@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-P5WMHCH5-S
  skos:definition """A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivative investment instruments. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation">https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation</a>)"""@en, """En mathématiques, une équation aux dérivées partielles linéaire du second ordre, dont la forme générale est donnée par :  <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\sum _{i,j=1}^{n}{a_{ij}(\\\\mathbf {x} ){\\\\dfrac {\\\\partial ^{2}f}{\\\\partial x_{i}\\\\partial x_{j}}}}+\\\\sum _{i=1}^{n}{b_{i}(\\\\mathbf {x} ){\\\\dfrac {\\\\partial f}{\\\\partial x_{i}}}}+c(\\\\mathbf {x} )f=h(\\\\mathbf {x} ),\\\\ \\\\ \\\\ \\\\mathbf {x} \\\\in U\\\\subset \\\\mathbb {R} ^{n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>             <mo>,</mo>             <mi>j</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <msub>             <mi>a</mi>             <mrow class="MJX-TeXAtom-ORD">               <mi>i</mi>               <mi>j</mi>             </mrow>           </msub>           <mo stretchy="false">(</mo>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="bold">x</mi>           </mrow>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mstyle displaystyle="true" scriptlevel="0">               <mfrac>                 <mrow>                   <msup>                     <mi mathvariant="normal">∂<!-- ∂ --></mi>                     <mrow class="MJX-TeXAtom-ORD">                       <mn>2</mn>                     </mrow>                   </msup>                   <mi>f</mi>                 </mrow>                 <mrow>                   <mi mathvariant="normal">∂<!-- ∂ --></mi>                   <msub>                     <mi>x</mi>                     <mrow class="MJX-TeXAtom-ORD">                       <mi>i</mi>                     </mrow>                   </msub>                   <mi mathvariant="normal">∂<!-- ∂ --></mi>                   <msub>                     <mi>x</mi>                     <mrow class="MJX-TeXAtom-ORD">                       <mi>j</mi>                     </mrow>                   </msub>                 </mrow>               </mfrac>             </mstyle>           </mrow>         </mrow>         <mo>+</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <msub>             <mi>b</mi>             <mrow class="MJX-TeXAtom-ORD">               <mi>i</mi>             </mrow>           </msub>           <mo stretchy="false">(</mo>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="bold">x</mi>           </mrow>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mstyle displaystyle="true" scriptlevel="0">               <mfrac>                 <mrow>                   <mi mathvariant="normal">∂<!-- ∂ --></mi>                   <mi>f</mi>                 </mrow>                 <mrow>                   <mi mathvariant="normal">∂<!-- ∂ --></mi>                   <msub>                     <mi>x</mi>                     <mrow class="MJX-TeXAtom-ORD">                       <mi>i</mi>                     </mrow>                   </msub>                 </mrow>               </mfrac>             </mstyle>           </mrow>         </mrow>         <mo>+</mo>         <mi>c</mi>         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">x</mi>         </mrow>         <mo stretchy="false">)</mo>         <mi>f</mi>         <mo>=</mo>         <mi>h</mi>         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">x</mi>         </mrow>         <mo stretchy="false">)</mo>         <mo>,</mo>         <mtext> </mtext>         <mtext> </mtext>         <mtext> </mtext>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">x</mi>         </mrow>         <mo>∈<!-- ∈ --></mo>         <mi>U</mi>         <mo>⊂<!-- ⊂ --></mo>         <msup>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="double-struck">R</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sum _{i,j=1}^{n}{a_{ij}(\\\\mathbf {x} ){\\\\dfrac {\\\\partial ^{2}f}{\\\\partial x_{i}\\\\partial x_{j}}}}+\\\\sum _{i=1}^{n}{b_{i}(\\\\mathbf {x} ){\\\\dfrac {\\\\partial f}{\\\\partial x_{i}}}}+c(\\\\mathbf {x} )f=h(\\\\mathbf {x} ),\\\\ \\\\ \\\\ \\\\mathbf {x} \\\\in U\\\\subset \\\\mathbb {R} ^{n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c31282cd31a495e7f3e7da051cda8d0d509e55bc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:64.651ex; height:7.176ex;" alt="{\\\\displaystyle \\\\sum _{i,j=1}^{n}{a_{ij}(\\\\mathbf {x} ){\\\\dfrac {\\\\partial ^{2}f}{\\\\partial x_{i}\\\\partial x_{j}}}}+\\\\sum _{i=1}^{n}{b_{i}(\\\\mathbf {x} ){\\\\dfrac {\\\\partial f}{\\\\partial x_{i}}}}+c(\\\\mathbf {x} )f=h(\\\\mathbf {x} ),\\\\ \\\\ \\\\ \\\\mathbf {x} \\\\in U\\\\subset \\\\mathbb {R} ^{n}}"></span></dd></dl> est dite <i>parabolique</i> en un point donné <span class="texhtml"><b>x</b></span> de l'ouvert <i>U</i> si la matrice carrée symétrique <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle A(\\\\mathbf {x} )=\\\\left(a_{ij}\\
