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xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N</ml:id> <ml:real>1000</ml:real> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="93" left="6" top="68.25" width="135" height="12" align-x="23.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Mathcad numerical solution: </text> </region> <region region-id="94" left="6" top="87" width="24" height="12.75" align-x="17.25" align-y="96" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="3222be27-766f-4168-a76f-30e54d20f6ff" version-id="870b8bca-0a04-40db-a6b6-846bfa5f82b2" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="true" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\wave.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10">Solving the wave equation using both a solve block and the numol solver.</originComment> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">21546c5319a511f4b5452a7b23aab828</contentHash> <ml:id xml:space="preserve">Given</ml:id> </ml:provenance> </math> <rendering item-idref="8"> <element-image-map> <box left="1.5" top="0.75" width="22.5" 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branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">17e8e432e5396a172b2e2905fa4b71b7</contentHash> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:provenance> </math> <rendering item-idref="12"> <element-image-map> <box left="1.5" top="0.75" width="39.75" height="11.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="141" left="306" top="96.75" width="155.25" height="30" align-x="317.25" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">u</ml:id> <ml:apply> <ml:Pdesolve method="5-pt-diff"/> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">x</ml:id> <ml:matrix rows="2" cols="1"> <ml:real>0</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:matrix> <ml:id xml:space="preserve">t</ml:id> <ml:matrix rows="2" cols="1"> <ml:real>0</ml:real> <ml:id xml:space="preserve">τ</ml:id> </ml:matrix> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="103" left="6" top="134.25" width="199.5" height="12" align-x="24" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Variables for generation of 2D isochrones: </text> </region> <region region-id="138" left="6" top="159" width="39" height="12.75" align-x="19.5" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">8fd54222a2e274065477a6550bbfbfea</contentHash> <ml:define> <ml:id xml:space="preserve">X</ml:id> <ml:real>1000</ml:real> </ml:define> </ml:provenance> </math> <rendering item-idref="14"> <element-image-map> <box left="1.5" top="0.75" width="37.5" height="11.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="137" left="78" top="159" width="28.5" height="12.75" align-x="90" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">T</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="136" left="138" top="159" width="35.25" height="12.75" align-x="146.25" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">X</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="135" left="186" top="159" width="34.5" height="12.75" align-x="195" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">T</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="134" left="252" top="152.25" width="33" height="27.75" align-x="264" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">2eb956dddf3e5af6939b47c0b977f164</contentHash> <ml:define> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:id xml:space="preserve">X</ml:id> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="18"> <element-image-map> <box left="1.5" top="0.75" width="31.5" height="26.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="133" left="300" top="151.5" width="33" height="28.5" align-x="312" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="139" left="378" top="159" width="59.25" height="18" align-x="400.5" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="124" left="6" top="188.25" width="286.5" height="12" align-x="26.25" align-y="198" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Analytical solution for given boundary and initial conditions: </text> </region> <region region-id="130" left="6" top="205.5" width="327.75" height="43.5" align-x="40.5" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef 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style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="v">c</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:real>100</ml:real> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="21"> <element-image-map> <box left="1.5" top="0.75" width="326.25" height="42" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="129" left="378" top="219" width="68.25" height="21" align-x="405" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="a">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="a">u</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="158" left="6" top="266.25" width="181.5" height="12" align-x="18" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Create 3D surfaces and 2D isochrones: </text> </region> <region region-id="157" left="198" top="266.25" width="120.75" height="13.5" align-x="213" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">M</ml:id> <ml:apply> <ml:id xml:space="preserve">CreateMesh</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">h</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="156" left="330" top="266.25" width="129.75" height="17.25" align-x="349.5" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="a">M</ml:id> <ml:apply> <ml:id xml:space="preserve">CreateMesh</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="a">u</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">h</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="171" left="240" top="282" width="215.25" height="222" align-x="346.5" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" 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xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N</ml:id> <ml:real>1000</ml:real> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="93" left="6" top="68.25" width="135" height="12" align-x="23.