Abstarct: An extended meshfree method is presented for treating boundary value problems, where the total solution is expressed by a combination of particular and homogeneous solutions, each of which is assigned a specific role. The particular solution is any analytical (or numerical) exxpression satisfying the governing differential equation containing the source term but not necessarily the boundary conditions. A general method is presented for constructing this solution. Thus, the problem is reduced to a homogeneous equation where the original boundary conditions are modified by the particular solution. Herein, the meshfree method is used to solve this homogeneous equation, which involves a lower order behavior so that a relatively coarse discretization is acceptable. Several boundary value problems from potential theory as well as from shear deformable plate theory are solved, where linear exactness [Int. J. Numer. Methods Engrg. 50 (2001) 435; Int. J. Numer. Methods Engrg. 53 (2002) 2587] and bending exactness [Comput. Methods Appl. Mech. Engrg. 193 (2004) 1065], respectively, are imposed in their meshfree approximation fields. Numerical results demonstrate that this extended meshfree approach significantly improves the solution accuracy with commensurately less computational effort compared to the conventional meshfree formulation.