@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-BLP2HLSP-6
  skos:prefLabel "calcul intégral"@fr, "integral calculus"@en ;
  a skos:Concept ;
  skos:narrower psr:-RC9PHH1R-2 .

psr:-TS5KRQ77-X
  skos:prefLabel "équation fonctionnelle"@fr, "functional equation"@en ;
  a skos:Concept ;
  skos:narrower psr:-RC9PHH1R-2 .

psr:-V80329XF-X
  skos:prefLabel "Fredholm integral equation"@en, "équation intégrale de Fredholm"@fr ;
  a skos:Concept ;
  skos:broader psr:-RC9PHH1R-2 .

psr:-RC9PHH1R-2
  skos:definition """In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});I^{1}(u),I^{2}(u),I^{3}(u),...,I^{m}(u))=0}">
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<br/>        <mo>.</mo>
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<br/>        <mo>,</mo>
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<br/>          <mi>I</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});I^{1}(u),I^{2}(u),I^{3}(u),...,I^{m}(u))=0}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8ed9630567ab406532c6ad8e656ee7b57d7e3b" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:75.928ex; height:3.176ex;" alt="{\\\\displaystyle f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});I^{1}(u),I^{2}(u),I^{3}(u),...,I^{m}(u))=0}"></div>where <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 I^{i}(u)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>I</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle I^{i}(u)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a21677bf66231212a292be861732ea60c056a998" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.155ex; height:3.176ex;" alt="{\\\\displaystyle I^{i}(u)}"></span> is an integral operator acting on <i>u.</i> Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general <b>integral equation</b> above with the general form of a differential equation which may be expressed as follows:<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});D^{1}(u),D^{2}(u),D^{3}(u),...,D^{m}(u))=0}">
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});D^{1}(u),D^{2}(u),D^{3}(u),...,D^{m}(u))=0}</annotation>
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<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9edab0dc296a1465a9576ef22d9d10e68e8816" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:78.759ex; height:3.176ex;" alt="{\\\\displaystyle f(x_{1},x_{2},x_{3},...,x_{n};u(x_{1},x_{2},x_{3},...,x_{n});D^{1}(u),D^{2}(u),D^{3}(u),...,D^{m}(u))=0}"></div>where <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 D^{i}(u)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>D</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
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<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28293a541622ce145ad81ac3a6c617779f1530d6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.863ex; height:3.176ex;" alt="{\\\\displaystyle D^{i}(u)}"></span> may be viewed as a differential operator of order <i>i</i>. Due to this close connection between differential and integral equations, one can often convert between the two. For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation. In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form. See also, for example, Green's function and Fredholm theory. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Integral_equation">https://en.wikipedia.org/wiki/Integral_equation</a>)"""@en, """Une équation intégrale est une équation où la fonction inconnue est à l'intérieur d'une intégrale. Elles sont importantes dans plusieurs domaines physiques. Les équations de Maxwell sont probablement leurs plus célèbres représentantes. Elles apparaissent dans des problèmes des transferts d'énergie radiative et des problèmes d'oscillations d'une corde, d'une membrane ou d'un axe. Les problèmes d'oscillation peuvent aussi être résolus à l'aide d'équations différentielles. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/%C3%89quation_int%C3%A9grale">https://fr.wikipedia.org/wiki/%C3%89quation_int%C3%A9grale</a>)"""@fr ;
  skos:broader psr:-BLP2HLSP-6, psr:-TS5KRQ77-X ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/%C3%89quation_int%C3%A9grale>, <https://en.wikipedia.org/wiki/Integral_equation> ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:prefLabel "équation intégrale"@fr, "integral equation"@en ;
  skos:narrower psr:-V80329XF-X .

psr: a skos:ConceptScheme .