<description/> <author>Константин Титов</author> <company/> <keywords/> <revisedBy>Константин Титов</revisedBy> </userData> <identityInfo> <revision>2</revision> <documentID>6c762cd2-b0f3-4e3f-8b2a-18b9e78f30b3</documentID> <versionID>242a4b33-6187-4113-bdb9-8de4c34513d6</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> </textStyles> </textRendering> <mathRendering equation-color="#000"> <operators multiplication="dot" derivative="derivative" literal-subscript="large" 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Приложение 2</inlineAttr> </f> <f family="Arial Cyr" charset="204"> Решение уравнения теплопроводности </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="10"> <inlineAttr line-through="false">методом сеток явной схемы.</inlineAttr> </f> </text> </region> <region region-id="766" left="6" top="62.25" width="449.25" height="57" align-x="6" align-y="72" 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"> <f family="Arial Cyr" charset="204"> <inlineAttr line-through="false"> <tab/>Для вычислений будем использовать следующее разностное<sp count="2"/>уравнение или схему:</inlineAttr> </f> <f family="Times New Roman Cyr" charset="204"> <inlineAttr line-through="false">(5)</inlineAttr> </f> <f family="Arial Cyr" charset="204"> <inlineAttr line-through="false"> <sp count="20"/> </inlineAttr> </f> <region region-id="53" 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<ml:minus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="no-space"/> <ml:id xml:space="preserve">δ</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</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:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <f family="Arial Cyr" charset="204" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="4"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> </text> </region> <region region-id="773" left="24" top="132.75" width="243.75" height="13.5" 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="true"> <f family="Arial Cyr" charset="204" size="12">Зададим параметры алгоритма (глобально):</f> </text> </region> <region region-id="1162" left="354" top="132.75" width="121.5" height="15.75" align-x="390.75" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">g</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:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>36</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:globalDefine> </math> <rendering item-idref="3"/> </region> <region region-id="786" left="102" top="160.5" width="396" height="339.75" align-x="146.25" align-y="180" 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:function> <ml:id xml:space="preserve">U</ml:id> <ml:boundVars> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:for> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> </ml:range> <ml:for> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">l</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <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:apply> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ϕ</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">δ</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:sequence> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">l</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">δ</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</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">g</ml:id> <ml:sequence> <ml:parens> <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:parens> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">δ</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:parens> </ml:sequence> </ml:apply> </ml:localDefine> </ml:program> </ml:for> </ml:for> <ml:for> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:for> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">l</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="no-space"/> <ml:apply> <ml:mult style="no-space"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</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:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="no-space"/> <ml:id xml:space="preserve">δ</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:sequence> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">X</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="no-space"/> <ml:id xml:space="preserve">δ</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:localDefine> </ml:for> </ml:for> <ml:apply> <ml:transpose/> <ml:id xml:space="preserve">X</ml:id> </ml:apply> </ml:program> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="787" left="18" top="240.75" width="47.25" height="13.5" align-x="18" align-y="252" 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"> <f family="Arial Cyr" charset="204" size="12">а так же:</f> </text> </region> <region region-id="788" left="12" top="265.5" width="79.5" height="21" align-x="37.5" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="no-space"/> <ml:id xml:space="preserve">x</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:globalDefine> </math> <rendering item-idref="5"/> </region> <region region-id="789" left="12" top="294" width="52.5" height="16.5" align-x="40.5" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">ϕ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="no-space"/> <ml:real>60</ml:real> <ml:id 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</ml:lambda> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="19"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <region region-id="795" left="229.5" top="804" width="47.25" height="15.75" align-x="264.75" align-y="815.