In this work a treatment of hypersingular integral equations, which have relevant applications in many problems of wave dynamics, elasticity and fluid mechanics with mixed boundary conditions, is presented. The first goal of the present study is the development of an efficient analytical and direct numerical collocation method. The second one is the application of the method to the porous elastic materials when a periodic array of co-planar cracks is present. Starting from Cowin- Nunziato model...

We define a mapping which with each function $u\in L^{\mathrm{\infty }}\left(0,T\right)$ and an admissible value of $r>0$ associates the function $\xi$ with a prescribed initial condition ${\xi }^0$ which minimizes the total variation in the $r$-neighborhood of $u$ in each subinterval $\left[0,t\right]$ of $\left[0,T\right]$. We show that this mapping is non-expansive with respect to $u$, $r$ and ${\xi }^0$, and coincides with the so-called play operator if $u$ is a regulated function.