General hyperbolic Kirchhoff model for the free vibration of elastic string on a flat surface
Fatima Ezahra Bentata, Євген Олександрович ЗАЙЦЕВ · Applied Mathematics in Science and Engineering · 2025
In this work, we investigate the existence of weak solutions for the general form of the hyperbolic Kirchhoff-type problem involving a free boundary modelling the free vibration of an elastic string on a flat surface. We employ the Discrete Morse Flow (DMF) approach, which reformulates the original problem as a sequence of minimization problems at discrete time intervals. This ensures the existence of a minimizer for the discretized functional, which in turn serves as a weak solution to the main problem. The presence of non-local terms, arising from the p-Kirchhoff term, introduces dependencies on the gradient norm [Formula: see text] across the entire domain, making the analysis more challenging. These non-local terms encapsulate the effect of the free boundary and influence the behaviour of the solution globally, rather than being determined solely by local values of v. Our study provides a rigorous treatment to overcome these difficulties. Furthermore, we present numerical simulations to illustrate the physical implications of our results.