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </math> <rendering item-idref="20"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> <region region-id="796" left="283.5" top="804" width="48" height="15.75" align-x="315.75" align-y="815.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </math> <rendering item-idref="21"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, с краевыми условиями</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">U(x,0)=f(x), U(0,t)=phi(t), U(l,t)=psi(t)<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">дана на рис.1 и 2.</f> </text> </region> <region region-id="735" left="54" top="876" width="181.5" height="210" align-x="144" align-y="876" 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="22" disable-calc="false"> <inputs> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:real>0.1</ml:real> <ml:id xml:space="preserve">T</ml:id> </ml:sequence> </ml:apply> </inputs> <outputs/> </component> <rendering item-idref="23"/> </region> <region region-id="737" left="300" top="876" width="181.5" height="210" align-x="390" align-y="876" 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="24" disable-calc="false"> <inputs> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:sequence> </ml:apply> </inputs> <outputs/> </component> <rendering item-idref="25"/> </region> <region region-id="749" left="54" top="1104.75" width="456.75" height="41.25" align-x="54" align-y="1116" 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"> <f family="Arial Cyr" charset="204" size="12">На рис.2 возможна анимация 20 - 40 кадров, при которой хорошо видно как зависит</f> <f family="Arial Cyr" charset="204" size="12">устойчивость разностной схемы от выбора шага </f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12">, определяемая к</f> <f family="Arial Cyr" charset="204" size="12">оэффициентом Куранта</f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="747" left="54" top="1156.5" width="390.75" height="45" align-x="54" align-y="1182" 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"> <f family="Arial Cyr" charset="204" size="12">Коэффициент Куранта требует, чтобы </f> <region region-id="748" left="264.75" top="1157.25" width="70.5" height="44.25" align-x="319.5" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="26"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ! Здесь же имеем:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> </text> </region> <region region-id="759" left="54" top="1219.5" width="74.25" height="21" align-x="69" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">δ</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.005</ml:real> </result> </ml:eval> </math> <rendering item-idref="27"/> </region> <region region-id="758" left="138" top="1224.75" width="49.5" height="15.75" align-x="150.75" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.01</ml:real> </result> </ml:eval> </math> <rendering item-idref="28"/> </region> <region region-id="757" left="228" top="1211.25" width="135.75" height="44.25" align-x="282.75" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.00495</ml:real> </result> </ml:eval> </math> <rendering item-idref="29"/> </region> <region region-id="756" left="402" top="1211.25" width="76.5" height="44.25" align-x="456.75" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:real>0.1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>8.6736173798840355E-19</ml:real> </result> </ml:eval> </math> <rendering item-idref="30"/> </region> <region region-id="765" left="48" top="1266.75" width="459" height="123.75" align-x="48" align-y="1278" 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"> <f family="Arial Cyr" charset="204" size="12">На рис.1 при выборе h=0.1 </f> <f family="Arial Cyr" charset="204" size="12">требование </f> <f family="Arial Cyr" charset="204" size="12">Куранта </f> <f family="Arial Cyr" charset="204" size="12"> выполняется и осциллирование</f> <f family="Arial Cyr" charset="204" size="12">исчезает - разностная схема становится устойчивой, что можно легко проверить.</f> <f family="Arial Cyr" charset="204" size="12">Очевидно, достоверный результат можно получить только с помощью устойчи-</f> <f family="Arial Cyr" charset="204" size="12">вой работы разностной схемы. </f> <f family="Arial Cyr" charset="204" size="12"> Задание </f> <f family="Arial Cyr" charset="204" size="12">. Построить плоские графики показывающие мгновенное распределение температур по сечениям </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> стержня в моменты времени: t=0.02, 0.04. А также график изменения температуры во времени в заданном сечении стержня с координатой х=0.3, 0.75.<sp count="2"/>Для этого выберем шаг </f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12">таким, чтобы разностная схема была устойчивой, например: hh=0.1.