dc.contributor.advisor Yan, Yubin dc.contributor.author Khan, Monzorul dc.date.accessioned 2021-03-18T09:13:18Z dc.date.available 2021-03-18T09:13:18Z dc.date.issued 2020-03 dc.identifier https://chesterrep.openrepository.com/bitstream/handle/10034/624359/PhD%20Dissertation%20%28Completed%29%20%281%29.pdf?sequence=1 dc.identifier.citation Khan, M. (2020). Numerical methods for deterministic and stochastic fractional partial differential equations (Doctoral dissertation). University of Chester, United Kingdom. en_US dc.identifier.uri http://hdl.handle.net/10034/624359 dc.description.abstract In this thesis we will explore the numerical methods for solving deterministic and stochastic space and time fractional partial differential equations. Firstly we consider Fourier spectral methods for solving some linear stochastic space fractional partial differential equations perturbed by space-time white noises in one dimensional case. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space fractional partial differential equations. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. en_US Secondly we consider Fourier spectral methods for solving stochastic space fractional partial differential equation driven by special additive noises in one dimensional case. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. The space-time noise is approximated by the piecewise constant functions in the time direction and by appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equation is then solved by using Fourier spectral methods. Thirdly, we will consider the discontinuous Galerkin time stepping methods for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in t of degree at most q−1, q ≥ 1, which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given. Finally, we consider error estimates for the modified L1 scheme for solving time fractional partial differential equation. Jin et al. (2016, An analysis of the L1 scheme for the subdiffifusion equation with nonsmooth data, IMA J. of Number. Anal., 36, 197-221) ii established the O(k) convergence rate for the L1 scheme for both smooth and nonsmooth initial data. We introduce a modified L1 scheme and prove that the convergence rate is O(k2−α=), 0 < α < 1 for both smooth and nonsmooth initial data. We first write the time fractional partial differential equations as a Volterra integral equation which is then approximated by using the convolution quadrature with some special generating functions. A Laplace transform method is used to prove the error estimates for the homogeneous time fractional partial differential equation for both smooth and nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results. dc.language.iso en en_US dc.publisher University of Chester en_US dc.rights Attribution-NonCommercial-NoDerivatives 4.0 International * dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ * dc.subject numerical methods en_US dc.subject differential equations en_US dc.subject Fourier spectral methods en_US dc.title Numerical methods for deterministic and stochastic fractional partial differential equations en_US dc.type Thesis or dissertation en_US dc.type.qualificationname PhD en_US dc.type.qualificationlevel Doctoral en_US
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