1,721,001 research outputs found
A Computational Approach to Exponential-Type Variable-Order Fractional Differential Equations
Full text linkWe investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterization of such operators is performed in the Laplace domain, it is necessary to resort to accurate numerical methods to derive the corresponding behaviours in the time domain. In this regard, we develop a computational procedure to solve variable-order fractional differential equations of this novel class. Furthermore, we provide some numerical experiments to show the effectiveness of the proposed technique
A computationally efficient strategy for time-fractional diffusion-reaction equations
No full textAn efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term. The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices. Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper
Stability of fractional-order systems with Prabhakar derivatives
No full textFractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields) showing a simultaneous nonlocal and nonlinear behaviour. In this paper we study the asymptotic stability of systems of differential equations with the Prabhakar derivative, providing an exact characterization of the corresponding stability region. Asymptotic expansions (for small and large arguments) of the solution of linear differential equations of Prabhakar type and a numerical method for nonlinear systems are derived. Numerical experiments are hence presented to validate theoretical findings
An Easy-To-Use Tool to Solve Differential Equations with the Fractional Laplacian
No full textNumerical simulation of fractional-order partial differential equations is a challenging task and the majority of computing environments does not provide support for these problems. In this paper we describe how to exploit some of the Matlab features (a programming language not supporting fractional calculus in a naive way) to solve partial differential equations with the spectral fractional Laplacian. For shortness we focus on fractional Poisson equations but the proposed approach can be extended, with just some technical difficulties, to more involved problems. This approach cannot be considered as a highly efficient and accurate way to solve fractional partial differential equations, but as an easy-to-use tool for non specialists in numerical computation to obtain solutions without having to produce sophisticated numerical codes
Evaluation of fractional integrals and derivatives of elementary functions: Overview and tutorial
Full text linkSeveral fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann-Liouville and Caputo's derivatives converge, on long times, to the Grunwald-Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications
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