Since 1994, two main meshless methods have been developed and widely used: these are the element free Galerkin method and the meshless local Petrov-Galerkin method. Both methods solve partial differential equations by posing a numerical approximation to the solution using the moving least squares technique. Using Bernstein polynomials as the shape functions of Galerkin weak form-based methods improves the numerical approximation achieved at boundaries without losing accuracy inside the domain.