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SUPERVISOR
Reza Mokhtari
رضا مختاری (استاد راهنما)
 
STUDENT
Arezoo Latif altojar
آرزو لطیف التجار

FACULTY - DEPARTMENT

دانشکده ریاضی
DEGREE
Master of Science (MSc)
YEAR
1393

TITLE

On the Finite Difference Schemes For Linear/Nonlinear Complex Reaction-Diffusion Processes
The well-known diffusion equation has several applications in different fields of science and engineering. Several important processes in image processing such as denoising, inpainting, stereo vision, or optical flow exploit linear or nonlinear and real or complex diffusion processes. The method of finite difference is undoubtedly one of the most successful and powerful numerical methods for solving partial differential equations (PDEs) involving diffusion term. Stability and convergence analysis of a finite difference scheme is an interesting issue for scientists and mathematicians. The finite difference method (FDM) has a long history and we mention some of more interesting works. Convergence of FDMs for systems of nonlinear reaction-diffusion equations with real variables was studied by Hoff in 1978. For the complex case, we refer to Wang's works where he considered the analysis of some conservative schemes for a coupled nonlinear Schrodinger system in 2010. Although the stability condition and the numerical analysis of finite difference schemes for real nonlinear diffusion and reaction-diffusion equations has been investigated extensively and is widely documented in the literature. rigorous proof of the stability and convergence of finite difference schemes for the general nonlinear complex reaction-diffusion equations refers to some works of Ara?jo et. al. which began from 2014. In this thesis, we aim to investigate some recent works of Ara?jo et. al. related to a general nonlinear complex reaction-diffusion equation and proof of the stability and convergence properties of a ltr"
معادله واکنش-انتشار مختلط خطی/غیرخطی یک معادله سهموی است که عموما در پردازش تصویر استفاده می‌شود. در این رساله ابتدا معادله واکنش-انتشار مختلط خطی/غیرخطی را بیان و دسته‌ای از طرح‌های تفاضل متناهی را بر آن اعمال می‌کنیم. سپس به بررسی پایداری و اثبات همگرایی پرداخته و با در نظر گرفتن گسسته‌سازی ضمنی و نیمه‌ضمنی، دقت جواب عددی را به دست می‌آوریم. سرانجام به منظور نمایش نتایج تحلیلی، مثال‌های عددی محاسبه‌شده با طرح نیمه‌ضمنی و به کار رفته برای معادله واکنش-انتشار مختلط غیرخطی/خطی را ارائه می‌کنیم.

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