</f> </text> </region> <region region-id="797" left="0" top="1407" width="6008.25" align-x="8.25" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="626.25"> <area is-collapsed="false" name="" show-name="false" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="797" bottom-lock-id="798" bottom-tag=""> <region region-id="804" left="30" top="1422.75" width="291.75" height="13.5" align-x="30" align-y="1434" 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"> <f family="Arial Cyr" charset="204" size="12">Введем новое обозначение матрицы решения U(h,T):</f> </text> </region> <region region-id="848" left="354" top="1422.75" width="44.25" height="15.75" align-x="372.75" align-y="1434" 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">hh</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="846" left="432" top="1422.75" width="67.5" height="15.75" align-x="447.75" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R</ml:id> <ml:apply> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">hh</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="1081" left="30" top="1446.75" width="474" height="13.5" align-x="30" align-y="1458" 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"> <f family="Arial Cyr" charset="204" size="12">Вновь запишем диапазоны изменения индексов соответственно: </f> </text> </region> <region region-id="1086" left="30" top="1482" width="87.75" height="14.25" align-x="30" align-y="1494" 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"> <f family="Arial Cyr" charset="204" size="12">по переменной </f> <f family="Arial Cyr" charset="204" size="12"> t </f> </text> </region> <region region-id="1087" left="156" top="1474.5" width="72" height="34.5" align-x="171.75" align-y="1494" 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">Jt</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="1088" left="246" top="1482.75" width="42.75" height="15.75" align-x="261" align-y="1494" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Jt</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>16</ml:real> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <region region-id="1089" left="294" top="1482.75" width="45.75" height="15.75" align-x="306" align-y="1494" 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>0</ml:real> <ml:id xml:space="preserve">Jt</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="1090" left="378" top="1477.5" width="74.25" height="21" align-x="393" align-y="1494" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">δ</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.005</ml:real> </result> </ml:eval> </math> <rendering item-idref="36"/> </region> <region region-id="866" left="30" top="1530" width="102.75" height="14.25" align-x="30" align-y="1542" 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"> <f family="Arial Cyr" charset="204" size="12">и по переменной </f> <f family="Arial Cyr" charset="204" size="12"> х </f> </text> </region> <region region-id="867" left="162" top="1522.5" width="76.5" height="33.75" align-x="179.25" align-y="1542" 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">Ix</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">l</ml:id> <ml:id xml:space="preserve">hh</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="868" left="258" top="1530.75" width="44.25" height="15.75" align-x="274.5" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ix</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>10</ml:real> </result> </ml:eval> </math> <rendering item-idref="38"/> </region> <region region-id="869" left="330" top="1530.75" width="45.75" height="15.75" align-x="340.5" align-y="1542" 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>0</ml:real> <ml:id xml:space="preserve">Ix</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="1180" left="30" top="1572.75" width="440.25" height="40.5" align-x="30" align-y="1584" 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"> <f family="Arial Cyr" charset="204" size="12">На рис.4 при анимации 20 кадров видно как исчезает осциллирование: h от 0.01 </f> <f family="Arial Cyr" charset="204" size="12">увеличивается до 0.101 и разностная схема становится устойчивой (требование</f> <f family="Arial Cyr" charset="204" size="12">Куронта выполняется).</f> </text> </region> <region region-id="872" left="6" top="1632" width="214.5" height="151.5" align-x="6" align-y="1632" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="40"/> <rendering item-idref="41"/> </region> <region region-id="874" left="276" top="1632" width="232.5" height="154.5" align-x="276" align-y="1632" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="42"/> <rendering item-idref="43"/> </region> <region region-id="882" left="12" top="1818" width="223.5" height="171" align-x="12" align-y="1818" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="44"/> <rendering item-idref="45"/> </region> <region region-id="883" left="312" top="1824" width="219.75" height="169.5" align-x="312" align-y="1824" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="46"/> <rendering item-idref="47"/> </region> </area> <rendering item-idref="48"/> </region> <region region-id="888" left="18" top="2064.75" width="480.75" height="27" align-x="18" align-y="2076" 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"> <f family="Arial Cyr" charset="204" size="12">Проведем проверку полученного решения, используя ШПО Mathcad. Найдем ШПО в</f> <f family="Arial Cyr" charset="204" size="12">Help\QuickSheets<sp count="2"/>в строке поиска, указав опцию pdesolve и перекопировав его.