IDENT: Identifying Differential Equations with Numerical Time Evolution

Numerical Methods for Evolution and Stationary (Integro-)Differential Equations

This study-research project is about numerical methods of differential equations (ordinary differential equations (ODE), partial differential equation (PDE), and integro-differential equations (IDE).) Required courses for this project are Math 212/213 (Multivariable Calculus), Math 302 (Differential Equations); and additional training in Math 345 (Introduction in Mathematical Biology), Math 413...

Partial Differential Equations with Numerical Methods

Numerical solutions to integral equations equivalent to differential equations with fractional time

This paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations wit...

Numerical simulations of time-dependent partial differential equations

When a time-dependent partial differential equation (PDE) is discretized in space with a spectral approximation, the result is a coupled system of ordinary differential equations (ODEs) in time. This is the notion of the method of lines (MOL), and the resulting set of ODEs is stiff; the stiffness may be even exacerbated sometimes. The linear terms are the primarily responsible for the stiffness...

Numerical solution of partial differential equations in time-dependent domains

Numerical solution of heat transfer and fluid flow problems in two spatial dimensions is studied. An arbitrary Lagrangian-Eulerian (ALE) formulation of the governing equations is applied to handle time-dependent geometries. A Legendre spectral method is used for the spatial discretization, and the temporal discretization is done with a semi-implicit multi-step method. The Stefan problem, a conv...