Development And Analysis Of Finite Difference Schemes For Solving Linear Time Fractional Diffusion And Advection Diffusion Equations

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dc.contributor.authorSaleh Mahmoud Al-Shibani, Faoziya
dc.date.accessioned2017-10-20T07:42:05Z
dc.date.available2017-10-20T07:42:05Z
dc.date.issued2014-10
dc.description.abstractIn recent decades, fractional partial differential equations have proved to be powerful and essential tools for modeling certain problems and have attracted the interest of scholars in areas such as physics, biology, fluid mechanics, engineering, finance and many others. Fractional diffusion and advection-diffusion equations describe anomalous diffusion process. They give a better understanding of anomalous phenomena which can be observed in certain complex systems that arise in biological cells, plasmas, epidemic spreading and several others. Most fractional differential equations cannot be solved analytically to yield the exact solution and so approximate analytical and numerical methods are used. In this thesis, the focus is on numerical methods. We construct finite difference methods for solving the one dimensional linear inhomogeneous time fractional diffusion and advection-diffusion equation with a linear source term. In particular, we formulate finite difference schemes of first, second and fourth order accuracy in space and first order accuracy in time. The time fractional derivative is Riemann-Liouville fractional derivative of order ∈ . The Grünwald-Letnikov formula is used to treat the time fractional derivative term in the time fractional diffusion and advection-diffusion equations. Compact finite difference methods are also introduced for these equations. In particular, compact finite difference schemes with three and five points which can provide accurate solutions are developed. The truncation errors of the finite difference methods developed are calculated. The stability and convergence of the schemes are analyzed by means of the Fourier method. Numerical experiments are conducted and the results indicate the feasibility and effectiveness of the finite difference schemes that have been developed.en_US
dc.identifier.urihttp://hdl.handle.net/123456789/5034
dc.language.isoenen_US
dc.publisherUniversiti Sains Malaysiaen_US
dc.subjectLinear Time Fractional Diffusion And Advection Diffusion Equationsen_US
dc.subjectGrünwald-Letnikov Formulaen_US
dc.titleDevelopment And Analysis Of Finite Difference Schemes For Solving Linear Time Fractional Diffusion And Advection Diffusion Equationsen_US
dc.typeThesisen_US
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