</f> </text> </region> <region region-id="940" left="0" top="2121" width="6008.25" align-x="8.25" align-y="2130" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="1706.25"> <area is-collapsed="false" name="" show-name="false" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="940" bottom-lock-id="941" bottom-tag=""> <region region-id="1091" left="42" top="2142.75" width="379.5" height="13.5" align-x="42" align-y="2154" 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"> <f family="Arial Cyr" charset="204" size="12">Запишем исходное дифференциальное уравнение теплопроводности</f> </text> </region> <region region-id="968" left="42" top="2169.75" width="168.75" height="44.25" align-x="90" align-y="2196" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative style="derivative"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">c</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="49"/> </region> <region region-id="1092" left="36" top="2226.75" width="448.5" height="13.5" align-x="36" align-y="2238" 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"> <f family="Arial Cyr" charset="204" size="12">Сформулируем задание на решение </f> <f family="Arial Cyr" charset="204" size="12">дифференциального</f> <f family="Arial Cyr" charset="204" size="12"> уравнения</f> </text> </region> <region region-id="970" left="36" top="2250.75" width="27" height="15.75" align-x="48.75" align-y="2262" 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="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\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 a parabolic Partial Differential Equation.</originComment> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">21546c5319a511f4b5452a7b23aab828</contentHash> <ml:id xml:space="preserve">Given</ml:id> </ml:provenance> </math> <rendering item-idref="50"> <element-image-map> <box left="1.5" top="0.75" width="25.5" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1107" left="42" top="2280.75" width="31.5" height="15.75" align-x="57" align-y="2292" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="51"/> </region> <region region-id="1106" left="102" top="2280.75" width="46.5" height="15.75" align-x="117" align-y="2292" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">T</ml:id> <ml:real>0.08</ml:real> </ml:define> </math> <rendering item-idref="52"/> </region> <region region-id="1105" left="180" top="2275.5" width="149.25" height="25.5" align-x="219.75" align-y="2292" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve" subscript="t">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="xx">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="53"/> </region> <region region-id="1172" left="24" top="2316.75" width="109.5" height="13.5" align-x="24" align-y="2328" 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"> <f family="Arial Cyr" charset="204" size="12">Начальное условие:</f> </text> </region> <region region-id="1171" left="186" top="2316.75" width="66.75" height="15.75" align-x="225.75" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="54"/> </region> <region region-id="1176" left="24" top="2346.75" width="109.5" height="13.5" align-x="24" align-y="2358" 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"> <f family="Arial Cyr" charset="204" size="12">Граничные условия:</f> </text> </region> <region region-id="1173" left="300" top="2346.75" width="69" height="15.75" align-x="337.5" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <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:apply> <ml:mult style="default"/> <ml:real>60</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="55"/> </region> <region region-id="1174" left="414" top="2346.75" width="53.25" height="15.75" align-x="452.25" align-y="2358" 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="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\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 a parabolic Partial Differential Equation.</originComment> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">14fb7769d7db0687b7d1164000b7c024</contentHash> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">L</ml:id> <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="56"> <element-image-map> <box left="1.5" top="0.75" width="51.75" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1179" left="18" top="2382" width="453" height="14.25" align-x="18" align-y="2394" 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"> <f family="Arial Cyr" charset="204" size="12">Запишем задание на вычисление функции </f> <f family="Arial Cyr" charset="204" size="12"> u </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> t </f> <f family="Arial Cyr" charset="204" size="12">), указывая то как изменяются </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> t </f> <f family="Arial Cyr" charset="204" size="12"> : </f> </text> </region> <region region-id="978" left="42" top="2426.25" width="214.5" height="36.75" align-x="55.5" align-y="2448" 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">L</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">T</ml:id> </ml:matrix> <ml:real>1000</ml:real> <ml:real>100</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="57"/> </region> <region region-id="1119" left="18" top="2472" width="483.75" height="42.75" align-x="18" align-y="2484" 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"> <f family="Arial Cyr" charset="204" size="12"> Примечание </f> <f family="Arial Cyr" charset="204" size="12">. Шестая позиция в [...] влияет на точность решения - это число точек </f> <f family="Arial Cyr" charset="204" size="12">отсчета по </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">. Седьмая позиция в [...] - это </f> <f family="Arial