برخي از روش هاي پتروف - گالركين موضعي در حل مسائل انتشار- انتقال كسري

چكيده انگليسي :

Some Meshless Local Petrov Galerkin Methods in Solving Fractional Advection De usion Equations Monireh Mojtabaei m mojtabaie@math iut ac ir 20 09 2014 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari mokhtari@cc iut ac ir Advisor Dr Davoud Mirzaei d mirzaei@sci ui ac ir 2010 MSC 65M99 Keywords Moving least squares moving Kriging interpolation Petrov Galerkin fractional advection di usion equation Abstract Recently many new applications in engineering and science are governed by the fractional partialdi erential equations FPDEs If a fractional derivative is included in the governing equation theequation will be usually called the fractional partial di erential equation FPDE We consider afractional advection di usion equation FADE with the Caputo derivative among di erent types offractional derivative At present most of the fractional di erential equations are solved numericallyusing the nite di erence method FDM nite element method FEM and spectral approximation These methods lead to inherited issues including di culty in handling a complicated problem domain di culty in handling irregular nodal distribution and low accuracy In the recent years the developments of the meshless methods have been more attractive Thiskind of methods use a set of nodes scattered within the problem domain Therefore they have manyadvantages over the conventional numerical methods At present there are many meshless methodssuch as the element free Galerkin EFG method meshless local Petrov Galerkin MLPG method reproducing kernel particle method RKPM and so on In a MLPG method it can be used one ofthe meshfree approximations and any convenient test function for the solution process Atluri andZu have been examined six di rent realizations of the MLPG concept in 1998 The MLPG methodworks with a local weak form instead of a global weak which formulated over all local subdomains Schaback and Mirzaei could develop a MLPG method by using the generalized moving least squares GMLS and called it direct Meshless local Petrov Galerkin DMLPG method In this method thenumerical integrations over the moving least squares MLS shape functions have been replaced bymore accurate and cheaper integrations over polynomials Several approximation have been used forconstruction of the meshless shape functions such as the MLS approximation and the moving Kriginginterpolation MKI The MLS is accurate and stable for arbitrarily distributed nodes but it has notthe Kronecker delta property The Kriging interpolation has the Kronecker delta property as well asthe consistency property In this thesis we develop MLPG2 5 and DMLPG2 5 methods for the numerical simulation ofthe FADE For solving FADE by MLPG2 the test function is the Dirac Delta function So at rst a nite di erence scheme for temporal variable is proposed where the stability and convergence analysisare proved then MLS and MKI shape functions are developed for trial functions Then a new nitedi erences scheme is considered to deal with the Caputo fractional derivative Then for solving theFADE using the MLPG5 method the local weak forms of the equation are constructed For thispurpose the Heaviside test functions and MLS trial function are employed At the end The FADEis solved by DMLPG2 and DMLPG5 methods by a discretization similar to the methods of MLPG2and MLPG5 In almost all of numerical examples both regular and random Halton points are tested Finally these methods are compared with the methods of MLPG2 and MLPG5 and we conclude thatfor solving such problems the DMLPG method is faster and more accurate than the MLPG method

مجتبايي، منيره

برخي از روش هاي پتروف - گالركين موضعي در حل مسائل انتشار- انتقال كسري

كمترين مربعات متحرك , درونياب كريجينگ متحرك