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.


Introduction and problem formulation
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. The grain boundary forms a groove in the surface, with a known angle and the groove becomes deeper with time.
If we suppose that the surface profile is defined by a function p(x, t), where x is the horizontal coordinate and t is time, Mullins [12] showed that the idealized small-slope approximation case 1 + (p x ) where B = D s γ s ω/(kτ ) is Mullins coefficient, with surface diffusivity D s , surface energy γ s , the atomic volume ω, Boltzmann constant k, and the absolute temperature τ . In most crystalline systems, γ s is much larger than typical interfacial energies γ i . It can be considered that the surface diffusion D s taking place in some chemical phenomenon may be depend on time.
The alternative way arriving to time dependent Mullins coefficient model can be taken by more standard non-linear form of Mullins' equation [3] where θ = p x . Here D and E are surface diffusivity and tension coefficients and they may be general depend on the angle between the material surface and the crystal planes for an anisotropic material. For a polycrystalline metal, the surface diffusivity, the solid-fluid interfacial tension and the grain-boundary tension depend on temperature. Therefore, it should be extended to allow for transport coefficients to depend explicitly on time, as well as on slope in classic Mullins' equation. An alternative interpretation of the equation (1.1) with constant coefficient is the case with both D and E are constants and with time dependent coefficient is the case with both D and E are solely time dependent coefficients. 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 then it allows the boundary conditions for (1.1) to be changed from finite spatial range x = l to infinite x → +∞. Mullins determined the symmetric semi-infinite groove profile due to a single interface at x = 0 by solving (1.1) in the quadrant x > 0, t > 0, subject to initial and boundray conditions where m is the slope of the groove root, is the equilibrium dihedral angle at the surface-interface junction given by m = γi 2γs . The condition (1.2) describes the surface evolves from a flat surface. The first condition (1.3) reflects the symmetry of the system when the grain boundary is vertical. The second condition (1.3) imposes the zero flux at the grain boundary.
There are plenty of studies focusing on the solution of linear Mullins' equation (1.1) with constant coefficient. Mullins [12] solved this problem using Laplace transforms with respect to t. However, it is much simpler to use a Fourier cosine transform with respect to x [11]. This paper also solved the problem for an infinite periodic row of grooves that develops the Fourier series solution of published in earlier paper [14], where a sinusoidal disturbance was considered, leading to a dispersion relation. A multiple integration technique of the integral-balance method also allows to solve this problem in [8]. In the papers [2, 6, 10] the Mullins' equation is considered in R/ {0} , t > 0 and using various self-similar solution conception they found solution for R/ {0} , t > 0. We can also mention [7,9] for the fractional sub-diffusion modelling of Mullins' equation.
The equation (1.1) and boundary conditions (1.3) are identical to those of Mullins. The boundary conditions can not be always changed from x = l to x → +∞ if the surface diffusion is not considered as very slow. In present case the boundary x = l can be supported by a zero position and zero curvature at that boundary: The Laplace method is a suitable method for a constant choice of Mullins coefficient B and also groove profile at the root has be a fixed slope m for all time, but generally they have no obligation to be constant with respect to time variable. The finite range consideration of space variable x allows to take Mullins coefficient and slope of the groove root time dependent.
The spatial and time finite range version of Mullins equation with Mullins grooving boundaries (1.3) at x = 0 and also zero position and zero curvature boundaries (1.4) at x = l can be formulated in following form: by the substation where v(x, t) is a function satisfying the conditions v The total mass integral condition will be used in inverse problem. DP: Let B(t), t ∈ [0, T ] be a known positive function. We call direct problem the problem of finding u(x, t) satisfying the equations (1.5)-(1.7).
IP: Let B(t) be unknown function. We call inverse problem the problem of finding the pair (B(t), u(x, t)) satisfying the equations (1.5)-(1.7) and (1. We establish conditions for the existence and uniqueness of a smooth (classical) solutions both direct and inverse problems. The finite range for x suggests the method of separation of variables. For the simplicity we will suppose that l = 1.
The auxiliary spectral problem of the IBVP (1. The paper is organized as follows. In Section 2, we recall the eigenvalues and eigen-functions of the auxiliary spectral problem and existence-uniqueness theorem for DP (1.5)-(1.7). In Section 3, the existence and uniqueness theorems of the IP (1.5)-(1.8) are proved.

DP: Fourier series representation of the solution
and the consistency conditions First, consider the spectral problem (1.9) which is generated by differential operator For arbitrary y, z ∈ D(L) holds. It means that operator L is self-adjoint then the eigenvalues are real and normalized eigenfunctions are orthonormal basis in the space L 2 [0, 1] [13]. Moreover, λ = (y ,y ) (y,y) > 0 for arbitrary eigenvalue λ and corresponding eigenfunction y that L(y) = λy, y ∈ D(L). Simple calculations yields the positive eigenvalues normed eigenfunctions The following lemma is useful for DP.
Proof. It is clear that By using Cauchy-Shwartz and Bessel inequalities The main result for DP is presented as follows.
Theorem 1. (Existence and uniqueness of DP) Let the following conditions be satisfied: Proof. To construct the formal solution of the problem (1.5)-(1.7) for arbitrary B(t) ∈ C[0, T ], we will use the Fourier series in terms of the eigenfunctions y n (x) = √ 2 cos 4 √ λ n x , n = 0, 1, 2, ... of the auxiliary spectral problem (1.9): To show that this series then satisfies all the conditions of the problem (1.5)-(1.7), we must prove that the the t-partial derivative and the xxxx-fourth order partial derivative of the function defined by (2.1) is continuous, satisfies equation (1.5) in the region 0 < x < 1, t > 0, and the function (2.1) its xxsecond and xxx-third order partial derivative at the boundary points of the region (for points t = 0, x = 0, x = 1) must be continuous. We show that the series arising by termwise differentiation The convergence of first two majorant series results from D'Alambert criterion and the last is result of Lemma 1. Hence, it follows that the series (2.2) and (2.3) for t ≥ ε > 0 are uniformly convergent. Further, we conclude from superposition principle that the function defined by the series (2.1) satisfies equation (1.5). Since t is arbitrary it holds for all t > 0. The function (2.1) and its t-first, xx-second and xxx-third order partial derivative must be continuous at boundary points. More precicisely, the series (2.1) must be continuous at t = 0, We therefore obtain a function u(x, t) ∈ C 4,1 (Ω T ) ∩ C 3,0 (Ω T )) which is a solution of the problem (1.5)-(1.7) given by the Fourier series (2.1), that it is unique from the unique Fourier representations of the functions. Let us introduce the following class of functions
Proof. It is clear that by the integration by parts, where z n = √ 2 sin 4 √ λ n x .