Cyr" charset="204" size="12">число точек отсчета по </f> <f family="Arial Cyr" charset="204" size="12"> t </f> <f family="Arial Cyr" charset="204" size="12">. Вычислим </f> <f family="Arial Cyr" charset="204" size="12"> u </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> t </f> <f family="Arial Cyr" charset="204" size="12">) в точке (0.1, 0.2) и зададим дискретность изменения </f> <f family="Arial Cyr" charset="204" size="12"> x и </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> t </f> </text> </region> <region region-id="980" left="42" top="2526.75" width="101.25" height="15.75" align-x="100.5" align-y="2538" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:provenance expr-id="1"> <originRef doc-id="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\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 a parabolic Partial Differential Equation.</originComment> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">257f34e29a63b2b11ee61644f2485509</contentHash> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:real>0.1</ml:real> <ml:real>0.2</ml:real> </ml:sequence> </ml:apply> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>8.96134941310936</ml:real> </result> </ml:eval> </math> <rendering item-idref="58"> <element-image-map> <box left="1.5" top="0.75" width="50.25" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="996" left="192" top="2526.75" width="67.5" height="15.75" align-x="205.5" align-y="2538" 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">x</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.1</ml:real> </ml:sequence> <ml:id xml:space="preserve">L</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="995" left="300" top="2526" width="57.75" height="16.5" align-x="311.25" align-y="2538" 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:range> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">δ</ml:id> </ml:sequence> <ml:id xml:space="preserve">T</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="60"/> </region> <region region-id="1120" left="42" top="2586.75" width="246.75" height="13.5" align-x="42" align-y="2598" 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"> <f family="Arial Cyr" charset="204" size="12">Графически представим начальные условия:</f> </text> </region> <region region-id="986" left="12" top="2634" width="210" height="151.5" align-x="12" align-y="2634" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="61"/> <rendering item-idref="62"/> </region> <region region-id="1121" left="42" top="2814.75" width="327" height="13.5" align-x="42" align-y="2826" 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"> <f family="Arial Cyr" charset="204" size="12">Графически представим динамику изменения температуры:</f> </text> </region> <region region-id="1061" left="288" top="2850" width="243" height="169.5" align-x="288" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="63"/> <rendering item-idref="64"/> </region> <region region-id="1060" left="6" top="2856" 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<ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </math> <rendering item-idref="21"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, с краевыми условиями</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">U(x,0)=f(x), U(0,t)=phi(t), U(l,t)=psi(t)<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">дана на рис.1 и 2.</f> </text> </region> <region region-id="735" left="54" top="876" width="181.5" height="210" align-x="144" align-y="876" 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="22" disable-calc="false"> <inputs> <ml:apply 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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"> <f family="Arial Cyr" charset="204" size="12">На рис.2 возможна анимация 20 - 40 кадров, при которой хорошо видно как зависит</f> <f family="Arial Cyr" charset="204" size="12">устойчивость разностной схемы от выбора шага </f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12">, определяемая к</f> <f family="Arial Cyr" charset="204" size="12">оэффициентом Куранта</f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="747" left="54" top="1156.5" width="390.75" height="45" align-x="54" align-y="1182" 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"> <f family="Arial Cyr" charset="204" size="12">Коэффициент Куранта требует, чтобы </f> <region region-id="748" left="264.75" top="1157.25" width="70.5" height="44.25" align-x="319.5" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="26"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ! Здесь же имеем:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> </text> </region> <region region-id="759" left="54" top="1219.5" width="74.25" height="21" align-x="69" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">δ</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.005</ml:real> </result> </ml:eval> </math> <rendering item-idref="27"/> </region> <region region-id="758" left="138" top="1224.75" width="49.5" height="15.75" align-x="150.75" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.01</ml:real> </result> </ml:eval> </math> <rendering item-idref="28"/> </region> <region region-id="757" left="228" top="1211.25" width="135.75" height="44.25" align-x="282.75" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.00495</ml:real> </result> </ml:eval> </math> <rendering item-idref="29"/> </region> <region