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چكيده انگليسي :

A robust local polynomial collocation method for time independent problems Kobra Lali Dehaghi kobra lali@math iut ac ir 2014 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari rmokhtari@cc iut ac ir Advisor Dr Mehdi Tatari mtatari@sci ui ac ir 2010 MSC 65R20 45L05 Keywords Moving least square collocation method partiah derivatives equation for time inde pendent problem Abstract In the most traditional polynomial collocation methods polynomials are just applied to approxi mate the solution piecewise around the scattered nodes These methods with the moving least squares MLS approach are unstable and usually can work only for problems with simple boundary conditionsand regular geometry Polynomials have been rarely used globally such as the basis functions being treated in the radial basisfunctions collocation methods Weak form methods with MLS approach which need background cellsfor local integration perform with higher stability and are applied more widely in many numerical im plementations although they are sometimes unstable Study on improving the collocation techniquesand the construction of local clouds which means the selection of neighboring nodes involved in thelocal approximation is important issue for increasing stability Besides the aforementioned meth ods one can also nd a few literatures about meshless methods employing the reproducing kernelapproximation These methods with complicated formulation accurately approximate solutions ofsome partial di erential equations as a linear combination of the shape functions in a global way Although in the polynomial collocation methods with MLS approach the global solution is also thelinear combination of the shape functions the basic concept di ers from the reproducing kernel ap proximation so that the way for collocation is not the same In methods with reproducing kernelapproximation such as nite cloud methods the partial derivatives of the solution are obtained fromthe di erentiation of the global shape functions whereas in polynomial collocations methods withMLS approach they are obtained by di erentiating the local basis functions In this thesis a robustlocal polynomial collocation method which is based on the strong weak form is dealt with for solvingsome time independent problems This method is similar to the nite point method in which polyno mials are localized by putting their origins at the collocation points On the basis of the collocationapproach this method is as simple and straightforward as other analogous methods The satisfactionof the governing equation is additionally required on the boundary collocation points in this newlyproposed polynomial collocation method Therefore the method is developed in a way that not onlythe governing equation is satis ed on the boundaries but also the problem boundary conditions aresatis ed The sensitivity of the shape parameter the local supporting range of the shape functionsin the MLS approach and the convergence of the nodal resolution are studied by solving some testproblems Proposed method is also compared with some conventional methods such as xed kerneland Hermit approximations The robust local polynomial collocation method is further veri ed byapplying it to a steady state convection di usion problem Finally the present method is applied tocalculate the velocity elds of two potential ow problems Results obtained show that this methodis more accurate and robust than the conventional collocations methods especially in estimating thepartial derivatives of the solution near the boundary In fact accurate partial derivatives of the solu tion can be obtained in the process of seeking the solution This method can be further developed forsolving not only some complicated problems but also some time dependent problems

لالي دهقي، كبري

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