Abstract
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.
Highlights
Introduction and problem formulationThe paper by Mullins [12] considers the problem of calculation of the time evolution of the free surface in the process when a vertical flat grain boundary meets a horizontal flat surface
We investigate the Mullins’ equation of single surface grooving [12] when surface diffusion is not considered as very slow
This problem can be formed by a surface grooving of profiles in finite space region, [1]
Summary
The paper by Mullins [12] considers the problem of calculation of the time evolution of the free surface in the process when a vertical flat grain boundary meets a horizontal flat surface. It should be mentioned that a non-linear forward problem for the grain boundary with time-dependent Mullins coefficient due to temperature changes, has been solved in [3, 14]. Mullins [12] considers (1.1) for a groove, located at x = 0 It is mentioned in [1] that, if the surface diffusion is considered as very slow it allows the boundary conditions for (1.1) to be changed from finite spatial range x = l to infinite x → +∞. The finite range consideration of space variable x allows to take Mullins coefficient and slope of the groove root time dependent.
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