region-id="756" left="402" top="1211.25" width="76.5" height="44.25" align-x="456.75" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:real>0.1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>8.6736173798840355E-19</ml:real> </result> </ml:eval> </math> <rendering item-idref="30"/> </region> <region region-id="765" left="48" top="1266.75" width="459" height="123.75" align-x="48" align-y="1278" 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"> <f family="Arial Cyr" charset="204" size="12">На рис.1 при выборе h=0.1 </f> <f family="Arial Cyr" charset="204" size="12">требование </f> <f family="Arial Cyr" charset="204" size="12">Куранта </f> <f family="Arial Cyr" charset="204" size="12"> выполняется и осциллирование</f> <f family="Arial Cyr" charset="204" size="12">исчезает - разностная схема становится устойчивой, что можно легко проверить.</f> <f family="Arial Cyr" charset="204" size="12">Очевидно, достоверный результат можно получить только с помощью устойчи-</f> <f family="Arial Cyr" charset="204" size="12">вой работы разностной схемы. </f> <f family="Arial Cyr" charset="204" size="12"> Задание </f> <f family="Arial Cyr" charset="204" size="12">. Построить плоские графики показывающие мгновенное распределение температур по сечениям </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> стержня в моменты времени: t=0.02, 0.04. А также график изменения температуры во времени в заданном сечении стержня с координатой х=0.3, 0.75.<sp count="2"/>Для этого выберем шаг </f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12">таким, чтобы разностная схема была устойчивой, например: hh=0.1.</f> </text> </region> <region region-id="797" left="0" top="1407" width="6008.25" align-x="8.25" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="626.25"> <area is-collapsed="false" name="" show-name="false" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="797" bottom-lock-id="798" bottom-tag=""> <region region-id="804" left="30" top="1422.75" width="291.75" height="13.5" align-x="30" align-y="1434" 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"> <f family="Arial Cyr" charset="204" size="12">Введем новое обозначение матрицы решения U(h,T):</f> </text> </region> <region region-id="848" left="354" top="1422.75" width="44.25" height="15.75" align-x="372.75" align-y="1434" 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">hh</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="846" left="432" top="1422.75" width="67.5" height="15.75" align-x="447.75" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R</ml:id> <ml:apply> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">hh</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="1081" left="30" top="1446.75" width="474" height="13.5" align-x="30" align-y="1458" 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"> <f family="Arial Cyr" charset="204" size="12">Вновь запишем диапазоны изменения индексов соответственно: </f> </text> </region> <region region-id="1086" left="30" top="1482" width="87.75" height="14.25" align-x="30" align-y="1494" 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"> <f family="Arial Cyr" charset="204" size="12">по переменной </f> <f family="Arial Cyr" charset="204" size="12"> t </f> </text> </region> <region region-id="1087" left="156" top="1474.5" width="72" height="34.5" align-x="171.75" align-y="1494" 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">Jt</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">δ</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="1088" left="246" top="1482.75" width="42.75" height="15.75" align-x="261" align-y="1494" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Jt</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>16</ml:real> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <region region-id="1089" left="294" top="1482.75" width="45.75" height="15.75" align-x="306" align-y="1494" 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>0</ml:real> <ml:id xml:space="preserve">Jt</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="1090" left="378" top="1477.5" width="74.25" height="21" align-x="393" align-y="1494" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">δ</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.005</ml:real> </result> </ml:eval> </math> <rendering item-idref="36"/> </region> <region region-id="866" left="30" top="1530" width="102.75" height="14.25" align-x="30" align-y="1542" 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"> <f family="Arial Cyr" charset="204" size="12">и по переменной </f> <f family="Arial Cyr" charset="204" size="12"> х </f> </text> </region> <region region-id="867" left="162" top="1522.5" width="76.5" height="33.75" align-x="179.25" align-y="1542" 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">Ix</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">l</ml:id> <ml:id xml:space="preserve">hh</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="868" left="258" top="1530.75" width="44.25" height="15.75" align-x="274.5" align-y="1542" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ix</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>10</ml:real> </result> </ml:eval> </math> <rendering item-idref="38"/> </region> <region region-id="869" left="330" top="1530.75" width="45.75" height="15.75" align-x="340.5" align-y="1542" 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>0</ml:real> <ml:id xml:space="preserve">Ix</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="1180" left="30" top="1572.75" width="440.25" height="40.5" align-x="30" align-y="1584" 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"> <f family="Arial Cyr" charset="204" size="12">На рис.4 при анимации 20 кадров видно как исчезает осциллирование: h от 0.01 </f> <f family="Arial Cyr" charset="204" size="12">увеличивается до 0.101 и разностная схема становится устойчивой (требование</f> <f family="Arial Cyr" charset="204" size="12">Куронта выполняется).