Problem 51625. Solve an ODE: diffusion problem 2

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Problem statement</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>In the solution of a problem involving diffusion, the following ordinary differential equation arises</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"f''+a(f+eta f') = 0\"/></w:customXmlPr><w:r><w:t>f\\prime\\prime+a(f+\\eta f\\prime)=0</w:t></w:r></w:customXml><w:r><w:t> </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"a\"/></w:customXmlPr><w:r><w:t>a</w:t></w:r></w:customXml><w:r><w:t> is a positive constant and primes denote differentiation with respect to the independent variable </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"eta\"/></w:customXmlPr><w:r><w:t>\\eta</w:t></w:r></w:customXml><w:r><w:t>. The function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"f\"/></w:customXmlPr><w:r><w:t>f</w:t></w:r></w:customXml><w:r><w:t> and its derivatives vanish at infinity, and it is subject to the constraint</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"integral(f,{eta,-inf,inf}) = 1\"/></w:customXmlPr><w:r><w:t>\\int_{-\\infty}^\\infty fd\\eta=1</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to solve this problem—that is, return values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"f\"/></w:customXmlPr><w:r><w:t>f</w:t></w:r></w:customXml><w:r><w:t> at specified values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"eta\"/></w:customXmlPr><w:r><w:t>\\eta</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Background</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. </w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 317px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 158.5px; transform-origin: 407px 158.5px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 7.91667px; transform-origin: 63.0083px 7.91667px; unicode-bidi: normal; "><span style="font-weight: 700; ">Problem statement</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; "><span style="">In the solution of a problem involving diffusion, the following ordinary differential equation arises</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="f''+a(f+eta f') = 0" style="width: 130px; height: 19px;" width="130" height="19"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; "><span style="">where </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">a</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; "><span style=""> is a positive constant and primes denote differentiation with respect to the independent variable </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">η</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.3333px 7.91667px; transform-origin: 44.3333px 7.91667px; unicode-bidi: normal; "><span style="">. The function </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">f</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style=""> and its derivatives vanish at infinity, and it is subject to the constraint</span></span></div><div style="block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-17px"><img src="data:image/png;base64,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" alt="integral(f,{eta,-inf,inf}) = 1" style="width: 78px; height: 44px;" width="78" height="44"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; "><span style="">Write a function to solve this problem—that is, return values of </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">f</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; "><span style=""> at specified values of </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">η</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; "><span style="">. </span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; "><span style="font-weight: 700; ">Background</span></span></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.967px 7.91667px; transform-origin: 349.967px 7.91667px; unicode-bidi: normal; "><span style="">The physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. </span></span></div></div></div>

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Problem 51625. Solve an ODE: diffusion problem 2

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Problem 51625. Solve an ODE: diffusion problem 2

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Last 200 Solutions

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Problem Recent Solvers4

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More from this Author147

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