IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i9p806-d262997.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/9/806/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/9/806/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ginkyu Choi & Soon-Mo Jung, 2019. "A Dilation Invariance Method and the Stability of Inhomogeneous Wave Equations," Mathematics, MDPI, vol. 7(1), pages 1-17, January.
    2. Abdellatif Benchaib & Abdelkrim Salim & Saïd Abbas & Mouffak Benchohra, 2023. "New Stability Results for Abstract Fractional Differential Equations with Delay and Non-Instantaneous Impulses," Mathematics, MDPI, vol. 11(16), pages 1-19, August.
    3. Soon-Mo Jung & Yang-Hi Lee, 2017. "A Fixed Point Approach to the Stability of a Mean Value Type Functional Equation," Mathematics, MDPI, vol. 5(4), pages 1-9, December.
    4. Abe, Takaaki & Nakada, Satoshi, 2023. "The in-group egalitarian Owen values," Games and Economic Behavior, Elsevier, vol. 142(C), pages 1-16.
    5. Soon-Mo Jung & Ji-Hye Kim, 2018. "Hyers-Ulam Stability of Lagrange’s Mean Value Points in Two Variables," Mathematics, MDPI, vol. 6(11), pages 1-8, October.
    6. Soon-Mo Jung & Ki-Suk Lee & Michael Th. Rassias & Sung-Mo Yang, 2020. "Approximation Properties of Solutions of a Mean Value-Type Functional Inequality, II," Mathematics, MDPI, vol. 8(8), pages 1-8, August.
    7. L. Chitra & K. Alagesan & Vediyappan Govindan & Salman Saleem & A. Al-Zubaidi & C. Vimala, 2023. "Applications of the Tarig Transform and Hyers–Ulam Stability to Linear Differential Equations," Mathematics, MDPI, vol. 11(12), pages 1-25, June.
    8. Jagan Mohan Jonnalagadda, 2016. "Hyers-Ulam Stability of Fractional Nabla Difference Equations," International Journal of Analysis, Hindawi, vol. 2016, pages 1-5, September.
    9. Daniela Marian & Sorina Anamaria Ciplea & Nicolaie Lungu, 2022. "Hyers–Ulam Stability of a System of Hyperbolic Partial Differential Equations," Mathematics, MDPI, vol. 10(13), pages 1-9, June.
    10. El-sayed El-hady & Janusz Brzdęk, 2022. "Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation," Mathematics, MDPI, vol. 10(10), pages 1-15, May.
    11. 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.
    12. Zada, Akbar & Ali, Wajid & Park, Choonkil, 2019. "Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 60-65.
    13. Chinnaappu Muthamilarasi & Shyam Sundar Santra & Ganapathy Balasubramanian & Vediyappan Govindan & Rami Ahmad El-Nabulsi & Khaled Mohamed Khedher, 2021. "The Stability Analysis of A-Quartic Functional Equation," Mathematics, MDPI, vol. 9(22), pages 1-16, November.
    14. Naveed Ahmad & Zeeshan Ali & Kamal Shah & Akbar Zada & Ghaus ur Rahman, 2018. "Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations," Complexity, Hindawi, vol. 2018, pages 1-15, February.
    15. Zahra Eidinejad & Reza Saadati & Radko Mesiar, 2022. "Optimum Approximation for ς –Lie Homomorphisms and Jordan ς –Lie Homomorphisms in ς –Lie Algebras by Aggregation Control Functions," Mathematics, MDPI, vol. 10(10), pages 1-17, May.
    16. Zhenyu Bai & Chuanzhi Bai, 2024. "Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps," Mathematics, MDPI, vol. 12(6), pages 1-14, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:806-:d:262997. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.