Wavelets and radial basis functions in scienti c computing

Abstract

The present work is an application of wavelets and radial basis functions to numerical computing. More specifically, we have used Haar and Legendre wavelet for applications of wavelets and multiquadric for applications of radial basis functions. The application areas considered in this thesis are the numerical solution of Integral Equations (IEs), various order Integrodifferential Equations (IDEs), systems of IEs, Elliptic Partial Differential Equations (EPDEs), Parabolic Partial Differential Equations (PPDEs) and highly oscillatory integrals. A few theoretical results are proved for efficient evaluation of some particular systems that arise when we apply one- or two-dimensional Haar wavelet in the wavelet collocation method. Based on these theoretical results new numerical methods based on Haar wavelet are developed for solution of IEs, IDEs and systems of IEs. EPDEs are solved numerically using collocation methods with Haar and Legendre wavelet. Legendre wavelet is also applied for the numerical solution of PPDEs. A new method based on multiquadric radial basis functions is introduced for numerical solution of highly oscillatory integrals. While applying Haar wavelet to numerical solution of IEs we have considered both nonlinear Fredholm and nonlinear Volterra IEs of the second kind. Similarly in case of IDEs a Haar wavelet based method is applied to find numerical solution of first and higher orders nonlinear Fredholm and nonlinear Volterra IDEs. The main advantage of this method is that it is generic as it can be applied to IEs, IDEs and systems of IEs. More specifically the new approach aims at the numerical solution of Fredholm, Volterra and Volterra-Fredholm types of IEs, IDEs and IDEs of higher orders including initial- as well as boundary-value problems. With a slight modification the method can also be applied to find numerical solution of two-dimensional IEs, system of IDEs and partial IDEs. Another distinguishing feature of themethod is that unlike many other existing methods in the literature it does not use any intermediate technique for numerical integration of the kernel function in IEs or IDEs. We have developed two new types of collocation methods based on Haar wavelet and Legendre wavelet for numerical solution of EPDEs. A modification of the collocation method based on Haar wavelet for elliptic differential equations is also introduced that improves the efficiency of the method. The collocation method based on Legendre wavelet is extended to find numerical solution of PPDEs. An advantage of the proposed methods is that it can be applied to different types of boundary conditions (BCs) with slight modifications. For highly oscillatory multidimensional integrals a new Levin’s type method based on multiquadric radial basis functions is developed. Levin method converts the numerical integration problem of highly oscillatory multidimensional integral to a PDE which is subsequently solved using meshless method. The proposed methods are validated on a variety of problems as well as numerical results of the proposed methods are compared with several existing methods from the literature. The numerical results show better performance of the proposed methods for several benchmark problems.