The choosing of reproducing kernel particle shape function with mathematic proof

Many mechanical problems can be induced from differential equations with boundary conditions; there exist analytic and numerical methods for solving the differential equations. Usually it is not so easy to obtain analytic solutions. So it is necessary to give numerical solutions. The reproducing kernel particle (RKP) method is based on the Garlerkin Meshless method. According to the Sobolev space and Fourier transform, the RKP shape function is mathematically proved in this paper.