1,720,995 research outputs found
Numerical approximation of matrix functions for fractional differential equations
No full textIn questo lavoro si presentano delle connessioni rilevanti tra le funzioni di matrice e la soluzione di equazioni differenziali di ordine frazionario. Questo nesso é stato notato solo recentemente ed ora riscuote notevole interesse. Si presenta qui una rassegna dei fondamenti del calcolo frazionario e della teoria dell'approssimazione di funzioni di matrice; si mostrano inoltre i contributiche, insieme ad i miei coautori, abbiamo recentemente elaborato su questo argomento [13, 14, 15, 16, 32]In this paper relevant insights are given on the connection between matrix functions and the solution of differential equations of fractional order. This nexus only recently has been disclosed and is gaining weight in the current research. We present here a review on the basics of fractional calculus and matrix function approximations, together with the main results my coauthors and me have given to the subject in the recent works [13,14,15, 16, 32
Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions
Full text linkMultiterm fractional differential equations (MTFDEs) nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods for fractional differential equations (FDEs) to this case. In this paper, we first transform the MTFDEs into equivalent systems of FDEs, as done by Diethelm and Ford; in this way, the solution can be expressed in terms of Mittag–Leffler (ML) functions evaluated at matrix arguments. We then propose to compute it by resorting to the matrix approach proposed by Garrappa and Popolizio. Several numerical tests are presented that clearly show that this matrix approach is very accurate and fast, also in comparison with other numerical methods
A matrix approach for partial differential equations with Riesz space fractional derivatives
No full textFractional partial differential equations are emerging in many scientific fields and their numerical solution is becoming a fundamental topic. In this paper we consider the Riesz fractional derivative operator and its discretization by fractional centered differences. The resulting matrix is studied, with an interesting result on a connection between the decay behavior of its entries and the short memory principle from fractional calculus. The Shift-and-Invert method is then applied to approximate the solution of the partial differential equation as the action of the matrix exponential on a suitable vector which mimics the given initial conditions. The numerical results confirm the good approximation quality and encourage the use of the proposed approach
On the Matrix Mittag–Leffler Function: Theoretical Properties and Numerical Computation
Full text linkMany situations, as for example within the context of Fractional Calculus theory, require computing the Mittag–Leffler (ML) function with matrix arguments. In this paper, we collect theoretical properties of the matrix ML function. Moreover, we describe the available numerical methods aimed at this purpose by stressing advantages and weaknesses
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