On multiple solutions for a fourth order nonlinear singular boundary value problems arising in epitaxial growth theory

2021 ◽  
Author(s):  
Amit Kumar Verma ◽  
Biswajit Pandit ◽  
Ravi P. Agarwal
Keyword(s):  
Boundary Value Problems ◽  
Epitaxial Growth ◽  
Fourth Order ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Growth Theory ◽  
Singular Boundary ◽  
Singular Boundary Value Problems

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

2020 ◽  
Vol 37 (7) ◽  
pp. 2539-2560 ◽  
Author(s):  
Amit K. Verma ◽  
Biswajit Pandit ◽  
Carlos Escudero
Keyword(s):  
Boundary Value Problems ◽  
Numerical Method ◽  
Epitaxial Growth ◽  
Multiple Solutions ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Content Type ◽  
Nonexistence Of Solutions

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.

scholarly journals Positive solutions of fourth-order superlinear singular boundary value problems

2002 ◽  
Vol 66 (1) ◽  
pp. 95-104 ◽  
Author(s):  
Guoliang Shi ◽  
Shaozhu Chen
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Closed Interval ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Singular Boundary ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
Necessary And Sufficient

This paper investigates fourth-order superlinear singular two-point boundary value problems and obtains necessary and sufficient conditions for existence of C2 or C3 positive solutions on the closed interval.

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