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Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method

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  • B. A. Jacobs
  • C. Harley

Abstract

This work considers a hybrid solution method for the time-fractional diffusion model with a cubic nonlinear source term in one and two dimensions. Both Dirichlet and Neumann boundary conditions are considered for each dimensional case. The hybrid method involves a Laplace transformation in the temporal domain which is numerically inverted, and Chebyshev collocation is employed in the spatial domain due to its increased accuracy over a standard finite-difference discretization. Due to the fractional-order derivative we are only able to compare the accuracy of this method with Mathematica ’s NDSolve in the case of integer derivatives; however, a detailed discussion of the merits and shortcomings of the proposed hybridization is presented. An application to image processing via a finite-difference discretization is included in order to substantiate the application of this method.

Suggested Citation

  • B. A. Jacobs & C. Harley, 2018. "Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method," Journal of Mathematics, Hindawi, vol. 2018, pages 1-9, September.
    • Handle: RePEc:hin:jjmath:8924547
    DOI: 10.1155/2018/8924547
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    Cited by:

    1. Jacobs, B.A. & Celik, T., 2022. "Unsupervised document image binarization using a system of nonlinear partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 418(C).

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