例: 単位を指定した pdesolve の使用

関数 > 求解と最適化 > ソルブブロック関数 > 例: 単位を指定した pdesolve の使用

熱方程式を解くには、pdesolve 関数を使用します。

1. 以下の入力パラメータを作成します。nTime および nX はグリッド点の数です。τMax は trange の上限で、a は係数です。

2. PDE ソルブブロックを設定します。熱方程式を使用し、一連の初期条件を追加します。ソルブブロック内で単位 (m) を使用します。

<region id="ID0EJXM6" actualWidth="306.13" actualHeight="229.95063991081344" top="76.800000000000011" left="76.800000000000011" height="229.95063991081344" width="306.13" xmlns="http://schemas.mathsoft.com/worksheet50">
  <solveblock>
    <regions>
      <region id="" actualWidth="182.17999999999998" actualHeight="29.943200000000004" top="38.40000000000002" left="38.4" solve-block-category="constraint">
        <math resultRef="32">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <minus />
              <apply>
                <id xml:space="preserve">
                  <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">T<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">t</Subscript></Span>
                </id>
                <sequence>
                  <id xml:space="preserve">x</id>
                  <id xml:space="preserve">t</id>
                </sequence>
              </apply>
              <apply>
                <mult />
                <apply>
                  <pow />
                  <id xml:space="preserve">a</id>
                  <real>2</real>
                </apply>
                <apply>
                  <id xml:space="preserve">
                    <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">T<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">xx</Subscript></Span>
                  </id>
                  <sequence>
                    <id xml:space="preserve">x</id>
                    <id xml:space="preserve">t</id>
                  </sequence>
                </apply>
              </apply>
            </apply>
            <real>0</real>
          </apply>
          <resultFormat>
            <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
          </resultFormat>
        </math>
      </region>
      <region id="" actualWidth="197.97000000000003" actualHeight="47.2" top="76.800000000000026" left="57.6" solve-block-category="constraint">
        <math resultRef="33">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <id xml:space="preserve">
                <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">T<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">x</Subscript></Span>
              </id>
              <sequence>
                <apply>
                  <scale />
                  <real>0</real>
                  <id xml:space="preserve">m</id>
                </apply>
                <id xml:space="preserve">t</id>
              </sequence>
            </apply>
            <apply>
              <scale />
              <apply>
                <neg />
                <parens>
                  <apply>
                    <scale />
                    <real>10</real>
                    <apply>
                      <pow />
                      <id xml:space="preserve">e</id>
                      <apply>
                        <scale />
                        <real>-0.1</real>
                        <id xml:space="preserve">t</id>
                      </apply>
                    </apply>
                  </apply>
                </parens>
              </apply>
              <apply>
                <div />
                <real>1</real>
                <id labels="UNIT" xml:space="preserve">m</id>
              </apply>
            </apply>
          </apply>
          <resultFormat>
            <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
          </resultFormat>
        </math>
      </region>
      <region id="" actualWidth="127.37333333333335" actualHeight="47.2" top="115.20000000000003" left="57.6" solve-block-category="constraint">
        <math resultRef="34">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <id xml:space="preserve">
                <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">T<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">x</Subscript></Span>
              </id>
              <sequence>
                <apply>
                  <scale />
                  <real>1</real>
                  <id labels="UNIT" xml:space="preserve">m</id>
                </apply>
                <id xml:space="preserve">t</id>
              </sequence>
            </apply>
            <apply>
              <scale />
              <real>0</real>
              <apply>
                <div />
                <real>1</real>
                <id labels="UNIT" xml:space="preserve">m</id>
              </apply>
            </apply>
          </apply>
          <resultFormat>
            <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
          </resultFormat>
        </math>
      </region>
      <region id="" actualWidth="80.710000000000008" actualHeight="25.6" top="153.60000000000008" left="57.6" solve-block-category="constraint">
        <math resultRef="35">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <id xml:space="preserve">T</id>
              <sequence>
                <id xml:space="preserve">x</id>
                <real>0</real>
              </sequence>
            </apply>
            <real>0</real>
          </apply>
          <resultFormat>
            <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
          </resultFormat>
        </math>
      </region>
      <region id="" actualWidth="267.43666666666672" actualHeight="44.2" top="192.00000000000006" left="38.4" solve-block-category="solver">
        <math resultRef="36">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id labels="VARIABLE" xml:space="preserve">T</id>
            <apply>
              <id labels="KEYWORD" xml:space="preserve">Pdesolve</id>
              <sequence>
                <id labels="VARIABLE" xml:space="preserve">T</id>
                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
                <matrix rows="2" cols="1">
                  <apply>
                    <scale />
                    <real>0</real>
                    <id labels="UNIT" xml:space="preserve" label-is-contextual="true">m</id>
                  </apply>
                  <apply>
                    <scale />
                    <real>1</real>
                    <id labels="UNIT" xml:space="preserve">m</id>
                  </apply>
                </matrix>
                <id xml:space="preserve">t</id>
                <matrix rows="2" cols="1">
                  <real>0</real>
                  <id labels="VARIABLE" xml:space="preserve">τMax</id>
                </matrix>
              </sequence>
            </apply>
          </define>
          <resultFormat>
            <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
          </resultFormat>
        </math>
      </region>
    </regions>
  </solveblock>
</region>

