用PDE Toolbox求解三类典型方程求解抛物型方程.docx

用PDE Toolbox求解三类典型方程求解抛物型方程.docx

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用PDEToolbox求解三类典型方程求解抛物型方程

实验二用PDEToolbox求解三类典型方程

----求解抛物型方程

一、实验目的

本次实验在掌握PDEToolbox的功能和PDE图形用户界面的基础上，求解抛物型方程。

二、实验内容

1、解抛物型方程的一个例子

热传导方程：

金属板的导热问题

考虑一个带有矩形孔的金属板上的热传导问题。

板的左边保持在100℃，板的右边热量从板向环境空气定常流动，其他边及内孔边界保持绝缘。

初始

时板的温度为0℃，于是概括为如下定解问题：

的外边界定点坐标为（-0.5，-0.8），（0.5，-0.8），（0.5，0.8），（-0.5，0.8），内边界顶点坐标为（-00.5，-0.4），（00.5，-0.4），（00.5，0.4），（-00.5，0.4）。

使用GUI求解这一问题。

在PDEToolbox窗口的工具栏中选择GenericScalar.

第一步：

区域设置单击工具

，在窗口拖拉出一个矩形，双击矩形区域，在对话框中输入Left为-0.5，Bottom为-0.8，Width为1，Height为1.6，单击OK按钮，显示矩形区域R1。

用同样方法做内孔R2，只要设置Left为-00.5，Bottom为-0.4，Width为0.1，Height为0.8即可。

然后在栏中键入R1-R2。

第二步：

设置边界条件单击工具

，图形边界变红，逐段双击边界，打开BoundaryCondition对话框，输入边界条件。

对于同一类型的边界，可按shift键，将多个边界同时选择，统一设置边界条件。

本题在左边界上选择Dirichlet条件，输入h为1，r为100；右边界上，选择Neumann条件，输入g为-1，q为0；其他边界上，选择Neumann条件，输入g为0，q为0。

第三步：

设置方程类型单击PDE菜单中PDESpecification…选项，打开PDESpecification对话框，选择方程类型。

本题单击Parabolic（抛物型），输入d为1,c为1，a为0，f为0，然后单击OK按钮。

第四步：

网格剖分单击工具

，或者单击Mesh菜单中InitializeMesh选项，可进行初始网格剖分，如果需要网格加密，再单击

，或者单击Mesh菜单中RefineMesh选项。

第五步：

初值和误差设置单击Solve菜单中Parameters…选项，打开SolveParameters…对话框，输入Time为0：

5，u（t0）为0，Relativetolerance为0.01，Absolutetolerance为0.001，然后单击OK按钮。

第六步：

解的图形单击Plot菜单中Parameters…选项，出现PlotSelection对话框，从中可以选择Color,Contour,Arrows,Deformedmesh,Height（3-Dplot），还可以设置等值线的数目等。

