روش گالركين نا پيوسته موضعي در حل برخي معادلات تحولي كسري- زماني

چكيده انگليسي :

Local discontinuous Galerkin method in solving some time fractional evolution equations Somayeh Fouladi s fouladi@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 AbstractThe basic idea behind fractional calculus has a history during three hundred years that is similar toand aligned with that of more classical calculus and the topic has attracted the interests of mathemati cians who contributed fundamentally to the development of classical calculus including L Hospital Leibniz Liouville Riemann Gr nward and Letnikov In spite of this the development and analysisof fractional calculus and fractional di erential equations are not as mature as that associated withclassical calculus However during the last decade this has begun to change as it has become clearthat fractional calculus naturally emerges as a model for a broad range of non classical phenomenain the applied sciences and engineering A striking example of this can be considered as a modelfor anomalous transport processes and di usion leading to partial di erential equations PDEs offractional type These models are found in a wide range of applications such as porous ows mod els of a variety of biological processes and transport in fusion plasmas to name a few With thisemerging range of applications and models based on fractional calculus a need for the development ofrobust and accurate computational methods for solving these equations will be necessary A funda mental di erence between problems in classical calculus and fractional calculus is the global nature ofthe latter formulations Nevertheless methods based on nite di erence methods and nite elementformulations have been developed and successfully applied The discontinuous Galerkin DG niteelement method is a very attractive method for solving PDEs because of its exibility and e ciencyin terms of mesh and shape functions and the higher order of convergence can be achieved withoutover many iterations The discontinuous Galerkin method is a well established method for classicalconservation laws However for equations containing higher order spatial derivatives discontinuousGalerkin methods cannot be directly applied A careless application of the discontinuous Galerkinmethod to a problem with high order derivatives could yield an inconsistent method The idea of localdiscontinuous Galerkin methods for a time dependent partial di erential equation with higher deriva tives is to rewrite the equation into a rst order system and then apply the discontinuous Galerkinmethod to the system A key ingredient for the success of this method is the correct design of in terface numerical uxes These uxes must be designed to guarantee stability and local solvabilityof all of the auxiliary variables introduced to approximate the derivatives of the solution In thisthesis after dealing with elementary concepts and explain a brief history of fractional calculus andmethods of DG and LDG we develop an implicit fully discrete local discontinuous Galerkin nite ele ment method for solving some time fractional equations such as time fractional Schr dinger equationand time fractional convection di usion equation and time fractional KBK equation This method isbased on a nite di erence method in time and the local discontinuous Galerkin method in space This development is based on the extensive work on DG for problems founded in classical calculus Stability is ensured by a careful choice of interface numerical uxes Unconditional stability of theschemes is investigated and some L2 error estimates are obtained

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روش گالركين نا پيوسته موضعي در حل برخي معادلات تحولي كسري- زماني

روش تفاضل متناهي , معادله شرودينگر كسري- زماني , معادله انتقال - انتشار كسري - زماني , معادله KBK كسري - زماني