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چكيده انگليسي :

Local discountinuous Galerkin method in solving some space fractional evolution equations Safoura Nouri s nouri@math iut ac ir 2013 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari mokhtari@cc iut ac irAdvisor Dr Mehdi Tatari mtatari@cc iut ac ir2013 MSC 65M12 65M60Keywords Local discontinuous Galerkin methods fractional derivative and integral space fractionaldi usion equation space fractional convection di usion equation AbstractVery recently the eld of fractional calculus a mathematical topic developed in the 17th century at tracts much attention and has been extensively investigated and applied for many real problems inrheology and mechanical systems and other areas of applications Since most fractional partial dif ferential equations do not have exact analytical solutions approximation and numerical techniquesmust be used Solving such equations and numerical schemes for fractional di erential equations havebeen stimulated due to their numerous applications in the areas of physics and engineering Withthis emerging range of applications and models based on fractional calculus a need for the develop ment of robust and accurate computational methods for solving these equations will be necessary A fundamental di erence between problems in classical calculus and fractional calculus is the globalnature of the latter formulations Nevertheless methods based on nite di erence methods and niteelement formulations have been developed and successfully applied Discontinuous Galerkin DG methods are a class of nite element methods using completely discontinuous basis functions whichare usually chosen as piecewise polynomials Since the basis functions can be completely discontin uous these methods have the exibility which is not shared by typical nite element methods Aclass of DG methods for solving time dependent partial di erential equations PDEs with higherderivatives are termed local discontinuous Galerkin LDG methods The idea of LDG methods isto suitably rewrite a higher order PDE into a rst order system then apply the DG method to thesystem A key ingredient for the success of such methods is the correct design of interface numerical uxes These uxes must be designed to guarantee stability and local solvability of all of the auxiliaryvariables introduced to approximate the derivatives of the solution In this thesis at rst we introducesome notation and useful subjects for de nition of DG mehods then we present the basic formulationof the discontinuous Galerkin method for scalar conservation laws We implement a LDG methodfor the fractional di usion problem characterized by having fractional derivatives parameterized by 1 2 We show through analysis that one can construct a numerical ux which results in a schemethat exhibit optimal order of convergence O hk 1 in the continuous range between pure advection 1 and pure di usion 2 and also we implement LDG methods for the fractional convection di usion with a fractional operator of order 1 2 de ned through the fractional Laplacian The fractional operator of order is expressed as a composite of rst order derivatives and fractionalintegrals of order 2 and the fractional convection di usion problem is expressed as a system oflow order di erential integral equations and a local discontinuous Galerkin method scheme is derivedfor the equations We also investigate the numerical stability and convergence of the methods

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روش هاي گالركين ناپيوسته ي موضعي در حل برخي از معادلات تحولي كسري مكاني

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