Direct and inverse problems for thermal grooving by surface diffusion with time dependent Mullins coefficient
We consider the Mullins’ equation of a single surface grooving when the surface diffusion is not considered as very slow. This problem can be formed by a surface grooving of profiles in a finite space region. The finiteness of the space region allows to apply the Fourier series analysis for one groove and also to consider the Mullins coefficient as well as slope of the groove root to be time-dependent. We also solve the inverse problem of finding time-dependent Mullins coefficient from total mass measurement. For both of these problems, the grooving side boundary conditions are identical to those of Mullins, and the opposite boundary is accompanied by a zero position and zero curvature which both together arrive at self adjoint boundary conditions.
This work is licensed under a .
T. Asai and Y. Giga. On self-similar solutions to the surface diffusion flow equations with contact angle boundary conditions. Hokkaido University Preprint Series in Mathematics, 1039:1–25, 2013.
P. Broadbridge and J.M. Goard. Temperature-dependent surface diffusion near a grain boundary. Journal of Engineering Mathematics, 66(1-3):87–102, 2010. http://doi.org/10.1007/s10665-009-9332-9
K. Cao, D. Lesnic and M.I. Ismailov. Determination of the time-dependent thermal grooving coefficient. Journal of Applied Mathematics and Computing, pp. 1–23, 2020. http://doi.org/10.1007/s12190-020-01388-7
E.A. Coddington and N. Levinson. Theory of Ordinary Differential Equations. McGraw Hill Book Co., Inc., New York, Toronto, London, 1955.
N. Hamamuki. Asymptotically self-similar solutions to curvature flow equations with prescribed contact angle and their applications to groove profiles due to evaporation-condensation. Advances in Differential Equations, 19(3-4):317–358, 2014.
M. Abu Hamed and A.A. Nepomnyashchy. Groove growth by surface subdiffusion. Physica D: Nonlinear Phenomena, 298-299:42–47, 2015. http://doi.org/10.1016/j.physd.2015.02.001
J. Hristov. Multiple integral-balance method: Basic idea and an example with Mullins model of thermal grooving. Thermal science, 21(3):1555–1560, 2017. http://doi.org/10.2298/TSCI170410124H
J. Hristov. Fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the Mullins mode. Mathematical Modelling of Natural Phenomen, 13(1):14, 2018. http://doi.org/10.1051/mmnp/2017080
H. Kalantarova and A. Novick-Cohen. Self-similar grooving solutions to the Mullins’ equation. Quarterly of applied Mathematics, 2019. http://doi.org/10.1090/qam/1570
P. Martin. Thermal grooving by surface diffusion: Mullins revisited and extended to multiple grooves. Quarterly of applied mathematics, 67(1):125–136, 2009. http://doi.org/10.1090/S0033-569X-09-01086-4
W.W. Mullins. Theory of thermal grooving. Journal of Applied Physics, 28(3):333–339, 1957. http://doi.org/10.1063/1.1722742
M. Naimark. Linear Differential Operators. Elementary theory of linear differential operators. New York: Frederick Ungar Publishing Co, 1967.
P. Tritscher and P. Broadbridge. Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 450(1940):569– 587, 1995. http://doi.org/10.1098/rspa.1995.0101