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Some Properties of Approximate Solutions of Linear Differential Equations

Author

Listed:
  • Ginkyu Choi

    (Department of Electronic and Electrical Engineering, College of Science and Technology, Hongik University, Sejong 30016, Korea
    These authors contributed equally to this work.)

  • Soon-Mo Jung

    (Mathematics Section, College of Science and Technology, Hongik University, Sejong 30016, Korea
    These authors contributed equally to this work.)

  • Jaiok Roh

    (Ilsong College of Liberal Arts, Hallym University, Chuncheon, Kangwon-Do 200-702, Korea
    These authors contributed equally to this work.)

Abstract

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ″ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ″ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

Suggested Citation

  • Ginkyu Choi & Soon-Mo Jung & Jaiok Roh, 2019. "Some Properties of Approximate Solutions of Linear Differential Equations," Mathematics, MDPI, vol. 7(9), pages 1-11, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:806-:d:262997
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    References listed on IDEAS

    as
    1. Soon-Mo Jung, 2011. "Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis," Springer Optimization and Its Applications, Springer, number 978-1-4419-9637-4, September.
    2. Yongjin Li & Yan Shen, 2009. "Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2009, pages 1-7, October.
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    Cited by:

    1. Ginkyu Choi & Soon-Mo Jung, 2020. "The Approximation Property of a One-Dimensional, Time Independent Schrödinger Equation with a Hyperbolic Potential Well," Mathematics, MDPI, vol. 8(8), pages 1-8, August.

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