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Finite Difference Methods for Fractional Ordinary and Partial Differential Equations
Green, Charles Wing Ho
Green, Charles Wing Ho
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2026-03
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Doctoral thesis
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Abstract
This thesis investigates finite difference methods for solving fractional ordinary and partial differential equations. It begins with a comprehensive error analysis of the fractional Adams method applied to the Caputo-Hadamard fractional differential equation. New theoretical results are presented to illustrate how the smoothness of the solution and the regularity of the input data affect the convergence rates. Error estimates are derived, demonstrating that optimal convergence rates can be achieved using graded meshes when the solutions are non-smooth. The focus then shifts to the finite difference method for solving the time-fractional wave equation. A novel high-order scheme is developed to approximate the Riemann-Liouville fractional derivative of order α ∈ (1, 2), using the Hadamard finite-part integral combined with quadratic interpolation polynomial approximations. This scheme is employed to construct a new high-order method for approximating semi-linear fractional differential equations of the same order. By applying this method in the temporal direction and using the central difference method in the spatial direction, a fully discrete scheme is formulated for the time-fractional wave equation. Detailed error estimates and stability analyses are provided for the proposed schemes. It is shown that the numerical error admits an asymptotic expansion. Richardson extrapolation is then applied to enhance the convergence order without incurring significant additional computational cost. Numerical experiments are presented to validate the theoretical results and to demonstrate the accuracy and efficiency of the proposed schemes.
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Green, C. W. H. (2026). Finite Difference Methods for Fractional Ordinary and Partial Differential Equations [Unpublished doctoral thesis]. University of Chester.
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University of Chester
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Thesis or dissertation
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en