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href="https://en.wikipedia.org/wiki/differential_equation" class="extiw" title="w:differential equation">differential equation</a> that contains unknown <a href="https://en.wikipedia.org/wiki/Multivariable_calculus" class="extiw" title="w:Multivariable calculus">multivariable functions</a> and their <a href="https://en.wikipedia.org/wiki/partial_derivatives" class="extiw" title="w:partial derivatives">partial derivatives</a>. (A special case are <a href="https://en.wikipedia.org/wiki/ordinary_differential_equation" class="extiw" title="w:ordinary differential equation">ordinary differential equations</a> (ODEs), which deal with <a href="https://en.wikipedia.org/wiki/function_(mathematics)" class="extiw" title="w:function (mathematics)">functions</a> of a single variable and their <a href="https://en.wikipedia.org/wiki/derivative" class="extiw" title="w:derivative">derivatives</a>.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant <a href="https://en.wikipedia.org/wiki/computer_model" class="extiw" title="w:computer model">computer model</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Quotes">Quotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=1" title="Edit section: Quotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>One morning early in my (Hersh's) years as a thesis student of <a href="/w/index.php?title=Peter_Lax&action=edit&redlink=1" class="new" title="Peter Lax (page does not exist)">Peter Lax</a>, I entered my mentor's office to find him glowing in smiles. “Louis is back!” he cried out to me. Louis, I wondered? Oh yes, <a href="/w/index.php?title=Louis_Nirenberg&action=edit&redlink=1" class="new" title="Louis Nirenberg (page does not exist)">Louis Nirenberg</a>, also one of the partial differential specialists on the faculty of NYU's Courant Institute. He had been on leave in England; now he was back home! At the time. I didn't get it. Louis Nirenberg and Peter Lax were grad students together at NYU. Then they both stayed on to become famous faculty members there—Louis, a world master at elliptic partial differential equations, and Peter, a world master of hyperbolic PDEs. They hardly ever collaborated or produced joint publications. But their conversations and their intellectual and emotional interactions were a vital part of their creativity and success. <ul><li><cite style="font-style:normal" class="book" id="CITEREFReuben_HershVera_John-Steiner2010">Reuben Hersh; Vera John-Steiner (13 December 2010). <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=gvsHAnAuIp4C&pg=138">Loving and Hating Mathematics: Challenging the Myths of Mathematical Life</a></i>. Princeton University Press. p. 138. <a href="/wiki/Special:BookSources/1-4008-3611-5" title="Special:BookSources/1-4008-3611-5">ISBN 1-4008-3611-5</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Loving+and+Hating+Mathematics%3A+Challenging+the+Myths+of+Mathematical+Life&rft.aulast=Reuben+Hersh&rft.au=Reuben+Hersh&rft.au=Vera+John-Steiner&rft.date=13+December+2010&rft.pages=p.%26nbsp%3B138&rft.pub=Princeton+University+Press&rft.isbn=1-4008-3611-5&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgvsHAnAuIp4C%26pg%3D138&rfr_id=info:sid/en.wikipedia.org:Partial_differential_equation"><span style="display: none;"> </span></span></li></ul></li></ul> <ul><li>Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probalistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods. <ul><li>J Lindenstrauss, L C Evans, A Douady, A Shalev and N Pippenger, Fields Medals and Nevanlinna Prize presented at ICM-94 in Zürich, Notices Amer. Math. Soc. 41 (9) (1994), 1103-1111.</li></ul></li></ul> <ul><li>The first systematic attack on a problem involving a partial differential equation was carried out in a sequence of 1746 papers by Jean Le Rond d'Alembert (1717-1783), who sought the fundamental modes of <a href="/w/index.php?title=Vibration&action=edit&redlink=1" class="new" title="Vibration (page does not exist)">vibration</a> of a vibrating <a href="/w/index.php?title=String&action=edit&redlink=1" class="new" title="String (page does not exist)">string</a>. <ul><li><cite style="font-style:normal" class="book" id="CITEREFPeter_V._O'Neil1999">Peter V. O'Neil (1999). <i><a rel="nofollow" class="external text" href="http://books.google.com/books?id=7oknR-deVPwC&pg=PA451">Beginning Partial Differential Equations</a></i>. John Wiley & Sons. p. 451. <a href="/wiki/Special:BookSources/978-0-471-23887-4" title="Special:BookSources/978-0-471-23887-4">ISBN 978-0-471-23887-4</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Beginning+Partial+Differential+Equations&rft.aulast=Peter+V.+O%27Neil&rft.au=Peter+V.+O%27Neil&rft.date=1999&rft.pages=p.