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A Review on a Class of Second Order Nonlinear Singular BVPs

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  • Amit K. Verma

    (Department of Mathematics, Indian Institute of Technology Patna, Patna 801106, Bihar, India
    Amit K. Verma dedicates this review to supervisor Prof. Rajni Kant Pandey, IIT Kharagpur.)

  • Biswajit Pandit

    (Department of Mathematics, Indian Institute of Technology Patna, Patna 801106, Bihar, India)

  • Lajja Verma

    (Department of Applied Sciences, Netaji Subhas Institute of Technology, Patna 801116, Bihar, India)

  • Ravi P. Agarwal

    (Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363-8202, USA)

Abstract

Several real-life problems are modeled by nonlinear singular differential equations. In this article, we study a class of nonlinear singular differential equations, explore its various aspects, and provide a detailed literature survey. Nonlinear singular differential equations are not easy to solve and their exact solution does not exist in most cases. Since the exact solution does not exist, it is natural to look for the existence of the analytical solution and numerical solution. In this survey, we focus on both aspects of nonlinear singular boundary value problems (SBVPs) and cover different analytical and numerical techniques which are developed to deal with a class of nonlinear singular differential equations − ( p ( x ) y ′ ( x ) ) ′ = q ( x ) f ( x , y , p y ′ ) for x ∈ ( 0 , b ) , subject to suitable initial and boundary conditions. The monotone iterative technique has also been briefed as it gained a lot of attention during the last two decades and it has been merged with most of the other existing techniques. A list of SBVPs is also provided which will be of great help to researchers working in this area.

Suggested Citation

  • Amit K. Verma & Biswajit Pandit & Lajja Verma & Ravi P. Agarwal, 2020. "A Review on a Class of Second Order Nonlinear Singular BVPs," Mathematics, MDPI, vol. 8(7), pages 1-50, June.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1045-:d:377269
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    References listed on IDEAS

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    1. Amit K. Verma & Lajja Verma, 2011. "Nonlinear Singular BVP of Limit Circle Type and the Presence of Reverse-Ordered Upper and Lower Solutions," International Journal of Differential Equations, Hindawi, vol. 2011, pages 1-13, September.
    2. Roul, Pradip & Madduri, Harshita & Kassner, Klaus, 2019. "A sixth-order numerical method for a strongly nonlinear singular boundary value problem governing electrohydrodynamic flow in a circular cylindrical conduit," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 416-433.
    3. Roul, Pradip & Prasad Goura, V.M.K., 2019. "B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 428-450.
    4. Goura, V.M.K. Prasad & Roul, Pradip, 2019. "Erratum to: B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 198-201.
    5. Ramos, J.I., 2008. "Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 400-408.
    6. Roul, Pradip & Prasad Goura, V.M.K. & Agarwal, Ravi, 2019. "A compact finite difference method for a general class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 283-304.
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    1. Kayenat, Sheerin & Verma, Amit Kumar, 2022. "On the choice of denominator functions and convergence of NSFD schemes for a class of nonlinear SBVPs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 263-284.

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