{"id":3649,"date":"2020-07-10T06:33:59","date_gmt":"2020-07-10T06:33:59","guid":{"rendered":"https:\/\/www.aiproblog.com\/index.php\/2020\/07\/10\/fourier-series-and-differential-equations-with-some-applications-in-r-and-python-part-2\/"},"modified":"2020-07-10T06:33:59","modified_gmt":"2020-07-10T06:33:59","slug":"fourier-series-and-differential-equations-with-some-applications-in-r-and-python-part-2","status":"publish","type":"post","link":"https:\/\/www.aiproblog.com\/index.php\/2020\/07\/10\/fourier-series-and-differential-equations-with-some-applications-in-r-and-python-part-2\/","title":{"rendered":"Fourier Series and Differential Equations with some applications in R and Python (Part 2)"},"content":{"rendered":"

Author: Sandipan Dey<\/p>\n

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This is the 2nd part of the article on a few applications of Fourier Series in solving differential equations<\/strong>. All the problems are taken from the edx Course: MITx – 18.03Fx: Differential Equations Fourier Series and Partial Differential Equations<\/strong>. The article will be posted in two parts (two separate blongs)<\/p>\n

We shall see how to solve the following ODEs \/ PDEs using Fourier series:<\/p>\n

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  1. Solve the PDE<\/em> for Heat<\/strong> \/ Diffusion using Neumann<\/strong> \/ Dirichlet<\/strong> boundary conditions<\/li>\n
  2. Solve Wave<\/strong> equation (PDE).<\/li>\n
  3. Remove noise from an audio file with Fourier transform<\/strong>.<\/li>\n<\/ol>\n

    Heat \/ Diffusion Equation<\/h2>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \"f15\"\"f17\"<\/p>\n

    The following animation shows how the temperature changes on the bar with time (considering only the first 100 terms for the Fourier series for the square wave).<\/p>\n

    \"heat3\"<\/p>\n

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    Let’s solve the below diffusion <\/span>PDE<\/em> <\/span>with the given <\/span>Neumann <\/span><\/em><\/strong>BCs<\/em><\/strong>.<\/p>\n

    \"f12\"<\/p>\n

    As can be seen from above, the initial condition can be represented as a 2-periodic triangle wave function (using even periodic extension), i.e.,<\/p>\n

    \"f13\"<\/p>\n

    The next animation shows how the different points on the tube arrive at the steady-state solution over time.<\/p>\n

     <\/p>\n<\/div>\n<\/div>\n

    \"heat\"<\/p>\n

    The next figure shows the time taken for each point inside the tube, to approach within 1% the steady-state solution.<\/p>\n

    \"steady\"<\/p>\n

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     <\/p>\n

    \"f14\"<\/a><\/p>\n<\/div>\n<\/div>\n<\/div>\n

    with the following<\/p>\n