Obviously all physical phenomena are time-dependent. For engineering design it is necessary to predict the results of such phenomena by solving the mathematical models. The development of highly accurate and efficient solvers for time-dependent problems remains as an important and challenging research topic in computational physics, even though there are many numerical methods available for solving time dependent partial differential equations (PDEs). This thesis is an effort to use exponential basis functions (EBFs) for solving 2D time-dependent PDEs directly. The PDEs are assumed to be of constant-coefficient type. The main feature of the presented method is that the solution of the PDE is expressed as a function in space and time without using the routine schemes such as Laplace transformatin or finite difference method. To this end, the semi-analytical solution of the PDE is expressed as a series of EBFs. The constant coefficients of the solution series are determined form the initial/boundary conditions. In this thesis the initial and boundary conditions are satisfied at the same time in a collocation scheme by the use of a discrete transformation technique. To solve problems in a long period of time, a time marching method is used. In this method the problems are solved in a sequence of time intervals. This is performed by choosing a small time interval and repeating the procedure in a step by step manner while using the information obtained at the end of each time interval as the initial values for the next step. To show the robustness of the proposed method a verity of problems such as heat conduction, wave propagation in membrane, elastodynamic and poroelastodynamic problems are solved. It has been shown that the proposed method is capable of solving various initial boundary value problems efficiently. Key Words Initial boundary value problems, exponential basis functions, heat conduction,wave propagation in membrane, elastodynamic, porous media, Biot's theory, meshless method