University of Macau / Faculty of Science and Technology. / Department of Mathematics

Lam, Wai Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 55-56). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

University of Macau / Faculty of Science and Technology / Department of Mathematics

This dissertation presents new numerical methods for the solution of fractional differential equations of single and distributed order that find application in the different fields of physics and engineering. We start by presenting the relationship between fractional derivatives and processes like anomalous diffusion, and, we then develop new numerical methods for the solution of the time-fractional diffusion equations. The first numerical method is developed for the solution of the fractional diffusion equations with Neumann boundary conditions and the diffusivity parameter depending on the space variable. The method is based on finite differences, and, we prove its convergence (convergence order of O(Δx² + Δt²^-α), 0 < α < 1) and stability. We also present a brief description of the application of such boundary conditions and fractional model to real world problems (heat flux in human skin). A discussion on the common substitution of the classical derivative by a fractional derivative is also performed, using as an example the temperature equation. Numerical methods for the solution of fractional differential equations are more difficult to develop when compared to the classical integer-order case, and, this is due to potential singularities of the solution and to the nonlocal properties of the fractional differential operators that lead to numerical methods that are computationally demanding. We then study a more complex type of equations: distributed order fractional differential equations where we intend to overcome the second problem on the numerical approximation of fractional differential equations mentioned above. These equations allow the modelling of more complex anomalous diffusion processes, and can be viewed as a continuous sum of weighted fractional derivatives. Since the numerical solution of distributed order fractional differential equations based on finite differences is very time consuming, we develop a new numerical method for the solution of the distributed order fractional differential equations based on Chebyshev polynomials and present for the first time a detailed study on the convergence of the method. The third numerical method proposed in this thesis aims to overcome both problems on the numerical approximation of fractional differential equations. We start by solving the problem of potential singularities in the solution by presenting a method based on a non-polynomial approximation of the solution. We use the method of lines for the numerical approximation of the fractional diffusion equation, by proceeding in two separate steps: first, spatial derivatives are approximated using finite differences; second, the resulting system of semi-discrete ordinary differential equations in the initial value variable is integrated in time with a non-polynomial collocation method. This numerical method is further improved by considering graded meshes and an hybrid approximation of the solution by considering a non-polynomial approximation in the first sub-interval which contains the origin in time (the point where the solution may be singular) and a polynomial approximation in the remaining intervals. This way we obtain a method that allows a faster numerical solution of fractional differential equations (than the method obtained with non-polynomial approximation) and also takes into account the potential singularity of the solution. The thesis ends with the main conclusions and a discussion on the main topics presented along the text, together with a proposal of future work.

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

This thesis explores higher order numerical methods for solving fractional differential equations.

Fractional Calculus can be thought of as a generalisation of conventional calculus in the sense that it extends the concept of a derivative (integral) to include non-integer orders. Effective mathematical modelling using Fractional Differential Equations (FDEs) requires the development of reliable flexible numerical methods. The thesis begins by reviewing a selection of numerical methods for the solution of Single-term and Multi-term FDEs. We then present: 1. a graphical technique for comparing the efficiency of numerical methods. We use this to compare Single-term and Multi-term methods and give recommendations for which method is best for any given FDE. 2. a new method for the solution of a non-linear Multi-term Fractional Dif¬ferential Equation. 3. a sequence of methods for the numerical solution of a Distributed Order Differential Equation. 4. a discussion of the problems associated with producing a computer program for obtaining the optimum numerical method for any given FDE.

