Existence and nonexistence results of radial solutions to singular BVPs arising in epitaxial growth theory

11/12/2019
by   Amit Kumar Verma, et al.
0

The existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. The fourth-order radial equation is non-self adjoint and has no exact solutions. Also, it admits multiple solutions. Furthermore, solutions depend on the size of the parameter. We show that solutions exist for small positive values of this parameter. For large positive values of this parameter, we prove the nonexistence of solutions. We establish the qualitative properties of the solutions and provide bounds for the values of this parameter, which help us to separate the existence from nonexistence. We propose a new numerical scheme to capture the radial solutions. The results show that the iterative method is of better accuracy, more convenient and efficient for solving BVPs, which have multiple solutions. We verify theoretical results by numerical results. We also see the existence of solutions for negative values of the same parameter.

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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 and etc. Epitaxial growth technique is used to produce the growth of semiconductor films and multilayer structures under high vacuum conditions ([5]). The major advantages of epitaxial growth are to reduces the growth time, better structural and superior electrical properties, eliminates the wastages caused during growth, wafering cost, cutting, polishing and etc. Several types of epitaxial growth techniques like the Hybrid vapor phase epitaxy ([16]), Chemical beam epitaxy ([11]), Molecular beam epitaxy (MBE), etc 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 is described by Carlos et. al. in [10, 8, 9, 7]. In these references, the mathematical description of epitaxial growth is carried out by means of a function

which describes the height of the growing interface in the spatial point at time . Authors ([10, 8, 9, 7]) shown that the function obeys the fourth order partial differential equation

(1)

where models the incoming mass entering the system through epitaxial deposition and measures the intensity of this flux. For simplicity they considered the stationary counterpart of the partial differential equation (1), which is given by

(2)

where they assumed that is a stationary flux. Again, they set this problem on the unit disk and considered two types of boundary conditions. Corresponding to (2) homogeneous Dirichlet boundary condition ([8]) is

(3)

where is unit out drawn normal to , and homogeneous Navier boundary condition is

(4)

By using the transformation and , the above partial differential equation (2

) is converted into a fourth order ordinary differential equation which reads

(5)

where .

The boundary conditions that correspond to (3) are

(6)

and the boundary conditions corresponding to (4)

(7)

Here, we impose another boundary conditions corresponding to (4)

(8)

The condition imposes the existence of an extremum at the origin. The conditions and are the actual boundary conditions. For simplicity we take , which physically means that the new material is being deposited uniformly on the unit disc. Now, by using , and integrating by parts ([8]) from equation (5), we have

(9)

By using the transformation and , it is posible to reduce the equation (9) into the following equation

(10)

Corresponding to (10), we define the following three boundary value problems:

(11)
(12)

and

(13)

The BVPs (11), (12) and (13) can equivalantly be described as the following integral equations (IE):

  • IE corresponding to Problem :

    (14)
  • IE corresponding to Problem :

    (15)
  • IE corresponding to Problem :

    (16)

We consider the function , where is defined as

In [8], Carlos et. al. proved the existence and nonexistence of solutions of Problem and Problem by using upper and lower solution techniques. Corresponding to Problem and , they have also provided the rigorous bounds of the values of the parameter , which helps us to separate the existence from nonexistence. We did not find any theoretical results corresponding to Problem . Again, equation (5) is a nonlinear, singular, non-self-adjoint and has no exact solutions. Moreover, it admits multiple solutions. Therefore discrete methods such as finite element method etc may not be applicable to capture all solutions together. These facts highlight the difficulties to deal with such BVPs both analytically and numerically. Furthermore, to the best of our knowledge, there are only a few research papers that address both theoretical and numerical results corresponding to BVPs (11), (12) and (13), and a lot of investigations are still pending.

In this work, basically, we extend the theoretical results, which is described by Carlos et. al. in [8]. We prove some qualitative properties of the solutions and provide the rigorous bounds of the same parameter corresponding to different problems. To prove the existence of solutions, here we use the monotone iterative technique ([21, 20, 23, 22, 24, 6, 27, 15]). Recently, many researchers applied this technique on the initial value problem (IVP) for the nonlinear noninstantaneous impulsive differential equation (NIDE) ([3]), p-Laplacian boundary value problems with the right-handed Riemann-Liouville fractional derivative ([28]), etc to prove the existence of the solution. Here, we also present numerical results to verify the theoretical results. We propose an iterative scheme to compute the approximate numerical solutions of the fourth-order differential equation (5) with by using equations (11), (12), (13) and it’s respective Green’s function. Recently, many authors have used numerical approximate methods like the Adomian decomposition method (ADM), homotopy perturbation method (HPM), etc to find approximate solution for different models involving differential equations ([18, 19]), integral equation ([17, 4, 1]), fractional differential equations ([14, 2]) etc. After that, Waleed Al Hayani ([13]) and Singh et. al. ([25]) applied ADM with Green’s function to compute the approximate solution. They focused on the BVPs which have a unique solution. The major advantage of our proposed technique is to capture multiple solutions together with desired accuracy.

The remainder of the paper has been focused on both theoretical and numerical results. We have proved some 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 of equation (5) corresponding to different types of boundary conditions are shown in section 4. In section 5, we apply our proposed technique on the integral equations and show a wide range of numerical results. Finally in section 6, we draw our main conclusions.

2 Preliminary

Corresponding to , we prove some basic qualitative properties of the solution , which satisfies the following inequality

(17)

Here, we omit the proof of lemma 2.0.1, lemma 2.0.2, lemma 2.0.3, corollary 2.0.1, lemma 2.0.4 which has been done by Carlos et. al. in [8].

