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

Author

Listed:
  • Amit K Verma

    (Department of Mathematics, Indian Institute of Technology Patna, Patna 801106, Bihar, India
    These authors contributed equally to this work.)

  • Biswajit Pandit

    (Department of Mathematics, Indian Institute of Technology Patna, Patna 801106, Bihar, India
    These authors contributed equally to this work.)

  • Ravi P. Agarwal

    (Department of Mathematics, Texas A & M University-Kingsville, 700 University Blvd., MSC 172, Kingsville, TX 78363-8202, USA)

Abstract

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is non-self adjoint, it does not have exact solutions, and it admits multiple solutions. Here, λ ∈ R measures the intensity of the flux and G is stationary flux. The solution depends on the size of the parameter λ . We use a monotone iterative technique and integral equations along with upper and lower solutions to prove that solutions exist. We establish the qualitative properties of the solutions and provide bounds for the values of the parameter λ , which help us to separate existence from nonexistence. These results complement some existing results in the literature. To verify the analytical results, we also propose a new computational iterative technique and use it to verify the bounds on λ and the dependence of solutions for these computed bounds on λ .

Suggested Citation

  • Amit K Verma & Biswajit Pandit & Ravi P. Agarwal, 2021. "Analysis and Computation of Solutions for a Class of Nonlinear SBVPs Arising in Epitaxial Growth," Mathematics, MDPI, vol. 9(7), pages 1-25, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:7:p:774-:d:529159
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    References listed on IDEAS

    as
    1. Agarwal, Ravi & O'Regan, D. & Hristova, S., 2017. "Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses," Applied Mathematics and Computation, Elsevier, vol. 298(C), pages 45-56.
    2. Ghorbani, Asghar, 2009. "Beyond Adomian polynomials: He polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1486-1492.
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    1. Pandit, Biswajit & Mainini, Pratikshya & Verma, Amit K. & Agarwal, Ravi P., 2026. "Existence and non-existence of radial solutions for a class of fourth order elliptic PDE arising in epitaxial growth theory," Applied Mathematics and Computation, Elsevier, vol. 509(C).

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