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fractional differential equations (3)

distributed order differential equations (1)multi-term fractional differential equations (1)numerical approach (1)parallel algorithms (1)time-fractional diffusion equations (1)View MoreAuthorsBanks, Nicola E. (1)Connolly, Joseph A. (1)Ferras, Luis L. (1)Types
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PhD (3)Thesis or dissertation (3)

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The numerical solution of fractional and distributed order differential equations

Connolly, Joseph A. (University of Liverpool (University College Chester), 2004-12)

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.

Insights from the parallel implementation of efficient algorithms for the fractional calculus

Banks, Nicola E. (University of Chester, 2015-07)

This thesis concerns the development of parallel algorithms to solve fractional differential equations using a numerical approach. The methodology adopted is to adapt existing numerical schemes and to develop prototype parallel programs using the MatLab Parallel Computing Toolbox (MPCT). The approach is to build on existing insights from parallel implementation of ordinary differential equations methods and to test a range of potential candidates for parallel implementation in the fractional case. As a consequence of the work, new insights on the use of MPCT for prototyping are presented, alongside conclusions and algorithms for the effective implementation of parallel methods for the fractional calculus. The principal parallel approaches considered in the work include: - A Runge-Kutta Method for Ordinary Differential Equations including the application of an adapted Richardson Extrapolation Scheme - An implementation of the Diethelm-Chern Algorithm for Fractional Differential Equations - A parallel version of the well-established Fractional Adams Method for Fractional Differential Equations - The adaptation for parallel implementation of Lubich's Fractional Multistep Method for Fractional Differential Equations An important aspect of the work is an improved understanding of the comparative diffi culty of using MPCT for obtaining fair comparisons of parallel implementation. We present details of experimental results which are not satisfactory, and we explain how the problems may be overcome to give meaningful experimental results. Therefore, an important aspect of the conclusions of this work is the advice for other users of MPCT who may be planning to use the package as a prototyping tool for parallel algorithm development: by understanding how implicit multithreading operates, controls can be put in place to allow like-for-like performance comparisons between sequential and parallel programs.

Numerical Solution of Fractional Differential Equations and their Application to Physics and Engineering

Ferras, Luis L. (University of Chester, 2018-12-03)

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<sup>-α</sup>), 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.

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