Lemma 2.0.1.

Let satisfy and equation (17), then .

Lemma 2.0.2.

Let satisfy , and equation (17), then for all .

Lemma 2.0.3.

Let satisfy , and equation (17), then for all .

Corollary 2.0.1.

Let satisfy , and equation (17), then if and only if .

Lemma 2.0.4.

Let satisfy . Then for every , we have

(18)
Lemma 2.0.5.

Let satisfy , and equation (17), then for all .

Proof.

First, we show that . Assume . Since , therefore we have there exist a such that . Now from (17), we have is increasing function on . Again by mean value theorem, we have

(19)

Since , therefore we have Hence we get , which is a contradiction. So, we have Furthermore, is a convex function along with . Also is increasing, which implies . Again is decreasing function on . Therefore and leads to on . ∎

Lemma 2.0.6.

Let be the solution of Problem , then satisfies the following integral equation

(20)

and

(21)
Proof.

The Green’s function of the Problem can be written as

(22)

Therefore from equation (22) and Problem , we can easily deduce the integral equation (20). Now, by using the result of Lemma 2.0.1, we have

(23)

Now, put

(24)

Therefore we get provided . Consequently, we have

(25)

Hence, from equation (23), we get the equation (21). ∎

Lemma 2.0.7.

Let be the solution of Problem , then can be written as in the following form

(26)

and also satisfies

(27)
Proof.

By using the boundary condition and properties of Green’s function, we have

(28)

Similarly, from equation (28) and Problem , we can easily derive the equation (26). Now, by using the result of Lemma 2.0.1, we have

(29)

Therefore, from equations (29) and by similar analysis as in Lemma 2.0.6, we can prove the result (27). ∎

Lemma 2.0.8.

Let be the solution of Problem , then can be written as in the following form

(30)

and satisfies

(31)
Proof.

The Green’s function of the Problem is given by

(32)

Again, from equation (32) and Problem , we derive integral equation (30). Furthermore, by using the result of Lemma 2.0.1, we have

(33)

Again, by similar analysis as in Lemma 2.0.7, we get the inequality (31). ∎

Remark 2.0.1.

From Lemma 2.0.8 (respectively Lemma 2.0.7 and Lemma 2.0.6), we can easily justify the result of Lemma 2.0.2 (respectively Lemma 2.0.5 and Lemma 2.0.3).

3 Existence of solutions

In this section, we apply the monotone lower and upper solution technique to prove the existence of at least one solution of Problem , Problem and Problem . For this purpose, we need to prove some lemmas, which help us to proof the main theorems.

3.1 Construction of Green’s function

To investigate the Problem , Problem and Problem , we consider the corresponding nonlinear singular boundary value problems, which are given by

(34)
(35)

and

(36)

where , and .

Lemma 3.1.1.

Let and be the solution of Problem , then

(37)

where Green’s function is given by

(38)

and for all and .

Proof.

By using the boundary condition of Problem and properties of Green’s function, we can easily prove the equation (38). Furthermore we have for all and . ∎

Lemma 3.1.2.

Let and be the solution of Problem , then

(39)

where Green’s function is given by

(40)

and for all and .

Proof.

In a similar manner as in Lemma 3.1.1, we can easily get the equation (40), and prove for all and . ∎

Lemma 3.1.3.

Let , and be the solution of Problem , then

(41)

where Green’s function is given by

(42)

and for all and .

Proof.

Again by similar analysis, we can easily derive the equation (42). Now,

(43)
(44)
(45)
(46)
(47)
(48)
(49)

Hence from (42), we have for all and . ∎

Lemma 3.1.4.

Let and be the solution of Problem , then

(50)

where Green’s function is given by

(51)

and for all and .

Proof.

Proof is similar as in Lemma 3.1.1. ∎

Lemma 3.1.5.

Let and be the solution of Problem , then

(52)

where Green’s function is given by

(53)

and for all and .

Proof.

Proof is similar as in Lemma 3.1.2. ∎

Lemma 3.1.6.

Let , and be the solution of Problem , then

(54)

where Green’s function is given by

(55)

and for all and .

Proof.

Proof is similar as in Lemma 3.1.3. ∎

3.2 Anti-maximum principle

Proposition 3.2.1.

Let and is such that , then the solutions of Problem and Problem are non positive.

Proposition 3.2.2.

Let , and is such that , then the solutions of Problem are non positive.

Proposition 3.2.3.

Let (respectively ) and is such that , then the solutions of Problem (respectively Problem ) are non positive.

Proposition 3.2.4.

Let , and is such that , then the solutions of Problem are non positive.

3.3 Reverse order lower and upper solutions

Here, we define lower and upper solutions corresponding to Problem , Problem and Problem .

Definition 3.3.1.

(lower solution) A function is the lower solution of Problem (respectively Problem and Problem ) if

(56)

with and (respectively and ).

Definition 3.3.2.

(upper solution) A function is the upper solution of Problem (respectively Problem and Problem ) if

(57)

with and (respectively and ).

Now, we construct two sequences and corresponding to Problem (respectively Problem and Problem ), which are defined by

(58)
(59)

and

(60)
(61)

. We assume the following properties:

  • : and satisfies

    (62)

    and

    (63)
  • : is continuous on where .

Theorem 3.3.1.

Assume , and there exist and are lower and upper solutions of Problem which satisfy the properties and