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Application of nonlinear time-fractional partial differential equations to image processing via hybrid Laplace transform method. (English) Zbl 07029805
Summary: 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.

MSC:
65-XX Numerical analysis
35-XX Partial differential equations
78-XX Optics, electromagnetic theory
Software:
NDSolve; Mathematica
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