VaYu : Spectral methods for integro-differential equations - A Python library
=============================================================================

**Version**: |version| [Under Heavy Development]

Fractional calculus is a generalised form of the integer-order calculus. While an integer-order derivative is a local operator, a fractional derivative is a non-local operator. Furthermore, general fractional calculus is a further generalisation, which admits an arbitray (singular) kernel.  

Such operators are rather cumbersome to solve, hence we introduce a new class of basis functions, namely, Jacobi Convolution Polynomails, which are suited for a large class of integro-differential operators (of convolution type). Indeed, project VaYu aims at building a bridge between theory and practice, 
by providing tools (python library) for numerical investigation (simulation) 

Features
--------

* Spectral (Element) Methods
* Phyics-infomed Neural Networks
* Fractional and Genereal Fractional Operators
* Reduced Order Modeling (under development)



Also visit our ongoing, `Fractional Calculus Seminars <https://mathlab.sissa.it/fractional-calculus-seminars>`_
and associated `YouTube <https://www.youtube.com/@FractionalCalculusSeminar>`_ channel 



.. _cards-clickable:

Navigate
--------

.. grid:: 3
    :gutter: 1

    .. grid-item-card:: Get started with VaYu
    	:link: examples/index
    	:link-type: doc

        Try our example / tutorials with additional notebooks

    .. grid-item-card:: Fractional Calculus
    	:link: fc/index
    	:link-type: doc

        What is Fractional Calculus and its application?
      	- An introduction to theory and numerical methods
    
    .. grid-item-card:: Installation, Configuration, and Usage 
    	:link: install/index
    	:link-type: doc

        Installation guide and its dependencies

.. grid:: 3
    :gutter: 1
    
    .. grid-item-card:: API Reference
    	:link: reference/index
    	:link-type: doc
    	
    	Documentation for Users

    .. grid-item-card:: Developer Guides
    	:link: contributing/index
    	:link-type: doc
    	
    	How to contribute to the projects

   
    .. grid-item-card:: Release Notes
    	:link: releases
    	:link-type: doc
	
	New features, upgrades, deprecation notes, and bug fixes




Author(s) / Developer(s)
------------------------

* Pavan Pranjivan Mehta, 
	MathLab, Mathematics Area,
	SISSA, International School of Advanced Studies, Trieste, Italy.
	Email : pavan.mehta@sissa.it
	Web : https://www.pavanpmehta.com/


Citations
---------

* Mehta, P. P. and Rozza, G (2025) Vayu : A Python library for integro-differential equations (in-preparation)


Licences
--------

* VaYu is freely available under the GNU LESSER GENERAL PUBLIC LICENSE Version 2.1.


Funding / Acknowledgement
-------------------------

.. list-table:: 
   :widths: 30 70
   :header-rows: 1 

   * - Dates (MM/YYYY)
     - Grant / funding details
   * - 10/2023 - 10/2025
     - PNRR Digitalisation, Italy. PI: `Prof. Gianluigi Rozza <https://people.sissa.it/~grozza/>`_.


Indices and tables
------------------

* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`


.. toctree::
   :maxdepth: 3
   :hidden:
   :titlesonly:
   
   fc/index
   install/index
   examples/index
   reference/index
   contributing/index
   releases