dc.contributor.advisor Yan, Yubin en dc.contributor.author Broadbent, Emma * dc.date.accessioned 2018-03-26T14:46:21Z dc.date.available 2018-03-26T14:46:21Z dc.date.issued 2017-10 dc.identifier.citation Broadbent, E. (2017). Numerical Methods for Solving Nonlinear Fractional Differential Equations with Non-Uniform Meshes. (Masters thesis). University of Chester, United Kingdom. en dc.identifier.uri http://hdl.handle.net/10034/621033 dc.description.abstract In this dissertation, we consider numerical methods for solving fractional differential equations with non-uniform meshes. We first introduce some basic definitions and theories for fractional differential equations and then we consider the numerical methods fro solving fractional differential equation. In the literature, the popular numerical methods for solving fractional differential equation include the rectangle method, trapezoid method and predictor-corrector methods. We reviewed such methods and the ways to prove the stability and the error estimates for these methods. Since the fractional differential equation is a nonlocal problem, the computation cost is very long compared with the local problem. Therefore it is very important to design some higher order numerical methods for solving fractional differential equation. In this dissertation, we introduce a new higher order numerical method for solving fractional differential equation which is based on the quadratic interpolation polynomial approximation to the fractional integral. To capture the singularity near the origin we also introduce the non-uniform meshes. The numerical results show that the optimal convergence order can be recovered by using non-uniform meshes even if the data are not sufficiently smooth. dc.language.iso en en dc.publisher University of Chester en dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ * dc.subject fractional differential equations en dc.subject Caputo derivative en dc.subject non-uniform meshes en dc.title Numerical Methods for Solving Nonlinear Fractional Differential Equations with Non-Uniform Meshes en dc.type Thesis or dissertation en dc.type.qualificationname MSc en dc.type.qualificationlevel Masters Degree en html.description.abstract In this dissertation, we consider numerical methods for solving fractional differential equations with non-uniform meshes. We first introduce some basic definitions and theories for fractional differential equations and then we consider the numerical methods fro solving fractional differential equation. In the literature, the popular numerical methods for solving fractional differential equation include the rectangle method, trapezoid method and predictor-corrector methods. We reviewed such methods and the ways to prove the stability and the error estimates for these methods. Since the fractional differential equation is a nonlocal problem, the computation cost is very long compared with the local problem. Therefore it is very important to design some higher order numerical methods for solving fractional differential equation. In this dissertation, we introduce a new higher order numerical method for solving fractional differential equation which is based on the quadratic interpolation polynomial approximation to the fractional integral. To capture the singularity near the origin we also introduce the non-uniform meshes. The numerical results show that the optimal convergence order can be recovered by using non-uniform meshes even if the data are not sufficiently smooth.
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