Applications Of Orthogonal Spline And Quasi-Wavelets Collocation Methods On Numerical Solution Of Fractional PDES

Because of lack of application back ground, fractional calculus was devel-oped very slowly. However,in the last few decades many authors pointed out that fractional calculus are very suitable for the description of memory and hereditary properties of various materials and processes. And fractional differ-ential equation have been used to simulate many phenomena in physics, biology, chemistry, engineering thermodynamics, fluid dynamics, heat transfer, mechan-ics of materials, environmental sciences, finance and other science. However numerical methods and theoretical analysis of fractional equations are very dif-ficult tasks. Theoretical analysis is different with classical numerical method. This motivates us to develop effective numerical methods and theoretical anal-ysis for the fractional differential equations.In this thesis, we study the applications of orthogonal spline and quasi-wavelets collocation methods on numerical solution of fractional PDES, respec-tively. It is composed of five chapters, which are in dependent and correlative to one another. The first chapter briefly reviews the definitions of fractional calculus and further analyzes some of their properties and some notation and basic lemmas of orthogonal spline collocation method and quasi-wavelets collo-cation method. The chapters2,3and4are the central part of our thesis. The last chapter is the conclusion of the present thesis.Sub-diffusion is perhaps the most frequently studied complex problem, and all of these equations have important background of practice applications, such as dispersion in fractals and porous media, semiconductor Physics, turbulence and condensed matter physics. In Chapter2, the aim is to develop a novel numerical techniques-orthogonal spline collocation (OSC)-for the solution of the two-dimensional fractional sub-diffusion equation. The proposed technique is based on OSC method in space and a finite difference method (FDM) in time. Stability and convergence of the proposed method are rigourously discussed and theoretically proven. We present the results of numerical experiments in one and two space variables, which confirm the predicted convergence rates and exhibit optimal accuracy in various norms. The result of this chapter has been published on Journal of Computational Physics.In Chapter3, we propose a novel numerical scheme-quasi-wavelet method to study the initial-boundary value problem of the fourth order fractional PDES. The main idea of quasi-wavelet algorithm:based on Mallat multi-resolution analysis, arbitrary wavelet subspace can be standardized by a group of or-thogonal wavelet generation, this group of wavelet-base can be regularized by its own corresponding combination of orthogonal scaling function to be nor-malized. However, standardization of the general orthogonal scaling function of the Fourier transform is not continuous, so there is no good standardized orthogonal local features, which is not conductive to numerical calculation. In order to improve standardization of orthogonal scaling function and grad-ual localization of features, we have its regular treatment. This is the main idea of quasi-wavelet algorithm. We use the forward Euler scheme for time discretization and the Quasi-Wavelets based numerical method for space dis-cretization. Detailed discrete formulations are given to the treatment of three different boundary conditions, including clamped type condition, simply sup-ported type condition and a transversely supported type condition. Some nu-merical experiments are included to demonstrate the validity and applicability of the discrete technique. The comparisons of present results with analytical solutions show that the Quasi-Wavelets based numerical method has distinctive local property. Especially, the method is easy to implement and produce very accurate results. The result of this chapter has been published on International Journal of Computer Mathematics.Our interest in the Chapter4is a continuation of the investigation in Chap-ter3, in this Chapter, we modify the scheme discretization in time. In the time direction, a Crank-Nicolson time-stepping is used to approximate the differen-tial term and the product trapezoidal method is employed to treat the inte-gral term, and the quasi-wavelets numerical method for space discretization. The comparisons of present results with the previous chapter3show that the present scheme is more stable and efficient for numerically solving the fourth order fractional PDES. Besides, for the high-frequency oscillation problems, this new method is very efficient and superior. In order to demonstrate the power of the quasi-wavelet method in comparison with standard discretization methods we also consider the high-frequency oscillation problems with the the integro-differential term. Moreover, it is easy to see that its code is easy to implement and more remarkable. The result of this chapter has been published on Journal of Computational Physics.