1. Introduction
Epitaxy means the growth of a single thin film on top of a crystalline substrate. It is crucial for semiconductor thin film technology, hard and soft coatings, protective coatings, optical coatings, etc. The epitaxial growth technique is used to produce the growth of semiconductor films and multilayer structures under high vacuum conditions [
1]. The major advantages of epitaxial growth are reducing the growth time, better structural and superior electrical properties, eliminating waste caused during growth, wafering cost, cutting, polishing, etc. Several types of epitaxial growth techniques, such as hybrid vapor phase epitaxy [
2], chemical beam epitaxy [
3], and molecular beam epitaxy (MBE), have been used for the growth of compound semiconductors and other materials. In this work, we strictly focus on MBE, and we restrict our attention to the differential equation model, which was proposed by Escudero et al. [
4,
5,
6,
7]. The mathematical description of epitaxial growth is carried out by means of a function
defined as
which describes the height of the growing interface in the spatial point
at time
. The authors in [
4,
5,
6,
7] show that the function
obeys the fourth-order partial differential equation
where
models the incoming mass entering the system through epitaxial deposition,
measures the intensity of this flux, and the determinant of Hessian matrix is
The stationary counterpart of the partial differential Equation (
1) subject to the homogeneous Dirichlet boundary condition (4) and homogeneous Navier boundary condition (5) is defined as (see [
6])
where
is a stationary flux, and
n is the unit out drawn normal to
.
Using the transformation
and
, as a result of symmetry, the above set of equations are transformed into the following set of equations:
where
.
In this paper, we also impose the following boundary conditions which complements the work in [
6]:
For simplicity, we take , which physically means that the new material is being deposited uniformly on the unit disc.
Now, using
,
and integrating parts from Equation (
6), we have
Using the transformation
and
, it is possible to reduce Equation (
10) into the following equation:
Corresponding to (
11), we define the following three boundary value problems:
The BVPs (
12), (13) and (14) can be equivalently described as the following integral equations (IE):
IE corresponding to Problem 1:
IE corresponding to Problem 2:
IE corresponding to Problem 3:
We assume that
, where
is defined as
In [
6], Escudero et al. proved the existence and nonexistence of solutions of Problems 1 and 3 using upper and lower solution techniques. Corresponding to Problems 1 and 3, they have also provided the rigorous bounds of the values of the parameter
, which helps us to separate existence from nonexistence. In [
8], Verma et al. provide numerical illustrations via VIM to verify the results of Escudero et al. [
6]. To verify their numerical results, they provided other iterative schemes based on homotopy [
9] and the Adomian decomposition method [
10].
Equation (13) has not been investigated theoretically in the existing literature to the best of our knowledge. Moreover, many investigations are still pending relating to BVPs (
12), (13), and (14). Here, we focus on both theoretical and numerical work. We derive the sign of the solution and prove its existence in continuous space. We also compute the bounds of the parameter
. The results of this paper complements existing theoretical results. We also 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
.
To prove the existence of the solutions, we use the monotone iterative technique [
11,
12,
13,
14,
15,
16,
17]. Recently, many researchers applied this technique on the initial value problem (IVP) for the nonlinear noninstantaneous impulsive differential equation (NIDE) [
18], p-Laplacian boundary value problems with the right-handed Riemann–Liouville fractional derivative [
19], etc. to prove the existence of the solution. Here, we also present numerical results to verify the theoretical results. To develop the iterative scheme based on Green’s function, we consider Equations (
12)–(14). Recently, many authors have used numerical approximate methods like the VIM [
8], the Adomian decomposition method (ADM), the homotopy perturbation method (HPM) etc. to find approximate solutions for different models involving differential equations [
20,
21], integral equations [
22,
23,
24], fractional differential equations [
25,
26], the Stefan problem [
27,
28,
29,
30], system of integral equations [
31], etc. Thereafter, Waleed Al Hayani [
32] and Singh et al. [
33] applied ADM with Green’s function to compute the approximate solution. Recently, Noeiaghdam et al. [
34] proposed a technique based on ADM for solving Volterra integral equation with discontinuous kernels using the CESTAC method. To find out more about this method, please see [
35,
36]. They focused on the BVPs which have a unique solution. The major advantage of our proposed technique is its ability to capture multiple solutions together with a desired accuracy.
The remainder of the paper is focused on both theoretical and numerical results. We prove some of the basic properties of the BVPs in
Section 2. The monotone iterative technique is presented in
Section 3 to prove the existence of a solution. A wide range of
for Equation (
6), corresponding to different types of boundary conditions, is shown in
Section 4. In
Section 5, we apply our proposed technique to the integral equations and show a wide range of numerical results. Finally, in
Section 6, we draw our main conclusions.