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In herein,We investigate the application of space-time localized collocation method based on the radial basis function developed in [1] to solve some PDEs with variable coefficients and different boundary conditions.Using the method there is no need to any discretization time method as it is usually done using any order difference formula as implicit,explicit,θ-method,method-of-line approach and others.Considering a partial differential equation in d-dimension domain in space,the technique is based on solving the problem as(d+1)-dimension without making difference between space and time variables.The main originality of our paper is the use of a local formulation of the Localized RBF collocation method on the space-time domain to solve parabolic and hyperbolic equations.Another originality of our work is the application of the developed technique to a hyperbolic problem with variables coefficient,by considering them as ill-posed problems.The developed technique leads always to a square algebraic system.The efficiency and accuracy of the proposed method is demonstrated by solving different examples of parabolic and hyperbolic equation in one-and two-dimensional domain.