Fractional calculus is the generalization of integer-order calculus to rational order. This subject has at least three hundred years of history. However, it was traditionally regarded as a pure mathematical field and lacked real world applications for a very long time.
In recent decades, fractional calculus has re-attracted the attention of scientists and engineers. For example, many researchers have found that fractional calculus is a useful tool for describing hereditary materials and processes. It has been used to model the properties of viscoelastic materials and anomalous diffusion. Other applications of fractional calculus include signal processing, control of dynamic system, fractal theory, finance.
In this thesis, we have investigated several applications of fractional calculus and the use of multi-core hardware architecture for solving fractional differential equations. Within heat theory, we have studied fractional generalized Cattaneo equations and pointed out that the fractional heat equations may give negative absolute temperatures. Related to elastography, we have investigated the use of a fractional wave equation to describe the shear propagation induced by radiation force. We have concluded that there is a possibility of biased estimation of shear modulus. Numerical simulation of fractional partial differential equations is a time-consuming task due to the non-local property of fractional derivatives. We have shown that optimization techniques and parallel computing can reduce the long simulation time. We have also developed performance models which can give deep understanding of the optimization techniques and predict the simulation time of both serial and parallel implementations. Last but not least, we have demonstrated that parallel solvers of three-dimensional time-fractional diffusion problems are well suited for cutting-edge parallel hardware.
List of papers I. - V. Papers III. to V. are removed due to publisher copyright restrictions
I. W. Zhang and X. Cai, "Efficient implementations of the Adams-Bashforth-Moulton method for solving fractional differential equations", Proceedings of the 5th International Federation of Automatic Control Symposium on Fractional Differentiation and its Applications, Nanjing, 2012.
II. W. Zhang, W. Wei and X. Cai, "Performance modeling of serial and parallel implementations of the fractional Adams-Bashforth-Moulton method", Fractional Calculus and Applied Analysis, Volume 17, Number 3, 2014. doi:10.2478/s13540-014-0189-x.
III. W. Zhang, X. Cai and S. Holm, "Time-fractional heat equations and negative absolute temperatures", Computers and Mathematics with Applications, Volume 67, Number 1, Pages 164-171, 2014. doi:10.1016/j.camwa.2013.11.007.
IV. W. Zhang and S. Holm, "Estimation of shear modulus in media using the fractional Kelvin-Voigt model", submitted to Computers and Mathematics with Applications.
V. W. Zhang and X. Cai, "Solving 3D time-fractional diffusion equations by high-performance parallel computing", submitted.