3. 3D プロットで解を視覚化するには、CreateMesh関数を使用します。

<region id="ID0EIZM6" actualWidth="360.09000000000003" actualHeight="25.6" top="345.6" left="76.800000000000011" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="37">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve">u</id>
      <apply>
        <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">CreateMesh</id>
        <sequence>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">T</id>
          <apply>
            <scale />
            <real>0</real>
            <id labels="UNIT" xml:space="preserve" label-is-contextual="true">m</id>
          </apply>
          <apply>
            <scale />
            <real>1</real>
            <id labels="UNIT" xml:space="preserve" label-is-contextual="true">m</id>
          </apply>
          <real>0</real>
          <id labels="VARIABLE" xml:space="preserve">τMax</id>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">nX</id>
          <id labels="VARIABLE" xml:space="preserve">nTime</id>
        </sequence>
      </apply>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

4. 3D プロットを使用して解をプロットします。

<region id="ID0EU2M6" actualWidth="266.63226521707361" actualHeight="257.01802056680197" top="518.40000000000009" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <plot3D>
      <traces>
        <trace3D num-of-points="41" resultRef="40" SurfaceFill="#FFC3D9A5">
          <traceStyle color="#FF2775A0">surface</traceStyle>
        </trace3D>
      </traces>
      <graph-size width="209.03226521707361" height="195.618020566802" />
      <axes>
        <xAxis start="0" end="1" rank="1">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <numberFormat>
            <general precision="3" />
          </numberFormat>
          <xyDomain scale-type="linear" allow-negative-values="true" auto-scale="true">
            <startValue />
            <endValue />
          </xyDomain>
        </xAxis>
        <yAxis start="0" end="20" rank="1">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <numberFormat>
            <general precision="3" />
          </numberFormat>
          <xyDomain scale-type="linear" allow-negative-values="true" auto-scale="true">
            <startValue />
            <endValue />
          </xyDomain>
        </yAxis>
        <zAxis start="-0.00055" end="0.00165" rank="1">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <numberFormat>
            <general precision="3" />
          </numberFormat>
          <zDomain scale-type="linear" allow-negative-values="true" auto-scale="true">
            <startValue />
            <endValue />
          </zDomain>
        </zAxis>
      </axes>
      <plotEquations>
        <plotEquation>
          <math>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">U</id>
          </math>
          <math>
            <placeholder xmlns="http://schemas.mathsoft.com/math50" />
          </math>
        </plotEquation>
      </plotEquations>
      <CameraTransformMatrix Rows="4" Columns="4">
        <Cells>
          <Cell>0.0011441619532079339</Cell>
          <Cell>0.091956017087953423</Cell>
          <Cell>0.9957624123327542</Cell>
          <Cell>0</Cell>
          <Cell>0.32809858802063024</Cell>
          <Cell>0.94060649816846953</Cell>
          <Cell>-0.087239510212507962</Cell>
          <Cell>0</Cell>
          <Cell>-0.94464279356394487</Cell>
          <Cell>0.32680805761879472</Cell>
          <Cell>-0.029094433198413844</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0.8341587442339774</Cell>
        </Cells>
      </CameraTransformMatrix>
      <CameraType>直交</CameraType>
    </plot3D>
  </plot>
</region>

5. 等高線図を使用して解をプロットします。