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Mathcad numerical solution: </text> </region> <region region-id="94" left="6" top="87" width="24" height="12.75" align-x="17.25" align-y="96" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="3222be27-766f-4168-a76f-30e54d20f6ff" version-id="870b8bca-0a04-40db-a6b6-846bfa5f82b2" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" 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</ml:apply> </ml:provenance> </math> <rendering item-idref="12"> <element-image-map> <box left="1.5" top="0.75" width="39.75" height="11.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="141" left="306" top="96.75" width="155.25" height="30" align-x="317.25" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">u</ml:id> <ml:apply> <ml:Pdesolve method="5-pt-diff"/> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">x</ml:id> <ml:matrix rows="2" cols="1"> <ml:real>0</ml:real> <ml:id xml:space="preserve">h</ml:id> </ml:matrix> <ml:id xml:space="preserve">t</ml:id> <ml:matrix rows="2" cols="1"> <ml:real>0</ml:real> <ml:id xml:space="preserve">τ</ml:id> </ml:matrix> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="103" left="6" top="134.25" width="199.5" height="12" align-x="24" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Variables for generation of 2D isochrones: </text> </region> <region region-id="138" left="6" top="159" width="39" height="12.75" align-x="19.5" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">8fd54222a2e274065477a6550bbfbfea</contentHash> <ml:define> <ml:id xml:space="preserve">X</ml:id> <ml:real>1000</ml:real> </ml:define> </ml:provenance> </math> <rendering item-idref="14"> <element-image-map> <box left="1.5" top="0.75" width="37.5" height="11.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="137" left="78" top="159" width="28.5" height="12.75" align-x="90" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">T</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="136" left="138" top="159" width="35.25" height="12.75" align-x="146.25" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">X</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="135" left="186" top="159" width="34.5" height="12.75" align-x="195" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">T</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="134" left="252" top="152.25" width="33" height="27.75" align-x="264" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">2eb956dddf3e5af6939b47c0b977f164</contentHash> <ml:define> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:id xml:space="preserve">X</ml:id> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="18"> <element-image-map> <box left="1.5" top="0.75" width="31.5" height="26.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="133" left="300" top="151.5" width="33" height="28.5" align-x="312" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="139" left="378" top="159" width="59.25" height="18" align-x="400.5" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="124" left="6" top="188.25" width="286.5" height="12" align-x="26.25" align-y="198" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Analytical solution for given boundary and initial conditions: </text> </region> <region region-id="130" left="6" top="205.5" width="327.75" height="43.5" align-x="40.5" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" branch-id="00000000-0000-0000-0000-000000000000" revision-num="19" is-modified="true" region-id="0" href="C:\Documents and Settings\tmevans\Desktop\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">b48ec291c914e37fc2513a463bcb2251</contentHash> <ml:define> <ml:function> <ml:id xml:space="preserve" subscript="a">u</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="0">u</ml:id> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="v">c</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">t</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:real>100</ml:real> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="21"> <element-image-map> <box left="1.5" top="0.75" width="326.25" height="42" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="129" left="378" top="219" width="68.25" height="21" align-x="405" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="a">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="a">u</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="158" left="6" top="266.25" width="181.5" height="12" align-x="18" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Create 3D surfaces and 2D isochrones: </text> </region> <region region-id="157" left="198" top="266.25" width="120.75" height="13.5" align-x="213" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">M</ml:id> <ml:apply> <ml:id xml:space="preserve">CreateMesh</ml:id> <ml:sequence> <ml:id xml:space="preserve">u</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">h</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="156" left="330" top="266.25" width="129.75" height="17.25" align-x="349.5" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="a">M</ml:id> <ml:apply> <ml:id xml:space="preserve">CreateMesh</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="a">u</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">h</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="171" left="240" top="282" width="215.25" height="222" align-x="346.5" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="25" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="a">M</ml:id> <ml:id xml:space="preserve">M</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="26"/> </region> <region region-id="162" left="6" top="288" width="220.5" height="117" align-x="6" align-y="288" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="27"/> <rendering item-idref="28"/> </region> <region region-id="174" left="6" top="414" width="215.25" height="222" align-x="112.5" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="29" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="a">M</ml:id> <ml:id xml:space="preserve">M</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="30"/> </region> <region region-id="196" left="6" top="656.25" width="246" height="12" align-x="23.25" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Generate solution manually using finite differences: </text> </region> <region region-id="190" left="30" top="692.25" width="51" height="18.75" align-x="45.75" align-y="702" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="5a96b335-618e-4a1c-84f7-da6cddaff7b3" version-id="e017f3c1-4568-4cc1-b6ed-559b1a4acedd" 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