School of Civil Engineering, Iran University of Science and Technology Narmak, Tehran, P.O. Box 16765-163, Iran
In this paper, the Mixed Discrete Least Squared Meshless (MDLSM) method is used for solving the quadratic partial differential equations (PDEs). In MDLSM method the domain is discretized only with nodes and a minimization of a least squares functional is carried out. The least square functional is defined as the sum of the residuals of the governing differential equation and its boundary condition at the nodal points. In MDLSM, the main unknown parameter and its first derivatives are approximated independently with the same Moving Least Squares (MLS) shape functions. The solution of the quadraticPDE does not, therefore, require the calculation of the complex second order derivatives of MLS shape functions. Furthermore, both the Neumann and Dirichlet boundary conditions can be treated and imposed as a Dirichlet type boundary condition which is applied using a penalty method. The accuracy and efficiency of the MDLSM method are tested against three numerical benchmark examples from one-dimensional and two-dimensional PDEs. The results are produced and compared with the irreducible DLSM method and exact analytical solutions indicating the ability and efficiency of the MDLSM method for the efficient and effective solution of quadratic PDEs.