5058776 -0.8333238     0.007942 0.022837 5 1.5 -0.0043732 0.0166968 0.0183297 0.0209327 0.6334442 1.7342442 -0.8746320     0.010218 0.028871 6 1.6 -0.0047259 0.0208847 0.0229976 0.0263166 0.8170864 2.0229526 -0.9451710 0.013052 0.036452 7 1.7 -0.0052358 0.0261847 0.0289340 0.0331959 1.0324338 2.3874707 -1.0471510 0.016628 0.046066 8 1.8 -0.0059241 0.0329409 0.0365359 0.0420448 1.2878085 2.8481297 -1.1848225 0.021184 0.058358 9 1.9 -0.0068233 0.0416123 0.0463348 0.0535006 1.5938057 3.4317108 -1.3646661 0.027041 0.074191 10 2 -0.0079790 0.0528152 0.0590466 0.0684249 1.9640181 4.1736160 -1.5958035 0.034628 0.094723