Features inside the living cell are complex and crowded; in such complex environments diffusion processes can be said to exhibit three distinct behaviours: pure or Fickian diffusion, superdiffusion and subdiffusion. Furthermore, the behaviour of biochemical processes taking place in these environments does not follow classical theory. Because of these factors, the task of modelling dynamical proceses in complex environments becomes very challenging and demanding and has received considerable attention from other researchers seeking to construct a coherent model. Here, we are interested to study the phenomenon of subdiffusion, which occurs when there is molecular crowding. The Reaction Diffusion Partial Differential Equations (RDPDEs) approach has been used traditionally to study diffusion. However, these equations have limitations due to their unsuitability for a subdiffusive setting. However, I provide models based on Fractional Reaction Diffusion Partial Differential Equations (FRDPDEs), which are able to portray intracellular diffusion in crowded environments. In particular, we will consider a class of continuous spatial models to describe concentrations of molecular species in crowded environments. In order to investigate the variability of the crowdedness, we have used the anomalous diffusion parameter $\alpha$ to mimic immobile obstacles or barriers. We particularly use the notation $D_t^{1-\alpha} f(t)$ to represent a differential operator of noninteger order. When the power exponent is $\alpha=1$, this corresponds to pure diffusion and to subdiffusion when $0<\alpha<1$. This thesis presents results from the application of fractional derivatives to the solution of systems biology problems. These results are presented in Chapters 4, 5 and 6. An introduction to each of the problems is given at the beginning of the relevant chapter. The introduction chapter discusses intracellular environments and the motivation for this study. The first main result, given in Chapter 4, focuses on formulating a variable stepsize method appropriate for the fractional derivative model, using an embedded technique~\cite{landman07,simpson07,simpson06}. We have also proved some aspects of two fractional numerical methods, namely the Fractional Euler and Fractional Trapezoidal methods. In particular, we apply a Taylor series expansion to obtain a convergence order for each method. Based on these results, the Fractional Trapezoidal has a better convergence order than the Fractional Euler. Comparisons between variable and fixed stepsizes are also tested on biological problems; the results behave as we expected. In Chapter 5, analyses are presented related to two fractional numerical methods, Explicit Fractional Trapezoidal and Implicit Fractional Trapezoidal methods. Two results, based on Fourier series, related to the stability and convergence orders for both methods have been found. The third main result of this thesis, in Chapter 6, concerns the travelling waves phenomenon modeled on crowded environments. Here, we used the FRDPDEs developed in the earlier chapters to simulate FRDPDEs coupled with cubic or quadratic reactions. The results exhibit some interesting features related to molecular mobility. Later in this chapter, we have applied our methods to a biological problem known as Hirschsprung's disease. This model was introduced by Landman~\cite{landman07}. However, that model ignores the effects of spatial crowdedness in the system. Applying our model for modelling Hirschsprung's disease allows us to establish an interesting result for the mobility of the cellular processes under crowded environmental conditions.

This thesis is concerned with a particular numerical approach for solving the fractional partial differential equations (PDEs). In the last two decades, it has been observed that many practical systems are more accurately described by fractional differential equations (FDEs) rather than the traditional differential equation approaches. Consequently, it has become an important research area to study the theoretical and numerical aspects of various types of FDEs. This thesis will explore high order numerical methods for solving FDEs numerically. More precisely, spectral methods which exhibits exponential order of accuracy will be investigated. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss quadrature points. In this work, Hermite and modified rational functions are employed to serve as basis functions for solutions that decay exponentially and algebraically, respectively. The main emphasis of this thesis is to propose the spectral collocation method for FDEs posed in unbounded domains. Components of the differentiation matrix involving fractional Laplacian are derived which can then be computed recursively using the properties of confluent hypergeometric function or hypergeometric function. The first part of the thesis introduces preliminaries useful for other parts of the thesis. Review of the relevant definitions and properties of special functions such as Hermite functions, Bessel functions, hypergeometric functions, Gegenbauer polynomials, mapped Jacobi polynomials, modified rational functions are presented. Fractional Sobolev space is introduced and some lemmas on interpolation approximation in the fractional Sobolev space are also included. In the second part of the thesis, we present the spectral collocation method based on Hermite functions. Two bases are used, namely, the over-scaled Hermite function and generalized Hermite function, which are orthogonal functions on the whole line with appropriate weight functions. We will show that the fractional Laplacian of these two kinds of Hermite functions can be represented by confluent hypergeometric function. Behaviors of the condition numbers for the resulting spectral differentiation matrices with respect to the number of expansion terms are investigated. Moreover, approximation in two-dimensional space using the tensorized bases, application to multi-term problems and use of scaling to match different decay rate are also considered. Convergence analysis for generalized Hermite function are derived and numerical errors for two bases are analyzed. The third part of the thesis deals with the spectral collocation method based on modified rational functions. We first give a brief introduction for computation of the fractional Laplacian using modified rational functions, which is represented by hypergeometric functions. Then the differentiation matrix involving the fractional Laplace operator is given. Convergence analysis for modified Chebyshev rational functions and modified Legendre rational functions are derived and numerical experiments are carried out.