例: Stiffb および Stiffr の拡大ヤコビ行列

関数 > 求解と最適化 > 微分方程式の解法 > 例: Stiffb および Stiffr の拡大ヤコビ行列

Jacob関数を使用して常微分方程式 (ODE) の拡大ヤコビ行列を計算し、次にそれをソルバーStiffbおよびStiffrに入力します。

1. 4 つの未知数を持つ系を定義します。

変数 yi は x の関数です。

2. 系に定数を定義します。

3. 初期値を定義します。

4. 系の右辺に対応するベクトル関数 D(x,y) を定義します。

5. augment関数を使用して拡大ヤコビ行列を次のように作成します。

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6. Stiffb 関数と Stiffr 関数を呼び出します。

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返された行列には点の数に対応する 5 列と 4 つの未知数の解が含まれます。

7. 返された行列 Stiffb と Stiffr から 4 つの未知数の解を抽出します。

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8. 2 つの関数で返された未知数の解をプロットし、比較します。

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 <startValue resultRef="225">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <neg />
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 <secondTickValue resultRef="227">
 <apply xmlns="http://schemas.mathsoft.com/math50">
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 <apply xmlns="http://schemas.mathsoft.com/math50">
 <mult />
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 <axisLine position="origin" positionticmark="1" legendWidth="61.7533333333333" />
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 <gridLabels display="true" />
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 <tickMarks display="true" />
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 <axisLabel />
 <markers />
 <numberFormat>
 <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
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 <plotEquations>
 <plotEquation>
 <math resultRef="248">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">n</id>
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 <math resultRef="249">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
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 <xyDomain scale-type="linear" auto-scale="true">
 <startValue>
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </startValue>
 <secondTickValue resultRef="247">
 <real xmlns="http://schemas.mathsoft.com/math50">5</real>
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 <endValue>
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </endValue>
 </xyDomain>
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 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-0.5" end="2.5">
 <axisLine position="origin" positionticmark="0" legendWidth="73.69" />
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 <gridFrequency>7</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <numberFormat>
 <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
 </numberFormat>
 <plotEquations>
 <plotEquation>
 <math resultRef="256">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 yb<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
 </id>
 </math>
 <math resultRef="257">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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 </plotEquation>
 <plotEquation>
 <math resultRef="258">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 yr<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">2</Subscript>
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 <math resultRef="259">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
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 <secondTickValue resultRef="253">
 <real xmlns="http://schemas.mathsoft.com/math50">0</real>
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 <endValue resultRef="255">
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 </yAxis>
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</region>

<region id="ID0ES5T5" top="2227.2000000000003" left="38.400000000000006" height="230.40000486373904" width="671.600008368492" xmlns="http://schemas.mathsoft.com/worksheet50">
 <plot background-type="white" origin-positioning="true">
 <xyPlot>
 <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
 <legend />
 <traces>
 <trace resultRef="270">
 <traceStyle color="#FFFF0000" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
 </trace>
 <trace resultRef="271">
 <traceStyle color="#FF000000" symbol="none" line-weight="1" line-style="DashDotDot">lines</traceStyle>
 </trace>
 </traces>
 <graph-size width="571.600008368492" height="172.800004863739" />
 <axes>
 <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="20">
 <axisLine position="ticknumberlock" positionticmark="0" legendWidth="61.7533333333333" />
 <axisGrid>
 <gridFrequency>5</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <numberFormat>
 <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
 </numberFormat>
 <plotEquations>
 <plotEquation>
 <math resultRef="274">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">n</id>
 </math>
 <math resultRef="275">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue>
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </startValue>
 <secondTickValue resultRef="273">
 <real xmlns="http://schemas.mathsoft.com/math50">5</real>
 </secondTickValue>
 <endValue>
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </endValue>
 </xyDomain>
 </xAxis>
 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="8E-06">
 <axisLine position="ticknumberlock" positionticmark="0" legendWidth="73.69" />
 <axisGrid>
 <gridFrequency>3</gridFrequency>
 <gridLabels display="true" />
 <gridLines />
 <tickMarks display="true" />
 </axisGrid>
 <axisLabel />
 <markers />
 <numberFormat>
 <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
 </numberFormat>
 <plotEquations>
 <plotEquation>
 <math resultRef="282">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 yb<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </math>
 <math resultRef="283">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 <plotEquation>
 <math resultRef="284">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
 yr<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">3</Subscript>
 </id>
 </math>
 <math resultRef="285">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue resultRef="277">
 <real xmlns="http://schemas.mathsoft.com/math50">0</real>
 </startValue>
 <secondTickValue resultRef="279">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <mult />
 <real>4</real>
 <apply>
 <pow />
 <real>10</real>
 <real>-6</real>
 </apply>
 </apply>
 </secondTickValue>
 <endValue resultRef="281">
 <apply xmlns="http://schemas.mathsoft.com/math50">
 <mult />
 <real>8</real>
 <apply>
 <pow />
 <real>10</real>
 <real>-6</real>
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 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

プロットには 2 つの ODE ソルバーが同じ解を返していることが示されています。