| Radial basis function method is known as a powerful tool to handle large scaled scattered data problems. During the last decades, meshless radial basis functions method has attracted attentions of many researchers. In this dissertation, some meshless methods based on radial basis functions are discussed, moreover, some applications are presented.Chapter 1 presents the background of this dissertation, furthermore, the definition of radial basis functions space and its properties are introduced. For the further research, chapter 2 includes some conclusions about radial basis functions methods.By considering the relationship between integral transform and reproducing kernel Hilbert space, chapter 3 discusses the radial basis function approximation problem in Sobolev space. The construction of the radial kernel shows that our kernel is optimal in some sense.In chapter 4, the radial basis functions interpolation method is employed to handle a class of multi-dimensional parabolic inverse problems. By comparing the results of collocation method, we conclude that our method is more efficient and stable.Chapter 5 discusses the approximation property of Multiquadric (MQ) quasi-interpolation to the k-th derivatives. Furthermore, two kinds of MQ quasi-interpolation schemes on finite interval are presented, whose derivatives converge to the corresponding derivatives of the approximated functions. At the end of this chapter, the numerical experiments are presented to confirm the accuracy of the presented schemes. Theoretical results and numerical examples show that these schemes provide good accuracy even if the data points are irregularly distributed.To avoid encountering the large scaled system of equations, chapter 6 employs MQ quasi-interpolation method to solve partial differential equations. Experimental results show MQ quasi-interpolation is an efficient tool to solve partial differential equations numerically.Chapter 7 proposes some topics for further research.
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