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Contributions aux équations aux dérivées fractionnaires et au traitement d'images

Abstract : In this thesis we study a nonlinear system of fractional differential equations with power nonlinearities; the solution of the system blows up in a finite time. We provide the profile of the blowing-up solutions of the system by finding upper and lower estimates of the solution. Moreover, bilateral bounds on the blow-up time are given.We consider the inverse problem concerning a linear time fractional diffusion equation for the determination of the source term (supposed to be independent of the time variable) and temperature distribution from initial and final temperature data. The uniqueness and existence of the continuous solution of the inverse problem is proved. We also consider the inverse source problem for a two dimensional fractional diffusion equation. The results about the existence, uniqueness and continuous dependence of the solution of the inverse problem on the data are presented.We apply the linear heat equation involving a fractional derivative in time for denoising (simplification, smoothing, restoration or enhancement) of digital images. The order of the fractional derivative has been used for controling the diffusion process, which in result preserves the fine structures in the image during denoising process. Furthermore, an improvement in the proposed model is suggested by using the structure tensor of the images.
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Salman Amin Malik. Contributions aux équations aux dérivées fractionnaires et au traitement d'images. Mathématiques générales [math.GM]. Université de La Rochelle, 2012. Français. ⟨NNT : 2012LAROS370⟩. ⟨tel-00825874⟩

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