%26nbsp%3B451&rft.pub=John+Wiley+%26+Sons&rft.isbn=978-0-471-23887-4&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7oknR-deVPwC%26pg%3DPA451&rfr_id=info:sid/en.wikipedia.org:Partial_differential_equation"><span style="display: none;"> </span></span></li></ul></li></ul> <ul><li>The development of the theory of P.D.E. is closely linked with advances in <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>; in fact, <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a>’s approach to the study of conformal mapping via the Dirichlet principle led to the systematic development of the theory of elliptic P.D.E. and associated variational problems. The application of these methods to the theory of several complex variables was initiated by <a href="/wiki/W._V._D._Hodge" title="W. V. D. Hodge">Hodge</a> in his theory of harmonic integrals on compact manifolds. It is this work that led <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">H. Weyl</a> to prove the fundamental hypoellipticity theorem, known as Weyl’s lemma, which in turn led to the development of the general theory of elliptic P.D.E. <ul><li><cite style="font-style:normal"> (January 2004)"<a rel="nofollow" class="external text" href="http://www.ams.org/notices/200401/mem-spencer.pdf">Donald C. Spencer (1912–2001)</a>" (PDF). <i>Notices of the American Mathematical Society</i> <b>51</b> (1).</cite></li></ul></li></ul> <ul><li>If the original work of <a href="/w/index.php?title=Alberto_Calder%C3%B3n&action=edit&redlink=1" class="new" title="Alberto Calderón (page does not exist)">Calderón</a> and <a href="/w/index.php?title=Antoni_Zygmund&action=edit&redlink=1" class="new" title="Antoni Zygmund (page does not exist)">Zygmund</a> was related to elliptic PDE's, later developments allowed applications of their theory to parabolic equations and to general <a href="/w/index.php?title=Hypoelliptic_operator&action=edit&redlink=1" class="new" title="Hypoelliptic operator (page does not exist)">hypoelliptic operators</a>, and the more recent explosion of interest in the theory of <a href="/wiki/Oscillatory_integral" title="Oscillatory integral">oscillatory integrals</a> and in problems involving curvature has much to do with hyperbolic equations. <ul><li><cite style="font-style:normal">Ricci, Fulvio (1999). "<a rel="nofollow" class="external text" href="http://www.ams.org/journals/bull/1999-36-04/S0273-0979-99-00792-2/">Review: <i>Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals</i>, by Elias Stein</a>". <i>Bull. Amer. Math. Soc. (N.S.)</i> <b>36</b> (4): 505–521. <a href="https://www.wikidata.org/wiki/Special:GoToLinkedPage/enwiki/Q25670" class="extiw" title="d:Special:GoToLinkedPage/enwiki/Q25670">DOI</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0273-0979-99-00792-2">10.1090/s0273-0979-99-00792-2</a>.</cite></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=2" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint" style="clear: right; border: solid #aaa 1px; margin: 0 0 1em 1em; font-size: 90%; background: #f9f9f9; width: 250px; padding: 4px; spacing: 0px; text-align: left; float: right;"> <div style="float: left;"><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/File:Wikipedia-logo-v2.svg" class="mw-file-description" title="Wikipedia"><img alt="Wikipedia" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/50px-Wikipedia-logo-v2.svg.png" decoding="async" width="50" height="46" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/75px-Wikipedia-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/100px-Wikipedia-logo-v2.svg.png 2x" data-file-width="103" data-file-height="94" /></a><figcaption>Wikipedia</figcaption></figure></div> <div style="margin-left: 60px;"><a href="/wiki/Wikipedia" title="Wikipedia">Wikipedia</a> has an article about: <div style="margin-left: 10px;"><i><b><a href="https://en.wikipedia.org/wiki/Partial_differential_equation" class="extiw" title="wikipedia:Partial differential equation">Partial differential equation</a></b></i></div> </div> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikiquote.org/w/index.php?title=Partial_differential_equation&oldid=3473836">https://en.wikiquote.org/w/index.php?title=Partial_differential_equation&oldid=3473836</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categories" title="Special:Categories">Categories</a>: <ul><li><a href="/wiki/Category:Mathematics" title="Category:Mathematics">Mathematics</a></li><li><a href="/wiki/Category:Partial_differential_equations" title="Category:Partial differential equations">Partial differential equations</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 28 February 2024, at 08:44.</li> <li id="footer-info-copyright">Text is available under the <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike License</a>; additional terms may apply. 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