本例中选择Color,Contour和Arrows三项，然后单击Plot按钮，窗口显示出时的解的彩色图形。

第七步：

数值解的输出单击Solve菜单中ExportSolution选项，在打开的Export对话框中输入u，单击OK按钮确定。

再在MATLAB命令窗口键入u，按回车键确定，即显示按节点编号排列的数值解

>>u

u=

079.629195.377498.038098.570698.6395

079.629295.377498.038098.570598.6395

100.0000100.0000100.0000100.0000100.0000100.0000

083.357796.385898.586999.027599.0845

083.365396.387498.587599.027999.0849

089.818897.811199.161399.431599.4664

089.811897.809699.160799.431099.4660

100.0000100.0000100.0000100.0000100.0000100.0000

079.332495.313998.013998.554498.6243

078.559295.146297.948698.509698.5822

077.504594.917197.859298.448298.5244

076.590594.720097.783398.396798.4760

076.247094.643297.751698.374098.4545

076.572894.716797.782598.396398.4757

077.499694.916097.858798.447898.5241

078.557595.145897.948598.509598.5821

079.331195.313698.013898.554398.6243

080.302095.620698.208698.726698.7936

082.123196.082598.440898.912898.9739

084.960796.739198.728899.127199.1786

088.503697.527199.051599.356599.3960

092.353498.364499.379899.583099.6093

096.194999.189099.694899.795999.8090

080.325695.716398.316798.837298.9046

078.183495.245098.127898.705098.7796

077.157895.020298.038498.642798.7209

077.155295.019698.038198.642598.7207

078.180195.244198.127398.704698.7792

080.324495.715998.316598.837198.9044

100.0000100.0000100.0000100.0000100.0000100.0000

100.0000100.0000100.0000100.0000100.0000100.0000

100.0000100.0000100.0000100.0000100.0000100.0000

100.0000100.0000100.0000100.0000100.0000100.0000

100.0000100.0000100.0000100.0000100.0000100.0000

100.0000100.0000100.0000100.0000100.0000100.0000

100.0000100.0000100.0000100.0000100.0000100.0000

100.0000100.0000100.0000100.0000100.0000100.0000

100.0000100.0000100.0000100.0000100.0000100.0000

096.194699.188999.694799.795999.8090

092.353298.364399.379799.582999.6092

088.510197.528299.051699.356599.3959

084.949196.736998.728399.126999.1784

082.118996.081798.440598.912798.9738

080.301895.620698.208698.726698.7936

092.781198.458999.418099.609899.6347

094.801098.899199.591299.729699.7476

095.720199.098699.669299.783299.7980

095.724399.099599.669599.783599.7982

094.808998.900899.591999.730199.7480

092.784298.459699.418299.610099.6348

096.123199.179199.695299.798499.8118

077.913995.144398.055698.638598.7139

096.236699.203499.704599.804799.8176

077.828795.123898.046198.631198.7068

081.275295.891498.360798.855098.9190

093.190298.544999.449599.630499.6539

093.123698.530699.443999.626799.6503

081.315595.901198.365398.858598.9224

097.027799.375099.771499.850699.8609

076.689794.877297.950398.565798.6452

086.442797.071398.866899.226299.2727

086.820097.155098.900999.250399.2955

078.450695.222498.056298.623598.6969

096.920299.346199.755899.837899.8484

096.958299.354299.758999.839899.8503

078.430095.217198.053498.621298.6947

077.254295.027098.030098.631398.7090

095.976399.151799.687999.795199.8090

077.304295.038498.034998.634998.7125

095.922999.140299.683599.792199.8062

083.671896.446598.604799.036799.0926

090.245097.907199.201499.460499.4940

090.022897.858899.182599.447399.4816

083.502696.407698.587799.024199.0806

097.579399.490099.812799.877299.8855

076.857094.884997.931098.540998.6198

076.715894.845897.909298.522698.6019

097.844499.546299.833699.891099.8985

080.115795.579298.191798.714798.7823

096.272899.205699.701099.800299.8130

096.292399.209899.702699.801299.8140

080.121395.580998.192798.715598.7832

093.695198.656299.494199.661899.6835

079.902395.607098.260498.791698.8604

093.743098.666599.498299.664599.6861

079.861295.597698.256498.788798.8576

086.528697.088398.872299.229399.2755

086.666397.118898.884699.238099.2837

089.667997.780399.150799.424999.4604

083.904696.502998.631399.057399.1125

083.778796.474598.619499.048799.1042

089.521997.748399.138099.416199.4521

096.770299.312499.741899.827799.8388

079.458595.432098.130798.670998.7408

096.750099.308199.740299.826699.8378

079.452295.430198.129698.670098.7399

098.211999.622199.860399.907999.9141

077.268694.939197.924898.522598.5999

098.129899.604799.853899.903699.9100

077.312794.951297.931598.528298.6054

076.406294.750197.849798.470398.5506

098.591699.704099.891899.929499.9342

096.088299.176999.698599.802799.8162

076.963194.968998.011498.620598.6993

078.597395.326698.153298.719198.7923

094.670598.869199.578299.720199.7385

078.554095.316898.149198.716198.7895

094.716098.879099.582199.722799.7409

081.384695.904598.357598.848598.9120

093.567798.625899.480299.651199.6733

081.356495.897598.354198.845898.9094

093.622598.637699.484799.654299.6762

086.337397.046998.856199.218299.2651

087.034297.201898.919499.263299.3077

094.409298.807699.550599.699299.7184

080.116295.627698.248298.772998.8407

080.138695.632998.250898.774898.8426

094.374298.800199.547699.697299.7166

079.788195.463098.111298.641398.7099

098.115999.598799.849199.899299.9057

098.119899.599599.849499.899499.9059

079.788895.463298.111498.641498.7100

087.959197.406099.001999.321299.3626

085.464996.851698.775299.160299.2100

099.107199.812299.931399.955199.9582

076.427994.728897.821298.440398.5204

082.053996.085798.456498.930998.9923

091.561898.192299.312299.536399.5653

082.013396.076598.452498.928098.9896

091.611398.202999.316499.539299.5681

083.967296.518298.638699.063099.1179

089.345197.708099.120799.403499.4400

083.906096.504498.632999.058999.1140

089.413697.723099.126799.407699.4439

>>