We consider a problem of stabilization of the linearized Schrödinger equation using boundary actuation and measurements. We propose two different control designs. First, a simple proportional collocated boundary controller is shown to exponentially stabilize the system. However, the decay rate of the closed-loop system cannot be prescribed. The second, full-state feedback boundary control desig...

The standard multigrid procedure performs poorly or may break down when used to solve certain problems. The wavelet multigrid method solves this problem by using a wavelet transform and Schur complements to determine the coarse grid and intergrid transfer operators. This method, previously applied to diffusion and advection-dominated advection-diffusion problems, here is modified for applicatio...

Aim of the paper is to investigate approximate analytical solution of time-dependent partial differential equation using a semi-analytical method, the Optimal Homotopy Asymptotic Method (OHAM). To show the efficiency of the proposed method, we consider one-dimensional heat and advection-diffusion equations. OHAM uses simple computations with pretty good enough approximate solution, which has an...

A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate approximations are used. Results of numerical experiments, which show approximation order O(h8) for the Newton potential in high dimensions, for example, for n = 200 000, are provided. The computation...

In this paper, we prove some qualitative properties of pushed fronts for the periodic reaction-diffusion-equation with general monostable nonlinearities. Especially, exponential behavior when they are approaching their unstable state. The proof also allows us to get pulsating speed c larger than minimal speed. Through behavior, finally stability fronts.

We study the effect of the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method on the discretization of optimal control problems governed by linear advection-diffusion equations. We compare two approaches for the numerical solution of such optimal control problems. In the discretize-then-optimize approach the optimal control problem is first discretized, using the SUPG met...

Saul’yev-type asymmetric schemes have been widely used in solving diffusion and advection equations. In this work, we show that Saul’yev-type schemes can be derived from the exponential splitting of the semidiscretized equation which fundamentally explains their unconditional stability. Furthermore, we show that optimal schemes are obtained by forcing each scheme’s amplification factor to match...