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How can one process a document using \documentclass{jarticle} on Overleaf? This class requires pLaTeX, which is not available from the web interface.

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Yes—by adding a custom latexmkrc file in your project, containing the following lines:

$latex = 'platex';
$bibtex = 'pbibtex';

You will also need to set the LaTeX engine of your project to LaTeX+dvipdf; this setting can be found by clicking on the gear icon on the upper right of the Overleaf editor.

Here's an example project. (After opening the template, you may need to click on the PROJECT button at the top, to toggle the file list open, to view the latexmkrc file.)

If you just need to typeset a Japanese article and are not restricted to the (u)pTeX classes, you may want to take a look at the ltjarticle class instead, from the luatexja bundle. It uses LuaLaTeX, and is more robust and up-to-date.

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