Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations

Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations

Abstract. We present an efficient algorithm for the evaluation of the Caputo fractional derivative C0D α t f(t) of order α ∈ (0, 1), which can be expressed as a convolution of f (t) with the kernel t. The algorithm is based on an efficient sum-of-exponentials approximation for the kernel t on the interval [∆t, T ] with a uniform absolute error ε, where the number of exponentials Nexp needed is ...

A numerical method for solving a class of distributed order time-fractional diffusion partial differential equations according to Caputo-Prabhakar fractional derivative

In this paper, a time-fractional diffusion equation of distributed order including the Caputo-Prabhakar fractional derivative is studied. We use a numerical method based on the linear B-spline interpolation and finite difference method to study the solutions of these types of fractional equations. Finally, some numerical examples are presented for the performance and accuracy of the proposed nu...

Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications

Fractional-order Legendre wavelets and their applications for solving fractional-order differential equations with initial/boundary conditions

In this manuscript a new method is introduced for solving fractional differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use fractional-order Legendre wavelets and operational matrix of fractional-order integration. First the fractional-order Legendre wavelets (FLWs) are presented. Then a family of piecewise functions is proposed, based on whi...

The operational matrix of fractional derivative of the fractional-order Chebyshev functions and its applications

In this paper, we introduce a family of fractional-order Chebyshev functions based on the classical Chebyshev polynomials. We calculate and derive the operational matrix of derivative of fractional order $gamma$ in the Caputo sense using the fractional-order Chebyshev functions. This matrix yields to low computational cost of numerical solution of fractional order differential equations to the ...