Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem

<p style='text-indent:20px;'>In this article we consider a singularly perturbed Allen-Cahn problem <inline-formula><tex-math id="M1">\begin{document}$ u_t = \epsilon^2(a^2u_x)_x+b^2(u-u^3) $\end{document}</tex-math></inline-formula>, for <inline-formula><tex-math id="M2">\begin{document}$ (x,t)\in (0,1)\times\mathbb{R}^+ $\end{document}</tex-math></inline-formula>, supplied with no-flux boundary condition. The novelty here lies in the fact that the nonnegative spatial inhomogeneities <inline-formula><tex-math id="M3">\begin{document}$ a(\cdot) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ b(\cdot) $\end{document}</tex-math></inline-formula> are allowed to vanish at some points in <inline-formula><tex-math id="M5">\begin{document}$ (0,1) $\end{document}</tex-math></inline-formula>. Using the variational concept of <inline-formula><tex-math id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence we prove that, for <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> small, such degeneracy of <inline-formula><tex-math id="M8">\begin{document}$ a(\cdot) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ b(\cdot) $\end{document}</tex-math></inline-formula> induces the existence of stable stationary solutions which develop internal transition layer as <inline-formula><tex-math id="M10">\begin{document}$ \epsilon\to 0 $\end{document}</tex-math></inline-formula>.</p>

Asymptotic solution of the inverse problem for restoring the modular type source in Burgers’ equation with modular advection

For a singularly perturbed Burgers’ type equation with modular advection that has a time-periodic solution with an internal transition layer, asymptotic analysis is applied to solve the inverse problem for restoring the function of the source using known information about the observed solution of a direct problem at a given time interval (period).

Element Based Force Analysis of a Single Input-Multiple Output Linkage System

Presented in this paper is a method for force analysis of a single-input multiple-output (SIMO) linkage array that is designed for curve morphing applications, such as morphing airfoils. Different from the existing force methods, this method is developed to determine the force of a single actuator at the front that is needed to resist the forces on each loop for the entire multiloop linkage system. The proposed method is based on a full force model of a single loop four-bar linkage. When this model is applied to a multiloop system, two force sources are considered for each loop, namely the external point force on the coupler and the internal transition torque from the proceeding loop. As a result, a recursive method is proposed to compute the force from the last loop through intermediate loops to the first loop. The force vector of the first loop represents the required force of the single actuator needed to counteract the forces experienced by all the coupler forces. A number of simulations are performed and compared with FEM results to prove its effectiveness.