بررسي طرح هاي تفاضلي فشرده تركيبي در حل معادلات انتقال - انتشار كسري

چكيده انگليسي :

On the combined compact di erence schemes in solving time fractional advection di usion equations Mansooreh Ebad mansooreh ebad@math iut ac ir 11 01 2017 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari mokhtari@cc iut ac ir Advisor Dr Mehdi Tatari mtatari@cc iut ac ir 2010 MSC 65M 06 68U 10 Keywords combined compact di erence scheme time fractional advection di usion equa tion time fractional telegraph equation AbstractAs we know applications of fractional di erential equations in di erent science have inspired scientists tryingto nd out some high order and more e cient methods for solving these equations Fractional derivative isan extension of the ordinary derivative that someone can choose a non integer number as the derivative order For most time fractional advection di usion equations there is not any analytical solution therefore nding ane ective numerical solution is very important Several numerical schemes such as implicit and explicit methodshave been proposed to solve time fractional advection di usion equations and their stability and e ciency havebeen surveyed But those types of these methods are acceptable that can eventual modify conditions and pre pare ways to acquire more accurate approximate solutions One of these processes to solve fractional di erentialequations lead to construct some new nite di erence schemes which called combined compact nite di erence CCD schemes These high order schemes are obtained by combining rst and second and even higher order derivatives of the solution and are more accurate than classical nite deference schemes A CCD scheme has usu ally unconditional stability and can be sixth order in space for some problems with periodic boundary conditionsand can be fth order in space for some problems with other types of boundary conditions In this thesis at rst some de nitions and conceptions such as Gamma function for its fundamental role in fractional calculus arerepresented Furthermore the Grunewald Letnikov as well as the Riemann Liouville fractional derivatives andintegrals are rede ned Since we are interested in dealing with some problems involving the Caputo fractionalderivative we next explain fractional derivative of Caputo type with the well known L1 formula for approximatingit as well as its numerical error For solvinga time fractional advection di usion equation with the aid of a CCDscheme solution of the equation and its rst and second order derivatives in all nodes of the mesh are counted andconsidered as the unknown variables which must be determined More important characteristics of such schemes which specify their advantages are increasing order of convergence without increasing the number of involvednodes obtaining noticeable accuracy in some nodes neighbor the boundary imposing various types of boundaryconditions and investigating their stability analysis easily using the Fourier method Furthermore these methodsare usually more compact compared to other nite di erence methods and their accuracy is somewhat better In order to con rm theoretical results obtained for these schemes some numerical examples prepared and testedsuccessfully Finally as a research work we use a CCD scheme to solve some time fractional telegraph equationscontaining two Caputo fractional derivatives of order 1 and 2 For this purpose we construct a CCD schemaand investigate its stability and convergence analysis Convergence order of the proposed method isat least ve which depends on the type of the boundary conditions inspace and min 2 1 3 2 in time In order tocon rm theoretical results obtained for this scheme some numerical examples prepared and tested successfully

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بررسي طرح هاي تفاضلي فشرده تركيبي در حل معادلات انتقال - انتشار كسري - زماني

روش تفاضل متناهي فشرده تركيبي , معادله انتقال - انتشار كسري - زماني , معادله تلگراف كسري - زماني