</f> </text> </region> <region region-id="872" left="6" top="1632" width="214.5" height="151.5" align-x="6" align-y="1632" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="40"/> <rendering item-idref="41"/> </region> <region region-id="874" left="276" top="1632" width="232.5" height="154.5" align-x="276" align-y="1632" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="42"/> <rendering item-idref="43"/> </region> <region region-id="882" left="12" top="1818" width="223.5" height="171" align-x="12" align-y="1818" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="44"/> <rendering item-idref="45"/> </region> <region region-id="883" left="312" top="1824" width="219.75" height="169.5" align-x="312" align-y="1824" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="46"/> <rendering item-idref="47"/> </region> </area> <rendering item-idref="48"/> </region> <region region-id="888" left="18" top="2064.75" width="480.75" height="27" align-x="18" align-y="2076" 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"> <f family="Arial Cyr" charset="204" size="12">Проведем проверку полученного решения, используя ШПО Mathcad. Найдем ШПО в</f> <f family="Arial Cyr" charset="204" size="12">Help\QuickSheets<sp count="2"/>в строке поиска, указав опцию pdesolve и перекопировав его.</f> </text> </region> <region region-id="940" left="0" top="2121" width="6008.25" align-x="8.25" align-y="2130" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="1706.25"> <area is-collapsed="false" name="" show-name="false" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="940" bottom-lock-id="941" bottom-tag=""> <region region-id="1091" left="42" top="2142.75" width="379.5" height="13.5" align-x="42" align-y="2154" 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"> <f family="Arial Cyr" charset="204" size="12">Запишем исходное дифференциальное уравнение теплопроводности</f> </text> </region> <region region-id="968" left="42" top="2169.75" width="168.75" height="44.25" align-x="90" align-y="2196" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative style="derivative"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">c</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="49"/> </region> <region region-id="1092" left="36" top="2226.75" width="448.5" height="13.5" align-x="36" align-y="2238" 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"> <f family="Arial Cyr" charset="204" size="12">Сформулируем задание на решение </f> <f family="Arial Cyr" charset="204" size="12">дифференциального</f> <f family="Arial Cyr" charset="204" size="12"> уравнения</f> </text> </region> <region region-id="970" left="36" top="2250.75" width="27" height="15.75" align-x="48.75" align-y="2262" 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="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\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 a parabolic Partial Differential Equation.</originComment> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">21546c5319a511f4b5452a7b23aab828</contentHash> <ml:id xml:space="preserve">Given</ml:id> </ml:provenance> </math> <rendering item-idref="50"> <element-image-map> <box left="1.5" top="0.75" width="25.5" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1107" left="42" top="2280.75" width="31.5" height="15.75" align-x="57" align-y="2292" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="51"/> </region> <region region-id="1106" left="102" top="2280.75" width="46.5" height="15.75" align-x="117" align-y="2292" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">T</ml:id> <ml:real>0.08</ml:real> </ml:define> </math> <rendering item-idref="52"/> </region> <region region-id="1105" left="180" top="2275.5" width="149.25" height="25.5" align-x="219.75" align-y="2292" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve" subscript="t">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="xx">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="53"/> </region> <region region-id="1172" left="24" top="2316.75" width="109.5" height="13.5" align-x="24" align-y="2328" 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"> <f family="Arial Cyr" charset="204" size="12">Начальное условие:</f> </text> </region> <region region-id="1171" left="186" top="2316.75" width="66.75" height="15.75" align-x="225.75" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="54"/> </region> <region region-id="1176" left="24" top="2346.75" width="109.5" height="13.5" align-x="24" align-y="2358" 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"> <f family="Arial Cyr" charset="204" size="12">Граничные условия:</f> </text> </region> <region region-id="1173" left="300" top="2346.75" width="69" height="15.75" align-x="337.5" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <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:apply> <ml:mult style="default"/> <ml:real>60</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="55"/> </region> <region region-id="1174" left="414" top="2346.75" width="53.25" height="15.75" align-x="452.25" align-y="2358" 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="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\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 a parabolic Partial Differential Equation.