Abstract
In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is non-self adjoint, it does not have exact solutions, and it admits multiple solutions. Here, λ∈R measures the intensity of the flux and G is stationary flux. The solution depends on the size of the parameter λ. We use a monotone iterative technique and integral equations along with upper and lower solutions to prove that solutions exist. We establish the qualitative properties of the solutions and provide bounds for the values of the parameter λ, which help us to separate existence from nonexistence. These results complement some existing results in the literature. To verify the analytical results, we also propose a new computational iterative technique and use it to verify the bounds on λ and the dependence of solutions for these computed bounds on λ.
Highlights
Epitaxy means the growth of a single thin film on top of a crystalline substrate
We strictly focus on molecular beam epitaxy (MBE), and we restrict our attention to the differential equation model, which was proposed by Escudero et al [4,5,6,7]
We noticed that the approximate solution uADM and φADM always lies between the lower sequence {βn} and upper sequence {αn}
Summary
Σ : Ω ⊂ R2 × R+ → R, which describes the height of the growing interface in the spatial point x ∈ Ω ⊂ R2 at time t ∈ R+. We provide an iterative scheme based on Green’s function to compute the bounds and solutions to demonstrate the existence and nonexistence, which is dependent on λ. To prove the existence of the solutions, we use the monotone iterative technique [11,12,13,14,15,16,17]. Noeiaghdam et al [34] proposed a technique based on ADM for solving Volterra integral equation with discontinuous kernels using the CESTAC method.
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