GPU Computing of Special Mathematical Functions used in Fractional Calculus

Fractional calculus is a field with growing interest for researchers due to its applications in the various fields like mathematical modeling, control systems, image processing, financial systems, etc. Special mathematical functions like Gamma functions, Mittag-Leffler function, Hypergeometric function, etc. play a crucial role in the analytical and numerical solutions of fractional differential equations. However, the numerical computation of these functions is a tedious and time-consuming task. This is due to the fact that these functions do not have straightforward definitions and are mostly represented by series expansions. This chapter attempts to exploit the parallel computing power of Graphics Processing Unit (GPU) for computing some of the well-known special functions used in fractional calculus. Using the numerical computational platform of MATLAB and its parallel computing toolbox, the chapter reports the use of GPU hardware to compute these functions using their series definitions. It is shown with the help of various case studies and for different parameter combinations that the implementation of parallel computation of special functions reduces the execution time. A comparative study showing the effect of function parameters on their parallel computation is also presented.

Keywords: Fractional Calculus, Special Functions, Parallel computation, GPU Computing.