Journal
JOURNAL OF SCIENTIFIC COMPUTING
Volume 82, Issue 2, Pages -Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10915-020-01150-y
Keywords
Mittag-Leffler functions; Fractional evolution equations; Rational approximation; Global Pade approximation; Matrix function
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The two-parametric Mittag-Leffler function (MLF), E-alpha,E-beta, is fundamental to the study and simulation of fractional differential and integral equations. However, these functions are computationally expensive and their numerical implementations are challenging. In this paper, we present a unified framework for developing global rational approximants of E-alpha,E-beta(-x), x>0, with {(alpha,beta):0<= 1,beta >=alpha,(alpha,beta)not equal(1,1)}. This framework is based on the series definition and the asymptotic expansion at infinity. In particular, we develop three types of fourth-order global rational approximations and discuss how they could be used to approximate the inverse function. Unlike existing approximations which are either limited to MLF of one parameter or of low accuracy for the two-parametric MLF, our rational approximants are of fourth order accuracy and have low percentage error globally. For efficient utilization, we study the partial fraction decomposition and use them to approximate the two-parametric MLF with a matrix argument which arise in the solutions of fractional evolution differential and integral equations.
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