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fractional differential equations (20)

numerical methods (6)boundary value problems (2)multi-term equations (2)Adams method (1)Adams–Bashforth–Moulton scheme (1)Bagley-Torvik equation (1)Caputo derivative (1)distributed order differential equations (1)distributed order equations (1)View MoreJournalComputers and Fluids (1)Computers and mathematics with applications (1)Journal of Integral Equations and Applications (1)Nonlinear dynamics (1)Numerical algorithms (1)AuthorsFord, Neville J. (16)Diethelm, Kai (7)Simpson, A. Charles (5)Connolly, Joseph A. (3)Morgado, Maria L. (3)Ferras, Luis L. (2)Freed, Alan D. (2)Banks, Nicola E. (1)Edwards, John T. (1)Ford, Judith M. (1)View MoreTypesArticle (7)Preprint (5)Doctoral (3)PhD (3)Thesis or dissertation (3)View More

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Numerical solutions to the solution of some fractional differential equations

Simpson, A. Charles (Lea Press, 2002)

Numerical approaches to the solution of some fractional differential equations

Ford, Neville J.; Simpson, A. Charles (Lea Press, 2002)

The numerical solution of fractional differential equations: Speed versus accuracy

Ford, Neville J.; Simpson, A. Charles (Manchester Centre for Computational Mathematics, 2003-05-23)

This paper discusses the development of efficient algorithms for a certain fractional differential equation.

Detailed error analysis for a fractional Adams method

Diethelm, Kai; Ford, Neville J.; Freed, Alan D. (Springer, 2004-05)

This preprint discusses a method for a numerical solution of a nonlinear fractional differential equation, which can be seen as a generalisation of the Adams–Bashforth–Moulton scheme.

Comparison of numerical methods for fractional differential equations

Ford, Neville J.; Connolly, Joseph A. (American Institute of Mathematical Sciences/Shanghai Jiao Tong University, 2006-06)

This article discusses and evaluates the merits of five numerical methods for the solution of single term fractional differential equations.

A predictor corrector approach for the numerical solution of fractional differential equations

Diethelm, Kai; Ford, Neville J.; Freed, Alan D. (Springer, 2002-07)

This article discusses an Adams-type predictor-corrector method for the numerical solution of fractional differential equations.

Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries.

Ferras, Luis L.; Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; McKinley, Gareth H.; Nobrega, Joao M. (Elsevier, 2018-07-12)

In this work we discuss the connection between classical and fractional viscoelastic Maxwell models,
presenting the basic theory supporting these constitutive equations, and establishing some
background on the admissibility of the fractional Maxwell model. We then develop a numerical
method for the solution of two coupled fractional differential equations (one for the velocity and
the other for the stress), that appear in the pure tangential annular
ow of fractional viscoelastic fluids. The numerical method is based on finite differences, with the approximation of fractional
derivatives of the velocity and stress being inspired by the method proposed by Sun and Wu for
the fractional diffusion-wave equation [ Z.Z. Sun, X. Wu, A fully discrete difference scheme for a
diffusion-wave system, Applied Numerical Mathematics 56 (2006) 193-209]. We prove solvability,
study numerical convergence of the method, and also discuss the applicability of this method for
simulating the rheological response of complex fluids in a real concentric cylinder rheometer. By imposing a torsional step-strain, we observe the different rates of stress relaxation obtained with
different values of \alpha and \beta (the fractional order exponents that regulate the viscoelastic response
of the complex fluids).

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.

Numerical solution of multi-order fractional differential equations

Diethelm, Kai; Ford, Neville J. (Elsevier, 2004)

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.

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