IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i1p91-d474362.html
   My bibliography  Save this article

Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

Author

Listed:
  • Costică Moroşanu

    (Department of Mathematics, “Alexandru Ioan Cuza” University, Bd. Carol I, 11, 700506 Iaşi, Romania)

  • Silviu Pavăl

    (Faculty of Automatic Control and Computer Engineering, Technical University “Gheorghe Asachi” of Iaşi, Dimitrie Mangeron, nr. 27, 700050 Iaşi, Romania)

Abstract

In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f ( t , x ) , w ( t , x ) and v 0 ( x ) , we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space W p 1 , 2 ( Q ) , facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks.

Suggested Citation

  • Costică Moroşanu & Silviu Pavăl, 2021. "Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation," Mathematics, MDPI, vol. 9(1), pages 1-23, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:1:p:91-:d:474362
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/1/91/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/1/91/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Dongsun Lee & Seunggyu Lee, 2019. "Image Segmentation Based on Modified Fractional Allen–Cahn Equation," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-6, January.
    2. Barbu, Tudor & Miranville, Alain & Moroşanu, Costică, 2019. "A qualitative analysis and numerical simulations of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy–Neumann boundary conditions," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 170-180.
    Full references (including those not matched with items on IDEAS)

    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. Sungha Yoon & Darae Jeong & Chaeyoung Lee & Hyundong Kim & Sangkwon Kim & Hyun Geun Lee & Junseok Kim, 2020. "Fourier-Spectral Method for the Phase-Field Equations," Mathematics, MDPI, vol. 8(8), pages 1-36, August.
    2. Mitică CRAUS & Silviu-Dumitru PAVĂL, 2020. "An Accelerating Numerical Computation of the Diffusion Term in a Nonlocal Reaction-Diffusion Equation," Mathematics, MDPI, vol. 8(12), pages 1-12, November.
    3. Ham, Seokjun & Kim, Junseok, 2023. "Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 453-465.

    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:9:y:2021:i:1:p:91-:d:474362. 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.