GLOBAL EXISTENCE IN REACTION DIFFUSION NONLINEAR PARABOLIC PARTAIL DIFFERENTIAL EQUATION IN IMAGE PROCESSING.

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

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].

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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,H}1_{(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.

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