微分方程数值解上机作业

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1、9、给出初边值问题d2ud2u3rdx2(0

2、A=o=w

3、x=1=O(t>0)u1=—sm7DC8(0

4、示差分格式为UJ=心爲-2U：+UQ+2U：-U：；1（1）2、对于叫_（）二0的处理采用差分近似otU（Xgk）-U（Xm，—k）二o2k~匕-乞＞0⑵在⑴中令n=0,得叽=r2（Ci-2比+C.）+2比-U：⑶联立⑵⑶得，消去得:2U；乞（U阳-2匕+U4+匕丁是可得:n+lm"(U：L—2U：+UR+2U：厂Un-1m1=—SAn7TX82U〉2（U；爲-2U：：+Ul）+匕U；；：'"（u：L1=—sin^x8U'm上（U。2'曲—2t/"+trj+2tr—t/zmm-丿mnt-2(/°+t/°J+〃°mm-/mU：=02、方程的解析解为u=—{sin[龙(x-

5、1)]+sin[龙(x+Z)]}3、数据:精确解在给定网格格点的值n二0n=ln=2n=3n=4n二5n二6m=00000000m二100.0119360.0227040.031250.0367370.0386270.036737m=200.0227040.0431860.0594410.0698770.0734730.069877m=300.031250.0594410.0818140.0961780.1011270.096178m二400.0367370.0698770.0961780.1130640.1188820.113064m二500.0386270.073473

6、0.1011270.1188820.1250.118882m=600.0367370.0698770.0961780.1130640.1188820.113064m=700.031250.0594410.0818140.0961780.1011270.096178m=800.0227040.0431860.0594410.0698770.0734730.069877m=900.0119360.0227040.031250.0367370.0386270.036737m=1002.07E-093.94E-095.42E-096.37E-096.7E-096.37E-09近似解

7、在给定网格格点的值n=0n二1n二2n二3n=4n=5n=6m=00000000m二10.0386270.0367370.031250.0227040.0119361.04E-09-0.01194m=20.0734730.0698770.0594410.0431860.0227041.97E-09-0.0227m=30.1011270.0961780.0818140.0594410.031252.71E-09-0.03125m二40.1188820.1130640.0961780.0698770.0367373.19E-09-0.07347m=50.1250.1188820

8、.1011270.0734730.038627-0.036749.56E-09m=60.1188820.1130640.0961780.069877-2.7E-090.038627-0.03674m=70.1011270.0961780.0818140.0227040.069877-2.7E-09-0.03125m=80.0734730.0698770.0227040.0818140.022704-2E-09-0.0227m二9000.0698770.0227040.011936-IE-09-0.01194m二106.7E-096.37E-0900000精确解与近似解在给定

9、网格格点的误差n=0n二1n二2n=3n=4n二5n=6m=00000000m=l0.0386270.02480.008546-0.00855-0.0248-0.03863-0.04867m=20.0734730.0471730.016255-0.01625-0.04717-0.07347-0.09258m=30.1011270.0649280.022373-0.02237-0.06493-0.10113-0.12743111=40.1188820.0763270.0263-0.0263-0.07633-0.118