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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
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    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.
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