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The Method of Regularized Sources(MRS)for computing axisymmetric incompressible Stokes flow problems with Dirichlet and Neumann boundary conditions is presented.MRS has a boundary meshless character.The delta function sources that govern the fundamental solution are replaced by a three dimensional source of the type εφ(p)=(15ε4/8π)(r2+ε2)-7/2 with ε representing a shape parameter that governs its width.From the shape of the rational source,an axisymmetric source is derived by integration of the related three dimensional source over symmetry coordinate.From this expression,the related axisymmetric Stokes flow and pressure around the source are calculated analytically.The nodes are put on the boundary and the solution is sought as a linear combination of the fields due to the regularized sources in the boundary nodes.The intensity of the regularized sources in the nodes of the boundary are adjusted by colocation of the flow magnitude as prescribed by the boundary conditions.No artificial boundary is needed.The strategy for development of extremely complicated analytical expressions,used in the axisymmetric version of the method is elaborated.