ight)_{1\\\\leq i,j\\\\leq n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>A</mi>         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">x</mi>         </mrow>         <mo stretchy="false">)</mo>         <mo>=</mo>         <msub>           <mrow>             <mo>(</mo>             <msub>               <mi>a</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>i</mi>                 <mi>j</mi>               </mrow>             </msub>             <mo>)</mo>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>1</mn>             <mo>≤<!-- ≤ --></mo>             <mi>i</mi>             <mo>,</mo>             <mi>j</mi>             <mo>≤<!-- ≤ --></mo>             <mi>n</mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle A(\\\\mathbf {x} )=\\\\left(a_{ij}\\
ight)_{1\\\\leq i,j\\\\leq n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5bf49c4b5673ae9819399607ef74a820b52ec9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.878ex; height:3.343ex;" alt="{\\\\displaystyle A(\\\\mathbf {x} )=\\\\left(a_{ij}\\
ight)_{1\\\\leq i,j\\\\leq n}}"></span> des coefficients du second ordre admet <i>n</i>–1 valeurs propres non nulles et de même signe et une valeur propre nulle, le vecteur propre associé à cette dernière,  noté <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbf {v} _{0}(\\\\mathbf {x} )}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="bold">v</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>0</mn>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">x</mi>         </mrow>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {v} _{0}(\\\\mathbf {x} )}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612e6b774845cf567a62ad476ebce53e5a13f615" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.685ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathbf {v} _{0}(\\\\mathbf {x} )}"></span>, étant tel que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbf {v} _{0}(\\\\mathbf {x} )\\\\cdot \\\\mathbf {b} (\\\\mathbf {x} )\\
eq 0}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="bold">v</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>0</mn>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">x</mi>         </mrow>         <mo stretchy="false">)</mo>         <mo>⋅<!-- ⋅ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">b</mi>         </mrow>         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">x</mi>         </mrow>         <mo stretchy="false">)</mo>         <mo>≠<!-- ≠ --></mo>         <mn>0</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {v} _{0}(\\\\mathbf {x} )\\\\cdot \\\\mathbf {b} (\\\\mathbf {x} )\\
eq 0}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74a4e99b21819b1c120b39689de153d980bd4c28" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.331ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathbf {v} _{0}(\\\\mathbf {x} )\\\\cdot \\\\mathbf {b} (\\\\mathbf {x} )\\
eq 0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbf {b} (\\\\mathbf {x} )}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">b</mi>         </mrow>         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="bold">x</mi>         </mrow>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {b} (\\\\mathbf {x} )}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/549e1dc3096ddf35d602ab02a941744a987724c4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.706ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathbf {b} (\\\\mathbf {x} )}"></span> désignant le vecteur des <i>n</i> coefficients du premier ordre </span>.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/%C3%89quation_aux_d%C3%A9riv%C3%A9es_partielles_parabolique">https://fr.wikipedia.org/wiki/%C3%89quation_aux_d%C3%A9riv%C3%A9es_partielles_parabolique</a>)"""@fr ;
  skos:prefLabel "parabolic partial differential equation"@en, "équation aux dérivées partielles parabolique"@fr ;
  skos:inScheme psr: ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/%C3%89quation_aux_d%C3%A9riv%C3%A9es_partielles_parabolique>, <https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation> ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:broader psr:-LM732D0H-P ;
  a skos:Concept .

psr:-LM732D0H-P
  skos:prefLabel "équation aux dérivées partielles"@fr, "partial differential equation"@en ;
  a skos:Concept ;
  skos:narrower psr:-P5WMHCH5-S .