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In this manuscript, a numerical method based on the conjunction of Paraskevopoulos's algorithm and operational matrices is developed to solve numerically the multi-order linear and nonlinear fractional differential equations. By means of this conjunction, the multi-order problem is decomposed into a system of differential equations of fractional order which are then solved by employing operational matrices approach. The accuracy and efficiency of the method is examined by taking some examples. In addition, the numerical results presented in Pak et al. [Analytical solutions of linear inhomogeneous fractional differential equation with continuous variable coefficients. Adv Differ Equ. 2019;2019(256):1–22] are improved in our work. [ABSTRACT FROM AUTHOR]

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial [ABSTRACT FROM AUTHOR]

We show that any given function can be approximated with arbitrary precision by solutions of linear, time-fractional equations of any prescribed order. This extends a recent result by Claudia Bucur, which was obtained for time-fractional derivatives of order less than one, to the case of any fractional order of differentiation. In addition, our result applies also to the ψ-Caputo-stationary case, and it will provide one of the building blocks of a forthcoming paper in which we will establish general approximation results by operators of any order involving anisotropic superpositions of classical, space-fractional and time-fractional diffusions. [ABSTRACT FROM AUTHOR]

In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operatorwhereand the fractional derivatives,,are in the Caputo sense. Applying integral transform methods, we describe a complete family of eigenfunctions and fundamental solutions of the operatorin classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag–Leffler function. From the family of fundamental solutions obtained, we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper. [ABSTRACT FROM AUTHOR]

In this paper, a series of new high-order numerical approximations toα-th Caputo derivatives (0<α<2) is derived based on a compound of shift operators and high-order approximations to Riemann–Liouville derivatives. The convergence order is independent of the derivative orderα, rather than the previous error estimates. Several numerical examples including the Caputo-type advection–diffusion equation are displayed, which support the derived numerical schemes. [ABSTRACT FROM PUBLISHER]

In this paper, an implicit finite difference scheme for the nonlinear time-space-fractional Schrödinger equation is presented. It is shown that the implicit scheme is unconditionally stable with experimental convergence order ofO(τ2−α+h2), where τ andhare time and space stepsizes, respectively, and α (0<α<1) is the fractional-order in time. In order to reduce the computational cost, the explicit–implicit scheme is proposed such that the nonlinear term is easily treated. Meanwhile, the implicit finite difference scheme for the coupled time-space-fractional Schrödinger system is also presented, which is unconditionally stable too. Numerical examples are given to support the theoretical analysis. [ABSTRACT FROM PUBLISHER]

It is well known that, under standard assumptions, initial value problems for fractional ordinary differential equations involving Caputo-type derivatives are well posed in the sense that a unique solution exists and that this solution continuously depends on the given function, the initial value, and the order of the derivative. Here, we extend this well-posedness concept to the extent that we also allow the location of the starting point of the differential operator to be changed, and we prove that the solution depends on this parameter in a continuous way too if the usual assumptions are satisfied. Similarly, the solution to the corresponding terminal value problems depends on the location of the starting point and of the terminal point in a continuous way too. [ABSTRACT FROM AUTHOR]