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[Hamzah., 3(5): May, 2016] ISSN 2349-0292 Impact Factor 2.365

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Global Journal of Advanced Engineering Technologies and Sciences

GLOBAL EXISTENCE IN REACTION DIFFUSION NONLINEAR

PARABOLIC PARTAIL DIFFERENTIAL EQUATION IN IMAGE

PROCESSING Bassam Al-Hamzah*, Naji Yebari

*University Abdelmalek Essaadi Faculty of sciences Department of Mathematics

Tetouan, Morocco

University Abdelmalek Essaadi Faculty of sciences Department of Mathematics

Tetouan, Morocco

Abstract This paper deals with the equation.

We give the proof of global existence of our nonlinear reaction diffusion of problem (P). In the first we define an

approximating scheme and by using Schauder fixed point theorem in ordered Banach spaces, we show the existence

of a solution for this approached problem. Finally by making some estimations we prove that the solution of the

truncated equation converge to the solution of our problem. We use a new technique recently introduced in order to

generalize some of interesting prior works that has been presented.

Keywords: Weak solutions, Non linear restoration diffusion, Image processing.

Introduction In recent years attention has been given to weak solutions of elliptic problems under linear boundary conditions, and

different methods for the existence problem have been used [1], [4], [5], [12] and [13]. The corresponding parabolic

case equations have also been studied by many authors, see for instance [1], [14] and [17].

Moreover, image processing is always a challenging problem, this topic has become “hot”, recently and a very active

field of computer applications and research [18]. Image is restored to its original quality by inverting the physical

degradation phenomenon such as defocus, linear motion, atmospheric degradation and additive noise. Partial

differential equation (PDE) methods in image processing have proven to be fundamental tools for image diffusion and

restoration [7], [8], [9] and [21].

In this paper, we prove the existence of solutions for the reaction- diffusion of the form:

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In the present paper: We give the proof of global existence of our nonlinear reaction diffusion of problem (P), this is

done in four steps: the first step we show the positive solution. In the second step is to truncate the equation and shows

that the problem obtained has a solution. In the third step we establish appropriate estimates on the approximate

solutions. In the last step, we show the convergence of the approximate equation. We use a new technique recently

introduced, in fact our results f = f(t, x, u, u) are a generalization of the work f = 0 presented by Catté [16], and the

work f = f(t, x, u) presented by Alaa [2]. Now we will recall some functional spaces that will be used throughout this

paper. L2(0, T,H1(Ω)) is the set of functions u such that, for all every t ϵ (0, T), u(t) belongs to H1(Ω) with the norm

THE MAIN RESULT

Our main result in this paper is the following existence theorem. Assume that (Hf )1, (Hf )2, (Hf )3, (Hf )4, and (Hg), and

that for all

Then

(i) Problem (P) admits a weak positive solution.

(ii) If moreover for all 1 dans QT.

A typical examples where the result of this paper can be applied are (i) thre of the diffusivity Perana and Malik [21]

are




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