Dynamic programming approach to structural optimization problem – numerical algorithm

In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed

Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit

The 1D discrete fractional Laplacian operator on a cyclically closed (periodic) linear chain with finitenumber $N$ of identical particles is introduced. We suggest a "fractional elastic harmonic potential", and obtain the $N$-periodic fractionalLaplacian operator in the form of a power law matrix function for the finite chain ($N$ arbitrary not necessarily large) in explicit form.In the limiting case $N\rightarrow \infty$ this fractional Laplacian matrix recovers the fractional Laplacian matrix ofthe infinite chain.The lattice model contains two free material constants, the particle mass $\mu$ and a frequency$\Omega\_{\alpha}$.The "periodic string continuum limit" of the fractional lattice model is analyzed where lattice constant $h\rightarrow 0$and length $L=Nh$ of the chain ("string") is kept finite: Assuming finiteness of the total mass and totalelastic energy of the chain in the continuum limit leads to asymptotic scaling behavior for $h\rightarrow 0$ of thetwo material constants,namely $\mu \sim h$ and $\Omega\_{\alpha}^2 \sim h^{-\alpha}$. In this way we obtain the $L$-periodic fractional Laplacian (Riesz fractional derivative) kernel in explicit form.This $L$-periodic fractional Laplacian kernel recovers for $L\rightarrow\infty$the well known 1D infinite space fractional Laplacian (Riesz fractional derivative) kernel. When the scaling exponentof the Laplacian takesintegers, the fractional Laplacian kernel recovers, respectively, $L$-periodic and infinite space (localized) distributionalrepresentations of integer-order Laplacians.The results of this paper appear to beuseful for the analysis of fractional finite domain problems for instance in anomalous diffusion (L\'evy flights), fractional Quantum Mechanics,and the development of fractional discrete calculus on finite lattices

Wave equation in higher dimensions - periodic solutions

We discuss the solvability of the periodic-Dirichlet problem for the wave equation with forced vibrations xtt(t, y) − ∆x(t, y) + l(t, y, x(t, y)) = 0 in higher dimensions with sides length being irrational numbers and superlinear nonlinearity. To this effect we derive a new dual variational method