Non-integer order derivatives

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2007Includes bibliographical references (leaves: 53-57)Text in English; Abstract: Turkish and Englishvii, 85 leavesThis thesis is devoted to integrals and derivatives of arbitrary order and applications of the described methods in various fields. This study intends to increase the accessibility of fractional calculus by combining an introduction to the mathematics with a review of selected recent applications in physics. It is described general definitions of fractional derivatives. This definitions are compared with their advantages and disadvantages. Fractional calculus concerns the generalization of differentiation and integration to non-integer (fractional) orders. The subject has a long mathematical history being discussed for the first time already in the correspondence of G. W. Leibnitz around 1690. Over the centuries many mathematicians have built up a large body of mathematical knowledge on fractional integrals and derivatives. Although fractional calculus is a natural generalization of calculus, and although its mathematical history is equally long, it has, until recently, played a negligible role in physics. In the first chapter, Grünwald-Letnikov approache to generalization of the notion of the differentation and integration are considered. In the second chapter, the Riemann Liouville definition is given and it is compared with Grünwald-Letnikov definition. The last chapter, Caputo.s definition is given. In appendices, two applications are given including tomography and solution of Bessel equation

Optimal approximation of fractional derivatives through discrete-time fractions using genetic algorithms

This study addresses the optimization of rational fraction approximations for the discrete-time calculation of fractional derivatives. The article starts by analyzing the standard techniques based on Taylor series and Padé expansions. In a second phase the paper re-evaluates the problem in an optimization perspective by tacking advantage of the flexibility of the genetic algorithms

Pole-zero approximations of digital fractional-order integrators and differentiators using signal modeling techniques

A novel strategy to the development of digital pole-zero approximations to fractional-order integrators and differentiators is presented here. The scheme is based in the signal modeling techniques applied to deterministic signals, namely the Padé, the Prony and the Shanks methods. It is shown that the illustrated algorithms yield good results both in the time and the frequency domains. Moreover, they are capable to give superior approximations than other existent approaches, namely the widely used CFE method. Several examples are given that demonstrate the effectiveness of the proposed techniques.N/

Seismic reverse-time migration in viscoelastic media

Seismic images are key to exploration seismology. They help identify structures in the subsurface and locate potential reservoirs. However, seismic images suffer from the problem of low resolution caused by the viscoelasticity of the medium. The viscoelasticity of the media is caused by the combination of fractured solid rock and fluids, such as water, oil and gas. This viscoelasticity of the medium causes attenuation of seismic waves, which includes energy absorption and velocity dispersion. These two attenuation effects significantly change the seismic data, and thus the seismic imaging. The aim of this thesis is to deepen the understanding of seismic wave propagation in attenuating media and to further investigate the method for high-resolution seismic imaging. My work, presented in this dissertation, comprises the following three parts. First, the determination of the viscoelastic parameters in the generalised viscoelastic wave equation. The viscoelasticity of subsurface media is succinctly represented in the generalised wave equation by a fractional temporal derivative. This generalised viscoelastic wave equation is characterised by the viscoelastic parameter and the viscoelastic velocity, but these parameters are not well formulated and therefore unfavourable for seismic implementation. The causality and stability of the generalised wave equation are proved by deriving the rate-of-relaxation function. On this basis, the viscoelastic parameter is formulated based on the constant Q model, and the viscoelastic velocity is formulated in terms of the reference velocity and the viscoelastic parameter. These two formulations adequately represent the viscoelastic effect in seismic wave propagation. Second, the development of a fractional spatial derivatives wave equation with a spatial filter. This development aims to effectively and efficiently solve the generalised viscoelastic wave equation with fractional temporal derivative, which is numerically challenging. I have transferred the fractional temporal derivative into fractional spatial derivatives, which can be solved using the pseudo-spectral implementation. However, this method is inaccurate in heterogeneous media. I introduced a spatial filter to correct the simulation error caused by the averaging in this implementation. The numerical test shows that the proposed spatial filter can significantly improve the accuracy of the seismic simulation and maintain high efficiency. Moreover, the proposed wave equation with fractional spatial derivatives is applied to compensate for the attenuation effects in reverse-time migration. This allows the dispersion correction and energy compensation to be performed simultaneously, which improves the resolution of the migration results. Finally, the development of reverse-time migration using biaxial wavefield decomposition to reduce migration artefacts and further improve the resolution of seismic images. In reverse-time migration, the cross-correlation of unphysical waves leads to large artefacts. By decomposing the wavefield both horizontally and vertically, and selecting only the causal waves for cross-correlation, the artefacts are greatly reduced, and the delicate structures can be identified. This decomposition method is also suitable for reverse-time migration with attenuation compensation. The migration results show that the resolution of the final seismic image is significantly improved, compared to conventional reverse-time migration.Open Acces

Least-squares design of digital fractional-order operators

In this paper we develop a method for obtaining digital rational approximations to fractional-order operators of type s^y, where y e R. The proposed method is based on the least-squares (LS) minimization between the impulse response of the fractional Euler/Tustin operators and the digital rational-fraction approximation. We make a comparison with other approaches and the results reveal that the LS method gives superior approximations. The effectiveness of the method is demonstrated both in the time and frequency domains through an illustrative example.N/

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8