In this thesis, we present an expanded account of the numerical solutions of KdV equation using collocation and radial basis functions, based on an article by Dag et al. (2007). This equation is a generic equation for the study of weakly nonlinear long waves. It is an important nonlinear evolution equation with numerous applications in Physics and Engineering. Its analytical solutions have been found in special cases. Hence, finding the numerical solutions of KdV equation is essential. The numerical solutions of the evolution (partial differential) equations can be found by using the techniques known as the method of finite element, finite difference or finite volume. The meshfree or meshless methods try to circumvent the cumbersome issue of the mesh generation. Remarkable appearance of meshless methods returns to the last decade. They have been applied rapidly in finding numerical solutions of partial differential equations, ordinary differential equations and etc. One of the meshless methods is due to the pioneering effort of Kansa who directly collocated the radial basis functions (RBFs) for obtaining the approximate solutions of the equations. Because of the several advantages in comparison with the traditional methods, the Kansa's method which is known as the unsymmetric RBF collocation method, has been applied successfully to obtain numerical solutions of various types of ordinary/partial differential equations. These methods are very simple to implement because they are truly meshless in the sense that the collocation points need not have any connectivity requirement as needed in the traditional methods. The structure of this thesis is as follows: I chapter one, we gather a brief history of KdV equation, definition of solitons and an introduction to numerical solution of PDEs using RBF collocation. Numerical solutions of KdV equation are obtained by using RBF collocation with the aid of three different kinds of linearization in chapter two. Chapter three is devoted to explore numerical solutions of some special PDEs with the help of method of RBF collocation. In chapters two and three, accuracy of each scheme is tested by investigating the L 2 , L ? and RMS errors, conservative property of invariant quantities, propagation of solitons, interaction of solitary waves and breakdown of initial conditions into a train of solitons.