Many systems of partial differential equations are posed on all of R" and have Euclidean symmetry. These include the Navier-Stokes equations, the Boussinesq equations, the Kuramoto-Sivashinsky equation and reaction-diffusion systems (with constant diffusion coefficients). In many applications, where these and related Euclidean equivariant equations are used, time independent, spatially periodic...

In this paper the Kuramoto-Sivashinsky equation is solved numerically by collocation method. The solution is approximated as a linear combination of septic B-spline functions. Applying the Von-Neumann stability analysis technique, we show that the method is unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions....

Physics-informed neural networks (PINNs) have recently become a popular method for solving forward and inverse problems governed by partial differential equations (PDEs). By incorporating the residual of PDE into loss function network-based surrogate model unknown state, PINNs can seamlessly blend measurement data with physical constraints. Here, we extend this framework to PDE-constrained opti...

We consider the numerical solution of nonlinear, dissipative partial differential equations using a pseudospectral method and the methodology of approximate inertial manifolds. Coarse and fine grids are employed, with a nonlinear mapping to relate the solutions computed on these grids. The approach is illustrated by consideration of the Kuramoto-Sivashinsky equation subject to periodic solution...

We study steady thin reaction fronts described by the Kuramoto-Sivashinsky equation that separates fluids of different densities. This system may lead to hydrodynamic instabilities as buoyancy forces interact with the propagating fronts in a two-dimensional slab. We use Darcy's law to describe the fluid motion in this geometry. Steady front profiles can be flat, axisymmetric, or nonaxisymmetric...

A robust approach for adjoint-based sensitivity analysis of chaotic dynamics based on unstable periodic orbits is proposed. We show that a careful reformulation of established variational techniques to such trajectories enables the sensitivity of time averages with respect to design parameters to be calculated exactly, regardless of the stability characteristics and length of the orbit. This ho...