Applying Multiquadric Quasi-interpolation To Solve KdV Equation

Korteweg-de Vries(KdV) is one kind of the important nonlinear partial differential equations(PDEs). This equation can be applied in many fields, so how to get it's solution has become one of the hot problems which are researched at home and abroad. The radial basis function(RBF) has played an important role in solving the partial differential equations recently. And the multiquadric(MQ) function is one important kind of the radial basis functions. In this thesis, the multiquadric(MQ) quasi-interpolation method is applied to find the numerical solution of the KdV equation, which is a third-order nonlinear equation. We discussed a numerical scheme. Since we do not require to solve any system of linear equations, so we can find our scheme is simple and easy to implement. From the results of the numerical examples, we conclude that our scheme is feasible and valid.There are four parts in this thesis:In chapter 1, we introduce the type of partial differential equations(PDEs) and the numerical methods for solving the partial differential equations briefly.In chapter 2, we introduce the theory of the radial basis function and four kinds of multiquadric(MQ) quasi-interpolation as the preliminary knowledge of this thesis.In chapter 3, as the basic theory of this thesis, we introduce the knowledge of the application of the radial basis functions for numerical solving the partial differential equa-tions. Besides, we introduce a kind of multiquadric(MQ) quasi-interpolation which is generalized from the LD operator. This operator is presented by Chen and Wu and used to solve the Burgers'equation by them.The chapter 4 is the core of the thesis. Chen and Wu presented a kind of multi-quadric(MQ) quasi-interpolation, which is generalized from the LD operator, and used it to solve the Burgers'equation. Based on Chen and Wu's method, a numerical scheme for solving the KdV equation is discussed in this thesis. The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative, and the forward divided difference to approximate the temporal derivative, where the spatial derivative is approximated by the derivative of the generalizedî–¸LD quasi-interpolation operator.In the end, we make the conclusions of this thesis. Our scheme is simple and easy to implement. And we can conclude that our scheme is feasible and valid. Moreover, we can improve our scheme in the future.