Development of parallel computing algorithms for special mathematical functions in fractional calculus: Implementation with GPU and CUDA

This work presents the development of parallel computing algorithms for special mathematical function used in fractional calculus. A systematic and step-by-step procedure is proposed for parallelization of computation of Gamma function, 1-, 2-, 3-, and 4-parameter Mittag-Leffler function, Confluent Hypergeometric function, and Generalized Hypergeometric function. These special functions play pivotal role in fractional calculus and in the solution of fractional-order differential equations. These functions are expressed using summation definition, and their computation is time consuming. The proposed procedure is based on parallelizing the mathematical expressions of these special functions. The algorithms are implemented in Compute Unified Device Architecture (CUDA) on a Graphics Processing Unit (GPU). It is shown that the implementation of these functions on a parallel computing platform accelerates their computation, achieving a substantial speed-up. From simpler functions like Gamma to more complex functions like Hypergeometric functions, the methodology presented in the work helps in their faster and accurate computation. The maximum speed-up is reported for the Confluent Hypergeometric function. Furthermore, the speed-up achieved is robust to parameter values of the functions. • Development of a parallel computing framework for the computation of special functions. • Implementation of parallel computation algorithms for special functions using the GPGPU platform. • Comparative analysis of parallel and sequential execution methodologies for a special function.