بررسي چند طرح تفاضل متناهي براي فرايندهاي واكنش

چكيده انگليسي :

On The Finite Di erence Schemes For Linear Nonlinear Complex Reaction Di usion Processes Arezoo Latif Altojar a latif@math iut ac ir 21 09 2016 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari mokhtari@cc iut ac ir 2010 MSC 65M 12 65M 06 Keywords Complex linear nonlinear reaction di usion equation implicit semi implicit nite dif ference method numerical stability convergence analysis AbstractThe well known di usion equation has several applications in di erent elds of science and engineer ing Several important processes in image processing such as denoising inpainting stereo vision oroptical ow exploit linear or nonlinear and real or complex di usion processes The method of nitedi erence is undoubtedly one of the most successful and powerful numerical methods for solving par tial di erential equations PDEs involving di usion term Stability and convergence analysis of a nite di erence scheme is an interesting issue for scientists and mathematicians The nite di erencemethod FDM has a long history and we mention some of more interesting works Convergenceof FDMs for systems of nonlinear reaction di usion equations with real variables was studied byHo in 1978 For the complex case we refer to Wang s works where he considered the analysis ofsome conservative schemes for a coupled nonlinear Schrodinger system in 2010 Although the sta bility condition and the numerical analysis of nite di erence schemes for real nonlinear di usionand reaction di usion equations has been investigated extensively and is widely documented in theliterature rigorous proof of the stability and convergence of nite di erence schemes for the generalnonlinear complex reaction di usion equations refers to some works of Ara jo et al which beganfrom 2014 In this thesis we aim to investigate some recent works of Ara jo et al related to a general nonlinearcomplex reaction di usion equation and proof of the stability and convergence properties of a class of nite di erence schemes applied to them For this purpose at rst some preliminaries which will beused in the sequel are prepared Some important preliminaries are some useful inequalities such asCauchy Schwarz inequality Young s inequality Gronwall inequality Poincar and Poincar Friedrichsinequalities and etc Then we consider a nonlinear complex reaction di usion equation with an initialcondition and some Dirichlet or Neumann boundary conditions After that by separating real andimaginary parts of the PDE a coupled reaction di usion system of real variables is obtained and forsolving it some implicit and semi implicit methods are constructed Then some stability conditionsfor implicit and semi implicit methods are represented and several particular cases are highlighted Following that the accuracy of the numerical solution considering implicit and semi implicit dis cretizations is investigated and some properties of approximate solutions such as error estimates andrates of convergence are dealt with by establishing a variational system for the error between theexact and numerical solutions Finally for con rming the theoretical results several various examplesare solved by using implicit semi implicit schemes applied to the nonlinear complex reaction di usionequation with without reaction term and di erent boundary conditions For this purpose severalsuitable MATLAB codes are prepared Numerical results con rm theoretical results and show thatthe numerical order of convergence in time and space are approximately one and two respectively

لطيف التجار، آرزو

بررسي چند طرح تفاضل متناهي براي فرايندهاي واكنش - انتشار مختلط خطي/ غير خطي

روش تفاضلات متناهي ضمني/ نيمه ضمني , پايداري عددي , آناليز همگرايي