</originComment> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">14fb7769d7db0687b7d1164000b7c024</contentHash> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">L</ml:id> <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="56"> <element-image-map> <box left="1.5" top="0.75" width="51.75" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1179" left="18" top="2382" width="453" height="14.25" align-x="18" align-y="2394" 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"> <f family="Arial Cyr" charset="204" size="12">Запишем задание на вычисление функции </f> <f family="Arial Cyr" charset="204" size="12"> u </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> t </f> <f family="Arial Cyr" charset="204" size="12">), указывая то как изменяются </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> t </f> <f family="Arial Cyr" charset="204" size="12"> : </f> </text> </region> <region region-id="978" left="42" top="2426.25" width="214.5" height="36.75" align-x="55.5" align-y="2448" 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">L</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">T</ml:id> </ml:matrix> <ml:real>1000</ml:real> <ml:real>100</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="57"/> </region> <region region-id="1119" left="18" top="2472" width="483.75" height="42.75" align-x="18" align-y="2484" 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"> <f family="Arial Cyr" charset="204" size="12"> Примечание </f> <f family="Arial Cyr" charset="204" size="12">. Шестая позиция в [...] влияет на точность решения - это число точек </f> <f family="Arial Cyr" charset="204" size="12">отсчета по </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">. Седьмая позиция в [...] - это </f> <f family="Arial Cyr" charset="204" size="12">число точек отсчета по </f> <f family="Arial Cyr" charset="204" size="12"> t </f> <f family="Arial Cyr" charset="204" size="12">. Вычислим </f> <f family="Arial Cyr" charset="204" size="12"> u </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> t </f> <f family="Arial Cyr" charset="204" size="12">) в точке (0.1, 0.2) и зададим дискретность изменения </f> <f family="Arial Cyr" charset="204" size="12"> x и </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> t </f> </text> </region> <region region-id="980" left="42" top="2526.75" width="101.25" height="15.75" align-x="100.5" align-y="2538" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:provenance expr-id="1"> <originRef doc-id="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\PDE.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="5f57d37b-a3aa-4d20-91ef-5b5d6a14b018" version-id="df4b0ed4-7fae-4c1a-b90f-62c658fc007f" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="false" region-id="0" href="C:\Program Files\Mathsoft\Mathcad 13\qsheet\calculus_des\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 a parabolic Partial Differential Equation.</originComment> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">257f34e29a63b2b11ee61644f2485509</contentHash> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:real>0.1</ml:real> <ml:real>0.2</ml:real> </ml:sequence> </ml:apply> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>8.96134941310936</ml:real> </result> </ml:eval> </math> <rendering item-idref="58"> <element-image-map> <box left="1.5" top="0.75" width="50.25" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="996" left="192" top="2526.75" width="67.5" height="15.75" align-x="205.5" align-y="2538" 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">x</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.1</ml:real> </ml:sequence> <ml:id xml:space="preserve">L</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="995" left="300" top="2526" width="57.75" height="16.5" align-x="311.25" align-y="2538" 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:range> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">δ</ml:id> </ml:sequence> <ml:id xml:space="preserve">T</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="60"/> </region> <region region-id="1120" left="42" top="2586.75" width="246.75" height="13.5" align-x="42" align-y="2598" 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"> <f family="Arial Cyr" charset="204" size="12">Графически представим начальные условия:</f> </text> </region> <region region-id="986" left="12" top="2634" width="210" height="151.5" align-x="12" align-y="2634" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="61"/> <rendering item-idref="62"/> </region> <region region-id="1121" left="42" top="2814.75" width="327" height="13.5" align-x="42" align-y="2826" 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"> <f family="Arial Cyr" charset="204" size="12">Графически представим динамику изменения температуры:</f> </text> </region> <region region-id="1061" left="288" top="2850" width="243" height="169.5" align-x="288" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="63"/> <rendering item-idref="64"/> </region> <region region-id="1060" left="6" top="2856" 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