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Fractionalization of a Class of Semi-Linear Differential Equations

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Document pages: 30 pages

Abstract: The dynamics of a fractionalized semi-linear scalar differential equation isconsidered with a Caputo fractional derivative. By using a symbolic operationalmethod, a fractional order initial value problem is converted into anequivalent Volterra integral equation of second kind. A brief discussion is includedto show that the fractional order derivatives and integrals incorporatea fading memory (also known as long memory) and that the order of the fractionalderivative can be considered to be an index of memory. A variation ofconstants formula is established for the fractionalized version and it is shownby using the Fourier integral theorem that this formula reduces to that of theinteger order differential equation as the fractional order approaches an integer.The global existence of a unique solution and the global Mittag-Lefflerstability of an equilibrium are established by exploiting the complete monotonicityof one and two parameter Mittag-Leffler functions. The method and theanalysis employed in this article can be used for the study of more generalsystems of fractional order differential equations.

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