Quasi-interpolation With Radial Basis Function And Application To Solve Partial Differential Equations

This dissertation consists of five chapters. In Chapter 1, we summarize the known numerical methods for solving the partial differential equations(PDE). Chapter 2 includes the preliminary knowledge. In this chapter, we introduce the theory of the radial basis function (RBF) and the hyperbolic partial differential equation. In addition to this, we summarize in detail the knowledge of the konwn four kinds of multiquadric(MQ) quasi-interpolation. Chapter 3 roots in our research. In which, we give the sufficient and necessary conditions for a kind of quasi-interpolation as it is polynomial reproducing. Furthermore, we construct a new univariate MQ quasi-interpolation which possesses the properties of the linear reproducing and preserving monotonicity. The numerical results show that it possesses higher accuracy. In Chapter 4, we discuss the application of the radial basis functions for numerical solving the partial differential equations (PDEs). In which, we give a detailed introduction for applying the MQ to solve PDEs. We investigate mainly solving the hyperbolic and parabolic equations by using the MQ quasi-interpolation. The underlying idea of our means is that: employing the derivative of the MQ quasi-interpolation to approximate the spatial derivative of the PDE, while the approach of the temporal derivative of the PDE is used a finite difference. Of course, we impose other techniques, such as, when we construct the numerical schemes, we import a function which is called "switch function" to damp the dispersion of the numerical schemes and so on. From the results of the numerical experiments given by us, we see that our method is valid. Chapter 5, named discussion. In this chapter, we give the elementary perspectives for...