Getting Started#
A series of notebooks are available to help run the code using VaYu. There are located within the notebook directory, additionally the python scripts are within the example directory.
Notebooks
Introductory examples#
We now introduce basic concepts, which helps in demonstaring the usage of library.
Spectral Methods#
We now dive into constrting spectral methods for integer-order differential equations.
Spectral methods for integer-order differential equations
- Function approximation using Convolution Orthogonality
- Function approximation using Convolution Orthogonality over Homogenous Basis
- Galerkin using Legendre, BC via Lifting
- Petrov-Galerkin using Legendre, BC via Lifting
- Petrov-Galerkin using Shifted Legendre, BC via Lifting
- Galerkin-Tau using chebyshev
- Galerkin-Tau using Shited Legendre
- allen_cahn_1D module
- allen_cahn_2D module
We now dive into constrting spectral methods for fractional differential equations via convolution orthogonality
Spectral methods for General fractional PDE via convolution orthogonality
- Fractional Derivative using Convolution Orthogonality over Homogeneous Basis
- General Fractional Derivative using Convolution Orthogonality over Homogeneous Basis
- Homogeneous basis for Fractional Derivative using inner product
- Petrov-Galerkin Tau for Fractional Derivative using inner product
- Petrov-Galerkin Tau for General Fractional Derivative using inner product
Reduced Order Modeling#
We now dive into Reduced order modeling (ROM). ROM aims at speeding up simulations, by dimensionality reduction of the problem.
Physics-informed Neural Networks#
We now dive into Physics-informed Neural Netorks (PINNs). PINNs solves differential equations by invoking a neural network. The governing equation is a part of the loss function, hence data is not required everywhere, only at the boundries.