By using Cauchy-Shwartz and Bessel inequalities
The main result for IP are presented as follows.
Proof. The series (2.1) can be termwise continuously differentiable by t that the series (2.2) is uniformly converges, since the series It is easy to show that √ 2 ∞ n=0 Let us introduce the following class of functions: It is clear that Φ maps the set M onto itself, i.e., Φ : M → M . Now, we will show that the operator Φ is uniformly bounded and equicontinuous. Let M 1 be any bounded subset of the set M. Since Φ(M 1 ) ⊂ M , it follows that Φ(M 1 ) is uniformly bounded. According to the Arzela theorem, we establish the equicontinuity of the set Φ. For this purpose, we take an arbitrary ε > 0 and establish the existence of δ > 0 such that Taking into account that By using the inequality |f n (t)| .

Direct and Inverse Problems for Thermal Grooving by Surface Diffusion 143
On the other hand, since K(t) is continuous in the closed interval [0, T ], for all ε > 0 there exists δ 1 = δ 1 (ε) > 0 such that for all t 1 and t 2 in [0, T ] for which |t 1 − t 2 | < δ 1 . By choosing  Proof. Assume that there exist two solutions (B i (t), u (i) (x, t)), i = 1, 2, of problem (1.5)-(1.8). For the difference of these solutions , we obtain the following problem: By applying 1 0 u(x, t)dx = 0, 0 ≤ t ≤ T and using boundary conditions in (3.4) we become to the equality: xxxx (x, t)dx.
Using the Fourier series representation of u xxx (1, t), we represent a solution of problem (3.4) in the form xxxx n e −λn t τ B1(s)ds dτ = B(t)u (2) xxx (1, t).
Representing B(t) in the form of Volterra integral equation by taking account that u (2) xxx (1, t) = 0. From the problem by u (2) (x, t) we get the equality The kernel K(t, τ ) is continuous. In view of the properties of Volterra integral equations of the second kind, (3.5) has only the trivial solution B(t) ≡ 0. Therefore, B(t) ≡ 0, t ∈ [0, T ], and u(x, t) ≡ 0, (x, t) ∈ Ω T , as a solution of the problem (3.4).
The inverse problem of finding Mullins coefficient has a solution iff consistency-type condition is satisfied. If the inverse problem has solution then it is not unique. It means that Mullins coefficient can not be uniquely controlled by the total mass in the case of two grooves.
The unique restoration of the time-dependent Mullins coefficient can be obtained by the the measurement of the profile u(x 0 , t) = E(t), t ∈ [0, T ] at a fixed point x 0 ∈ [0, 1], in [4].

Remark 2.
The grooving boundary conditions proposed in Amram et al. [1], namely u x | x=0 = 0, u xx | x=0 = 0 as zero flux and zero curvature at the root. The relevant spectral problem can not be self-adjoint for any given boundary condition at x = 1. By the way the classical theory [5] of expansion in terms of eigenfunctions can not be applicable. The Sturm-type boundary conditions at x = 1 as ∂ m 1 [13], together with the Amram conditions at x = 0 that the system of eigenfunctions and associated functions is Riesz basis in L 2 [0, 1]. The Fourier method approach to this problem needs expansion of given functions in terms of eigenfunctions and associated functions of relevant spectral problem.

Conclusions
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]. The finiteness of the space region allows to apply the Fourier series analysis of finding profile and also to consider the Mullins coefficient time-dependent. We solve the inverse problem of finding time-dependent Mullins coefficient from the total mass data. The grooving boundary are supported by the conditions that are identical to those of Mullins [12] and the opposite boundary supported by the boundary conditions (1.4) which the auxiliary spectral problem is self-adjoint.
In practice, surface diffusivity D s is measured using sophisticated methods of characterisation based on radiotracers, field ion microscopy or topographic techniques. However, if an additional chemical phenomenon is taking place then, the surface diffusion Ds depends on time and becomes unknown. In such a situation, the measurement of surface diffusivity depending on time becomes unfeasible using the current state-of-the-art experimental procedures, but instead, one can consider the computational mathematics inversion for its determination.We also mention that, a more fulsome account of the inverse problem would use the small−t part of such a solution, perhaps with random errors added on, as a bench test.
Mullins' condition that there is no diffusion into the grain boundary (zero flux u xxx | x=0 = 0 at the grooving root), is replaced in [1] by a zero curvature at the root ( u xx | x=0 = 0). Because the relevant spectral problem can not be selfadjoint for any given boundary condition at x = 1, the Sturm-type boundary condition at x = 1 can be accompanied and the Mullins coefficient can also be determined in this case, which suggests a line for further investigation.