This paper presents the Elzaki homotopy perturbation transform scheme (EHPTS) to analyze the approximate solution of the multi-dimensional fractional diffusion equation. The Atangana-Baleanu derivative is considered in the Caputo sense. First, we apply Elzaki transform (ET) to obtain a recurrence relation without any assumption or restrictive variable. Then, this relation becomes very easy to handle for the implementation of the homotopy perturbation scheme (HPS). We observe that HPS produces the iterations in the form of convergence series that approaches the precise solution. We provide the graphical representation in 2D plot distribution and 3D surface solution. The error analysis shows that the solution derived by EHPTS is very close to the exact solution. The obtained series shows that EHPTS is a very simple, straightforward, and efficient tool for other problems of fractional derivatives.

Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana-Baleanu Caputo derivative.

We study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls–Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation of the dislocation function at these points, the potential may be either attractive or repulsive, hence collisions may occur in the latter case and, at the collision time, the dislocation function does not disappear. The goal of this paper is to provide accurate estimates on the relaxation times of the system after collision. More precisely, we take into account the case of two and three colliding points, and we show that, after a small transition time subsequent to the collision, the dislocation function relaxes exponentially fast to a steady state. In this sense, the system exhibits two different decay behaviors, namely an exponential time decay versus a polynomial decay in the space variables (and these two homogeneities are kept separate during the time evolution).

Relaxation times for atom dislocations in crystals

We focus on existence and rigidity problems of the vectorial Peierls-Nabarro (PN) model for dislocations. Under the assumption that the misfit potential on the slip plane only depends on the shear displacement along the Burgers vector, a reduced non-local scalar Ginzburg-Landau equation with an anisotropic positive (if Poisson ratio belongs to $(-1/2,1/3)$) singular kernel is derived on the slip plane. We first prove that minimizers of the PN energy for this reduced scalar problem exist. Starting from $H^{1/2}$ regularity, we prove that these minimizers are smooth 1D profiles only depending on the shear direction, monotonically and uniformly converge to two stable states at far fields in the direction of the Burgers vector. Then a De Giorgi-type conjecture of single-variable symmetry for both minimizers and layer solutions is established. As a direct corollary, minimizers and layer solutions are unique up to translations. The proof of this De Giorgi-type conjecture relies on a delicate spectral analysis which is especially powerful for nonlocal pseudo-differential operators with strong maximal principle. All these results hold in any dimension since we work on the domain periodic in the transverse directions of the slip plane. The physical interpretation of this rigidity result is that the equilibrium dislocation on the slip plane only admits shear displacements and is a strictly monotonic 1D profile provided exclusive dependence of the misfit potential on the shear displacement.

Existence and rigidity of the Peierls-Nabarro model for dislocations in high dimensions

We study the well-posedness of the vector-field Peierls-Nabarro model for curved dislocations with a double well potential and a bi-states limit at far field. Using the Dirichlet to Neumann map, the 3D Peierls-Nabarro model is reduced to a nonlocal scalar Ginzburg-Landau equation. We derive an integral formulation of the nonlocal operator, whose kernel is anisotropic and positive when Poisson's ratio $\nu\in(-\frac12, \frac13)$. We then prove that any bounded stable solutions to this nonlocal scalar Ginzburg-Landau equation has a 1D profile, which corresponds to the PDE version of flatness result for minimal surfaces with anisotropic nonlocal perimeter. Based on this, we finally obtain that steady states to the nonlocal scalar equation, as well as the original Peierls-Nabarro model, can be characterized as a one-parameter family of straight dislocation solutions to a rescaled 1D Ginzburg-Landau equation with the half Laplacian.

Existence and uniqueness of bounded stable solutions to the Peierls-Nabarro model for curved dislocations

. We consider periodic piecewise aﬃne functions, deﬁned on the real line, with two given slopes and prescribed length scale of the regions where the slope is negative. We prove that, in such a class, the minimizers of s -fractional Gagliardo seminorm densities, with 0 < s < 1, are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals where the slope is constant vanishes. Our results, for s = 12 , have relevant applications to the van der Merwe theory of misﬁt dislocations at semi-coherent straight interfaces. We consider two elastic materials having diﬀerent elastic coeﬃcients and casting parallel lattices having diﬀerent spacing. As a byproduct of our analysis, we prove the periodicity of optimal dislocation conﬁgurations and we provide the sharp asymptotic energy density in the semi-coherent limit as the ratio between the two lattice spacings tends to one.

pdf/two-slope-functions-minimizing-fractional-seminorms-and-21gt0i69.pdf

Two slope functions minimizing fractional seminorms and applications to misfit dislocations

In this paper, we perform mathematical validation of the Peierls--Nabarro (PN) models, which are multiscale models of dislocations that incorporate the detailed dislocation core structure. We focus on the static and dynamic PN models of an edge dislocation. In a PN model, the total energy includes the elastic energy in the two half-space continua and a nonlinear potential energy across the slip plane, which is always infinite. We rigorously establish the relationship between the PN model in the full space and the reduced problem on the slip plane in terms of both governing equations and energy variations. The shear displacement jump is determined only by the reduced problem on the slip plane while the displacement fields in the two half spaces are determined by linear elasticity. We establish the existence and sharp regularities of classical solutions in Hilbert space. For both the reduced problem and the full PN model, we prove that a static solution is a global minimizer in perturbed sense. We also show that there is a unique classical, global in time solution of the dynamic PN model.

Mathematical validation of the Peierls--Nabarro model for edge dislocations

In this paper, we revisit the mathematical validation of the Peierls–Nabarro (PN) models, which are multiscale models of dislocations that incorporate the detailed dislocation core structure. We focus on the static and dynamic PN models of an edge dislocation in Hilbert space. In a PN model, the total energy includes the elastic energy in the two half-space continua and a nonlinear potential energy, which is always infinite, across the slip plane. We revisit the relationship between the PN model in the full space and the reduced problem on the slip plane in terms of both governing equations and energy variations. The shear displacement jump is determined only by the reduced problem on the slip plane while the displacement fields in the two half spaces are determined by linear elasticity. We establish the existence and sharp regularities of classical solutions in Hilbert space. For both the reduced problem and the full PN model, we prove that a static solution is a global minimizer in a perturbed sense. We also show that there is a unique classical, global in time solution of the dynamic PN model.

https://www.aimsciences.org/article/exportPdf?id=da395b63-3a4c-4df3-a5bc-fac4249a0bf9

Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space

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Mathematical validation of the Peierls--Nabarro model for edge dislocations

Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space