The current study introduces a fast and accurate method based on a novel adaptive meshfree technique to solve time-dependent partial differential equations with solutions representing high gradients or quick changes in several local areas of the domain. Utilizing uniform grids for these problems is prohibitive computationally, since the solution reaches singularity. This study aims to suggest an adaptive strategy to produce a suitable and cost-effective irregular node refinement. For this purpose, a dynamic algorithm is proposed that finds areas with quick changes and applies a local node adaptive approach merely in those nearly singular areas. Additionally, within this algorithm, unlike the Kansa technique, the radial basis function collocation technique was mixed with a finite difference scheme. According to this approach, in place of using the adaptive algorithm on the complete domain of the problem, it can be used only on time steps. Therefore, we need only to solve small systems of linear equations on each time step instead of large systems on the entire domain. Besides performing a stability analysis of the numerical scheme, the new algorithm is tested on parabolic (heat equation) and hyperbolic (wave equation) PDEs over regular and irregular two-dimensional domains. The attained results prove the accuracy and effectiveness of the proposed technique. Especially, our computational method is able to reduce the nodes in the domain with no impairment in terms of accuracy, thus turning out to be effective in the localization of oscillations owing to sharp gradients in the solution.
