Kernel-based Approximation for Partial Differential Equations of Fractional Order

Partial differential equations of fractional order have been the focus of many studies because of their various applications in fluid mechanics, viscoelasticity, biology, physics, and engineering. Fractional calculus in mathematics is a natural extension of integer-order calculus. It has been used for modelling many physical processes arisen from real-life problems, for instance, the modelling on the transport of passive tracers carried by fluid flows in a porous medium under groundwater hydrology. Studies of the complicated phenomena of the interstitial fluid flows are still under intensive researches and are particularly challenging for quantitative analyses and modelling. Some kinds of numerical techniques are indispensable for industrial and engineering applications.The aim of this project is to provide a theoretical background and justification on developing efficient computational methods for solving both well-posed and ill-posed partial differential equations of fractional order. To speed up the time of computations, modern numerical methods, mainly meshless kernel-based approximation methods combined with the methods of fundamental solutions and particular solutions; Laplace Transform and regularization techniques, will be employed and implemented.