Exact solution of a class of Riemann problems for a function system (Q1594258)

A special vector-matrix Riemann problem arising in the delamination analysis of composites is considered, namely one has to find two functions of the type \[ \begin{aligned} F_{-}(p)&= \frac{1}{A} \int_{0}^{\infty} \frac{\partial}{\partial x} \left\{ \begin{matrix} [w]\\ [u_1] \end{matrix} \right\} e^{px} x, \\ F_{+}(p) &= \int_{-\infty}^{0} \left\{ \begin{matrix} [\sigma] \\ [\tau_1] \end{matrix} \right\} e^{px} dx \end{aligned} \] which are analytic in the left and right half-plane respectively and satisfy the following boundary condition on the imaginary axis: \[ F_{-}(t) = G(t) F_{+}(t),\quad G(t) = \|g_{ij}\|, \] where \[ \begin{aligned} g_{11}(t) &= \frac{t+\sin t\cos t }{\sin^{2}t - t^{2}} + \Psi(t),\\ g_{21}(t) &= - g_{12}(t) = \frac{t^2 \gamma}{\sin^{2}t - t^{2}}, \\ g_{22}(t) &= \frac{-t+\sin t\cos t }{\sin^{2}t - t^{2}} + \Psi(t), \end{aligned} \] \(\gamma\) is a function having the prescribed asymptotic behaviour. Closed form factorization of the matrix \(G(t)\) is found. This leads to the exact solution of the above problem. An asymptotic analysis of the solution is presented.