Partial differential equations are needed in most of the engineering fields. Analytical solutions to these equations cannot be derived except in some very special cases, making numerical methods more important. Alongside advances in science and technology, new methods have been proposed for solution of partial differential equations, such as meshless methods. Recently, the generalized exponential basis function (GEBF) meshless method has been introduced. In this method the unknown function is approximated as a linear combination of exponential basis functions. In linear problems, the unknown coefficients are calculated such that the homogenous form of main differential equation is satisfied in all points of the grid. In order to solve nonlinear equations, Newton-Kantorovich scheme is first used to linearize them. The linearized equations are then solved iteratively to obtain the result. In this paper, time dependent problems in solid mechanics have been investigated. In order to examine performance of the proposed method, linear and non-linear problems in solid mechanics are considered and the results are compared with analytical solutions. The results show good accuracy (less than 1 percentage error) of the presented method.