This thesis deals with the methods for solving PDEs on the basis of the reproducing kernel Hilbert space. These methods are In the symbolic methods, the analytical solution is shown in a series form and the approximate solution is constructed by truncating the series to finite number of terms. We followed two approaches in working with the symbolic methods. The first one is to approximate the solution by orthonormal basis functions derived from the Gram-Schmidt orthogonalization process. The second one is to omitting the Gram-Schmidt process and finding the approximate solution through the system of normal equations based on the best approximation theory. We applied both approaches for solving the generalized regularized long wave (GRLW) equation, nonlinear differential-difference equations, inverse problems with nonlocal boundary conditions, a In the numerical methods, the Galerkin and collocation techniques are implemented as meshless methods based on spatial trial spaces spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. For the time-dependent PDEs they lead to a system of ordinary differential equations (ODEs). This is the well-known method of lines, and it turns out to be accurate in several problems. We implemented both Galerkin and collocation methods accompanied with the method of lines for solving two-dimensional nonlinear coupled Burgers' equations and Brusselator reaction-diffusion system.