Numerical solutions for a class of singular boundary value problems arising in the theory of epitaxial growth

Abstract

Purpose This paper aims to apply an iterative numerical method to find the numerical solution of the nonlinear non-self-adjoint singular boundary value problems that arises in the theory of epitaxial growth. Design/methodology/approach The proposed problem has multiple solutions and it is singular too; so not every technique can capture all the solutions. This study proposes to use variational iterative numerical method and compute both the solutions. The computed solutions are very close to the exact result. Findings It turns out that the existence or nonexistence of numerical solutions fully depends on the value of a parameter. The authors show that numerical solutions exist for small positive values of this parameter. For large positive values of the parameter, they find nonexistence of solutions. They also observe existence of solutions for negative values of the parameter and determine the range of parameter values which separates existence and nonexistence of solutions. This parameter has a clear physical meaning, as it describes the rate at which new material is deposited onto the system. This fact allows interpreting the physical significance of the results. Originality/value The authors could capture both the solutions and got accurate estimation of the parameter. This method will be a great tool to handle such types of nonlinear non-self-adjoint equations that have multiple solutions in engineering and mathematical sciences. The results in this paper will have an impact on the understanding of theoretical models of epitaxial growth in near future.

Analysis and Computation of Solutions for a Class of Nonlinear SBVPs Arising in Epitaxial Growth

“…Therefore, from equations (30) and by similar analysis as in Lemma 1.6, we 93 can prove the result (28).…”

“…Proof. From Lemma 7.7 in [10] and Lemma 3.5, we get the equation ( To find the approximate solutions, we develop the iterative numerical schemes with the help of the Fredholm integral equations (15), (16) and 17respectively. Now, we decompose the solution u(t) of the form u(t) = ∑ ∞ i=0 u i (t), and approximate the nonlinear term in terms of Adomian's polynomials ( [13]) which is given by…”

“…Step 4. Approximate the term Again, we apply the above algorithm on equations (16) and 17, and we define the following iterative schemes:…”

“…Approximate solutions of equations (16) and 17can be written as u(t) = 233 n+1 ∑ i=0 u i (t), provided the series is convergent for n → ∞. Recently, the conver-234 gence of ADM was established by Verma et al in [28].…”

“…Approximate solutions of equations (16) and 17can be written as u(t) = 233 n+1 ∑ i=0 u i (t), provided the series is convergent for n → ∞. Recently, the conver-234 gence of ADM was established by Verma et al in [28]. Now by using the 235 transformation t = r 2 2 , u(t) = w(r), w(r) = rφ (r) and φ(1) = 0, we get the 236 solutions of equation (6).…”