dc.contributor.advisor Yan, Yubin en dc.contributor.author Leedle, Natasha * dc.date.accessioned 2018-03-26T14:45:42Z dc.date.available 2018-03-26T14:45:42Z dc.date.issued 2017-10-09 dc.identifier.citation Leedle, N. (2017). A New Predictor-Corrector Method for Solving Nonlinear Fractional Differential Equations with Graded Meshes. (Masters thesis). University of Chester, United Kingdom. en dc.identifier.uri http://hdl.handle.net/10034/621031 dc.description.abstract In this dissertation we consider the numerical methods for solving non-linear fractional differential equations. We first review the predictor-corrector methods for solving the nonlinear fractional differential equation with uniform meshes and discussed in detail how to prove the error estimates. The convergence orders of the predictorcorrector methods for solving nonlinear fractional differential equations available in the literature are only O(h1+α ), where α ∈ (0, 1) denotes the fractional order and h is the step size. It will take a long time to obtain the good approximate solutions by using such method. Therefore it is necessary to construct some higher order numerical methods to solve the nonlinear fractional differential equations. We construct a higher order numerical method with the convergence order O(h1+2α) by approximating the Riemann-Liouville fractional integral with the quadratic interpolation polynomials. The graded meshes can be used in the numerical methods to capture the singularity of the problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results. 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 Predictor-corrector method en dc.subject Nonlinear fractional differential equation en dc.subject Graded mesh en dc.title A New Predictor-Corrector Method for Solving Nonlinear Fractional Differential Equations with Graded 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 the numerical methods for solving non-linear fractional differential equations. We first review the predictor-corrector methods for solving the nonlinear fractional differential equation with uniform meshes and discussed in detail how to prove the error estimates. The convergence orders of the predictorcorrector methods for solving nonlinear fractional differential equations available in the literature are only O(h1+α ), where α ∈ (0, 1) denotes the fractional order and h is the step size. It will take a long time to obtain the good approximate solutions by using such method. Therefore it is necessary to construct some higher order numerical methods to solve the nonlinear fractional differential equations. We construct a higher order numerical method with the convergence order O(h1+2α) by approximating the Riemann-Liouville fractional integral with the quadratic interpolation polynomials. The graded meshes can be used in the numerical methods to capture the singularity of the problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
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