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

TL;DR

This paper investigates the existence of numerical solutions for a singular boundary value problem in epitaxial growth, revealing how solutions depend on a key physical parameter and identifying parameter ranges for existence.

Contribution

It introduces an iterative numerical method for a reduced second order problem and analyzes the parameter-dependent existence of solutions in the context of epitaxial growth.

Findings

Solutions exist for small positive parameters.

Solutions do not exist for large positive parameters.

Existence occurs for negative parameter values.

Abstract

The existence of numerical solutions to a fourth order singular boundary value problem arising in the theory of epitaxial growth is studied. An iterative numerical method is applied on a second order nonlinear singular boundary value problem which is the exact result of the reduction of this fourth order singular boundary value problem. It turns out that the existence or nonexistence of numerical solutions fully depends on the value of a parameter. We show that numerical solutions exist for small positive values of this parameter. For large positive values of the parameter, we find nonexistence of solutions. We 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 us to interpret the physical significance of our results.