Solving fractional differential equations by the ultraspherical integration method

In this paper, we present a numerical method to solve a linear fractional differential equations. This new investigation is based on ultraspherical integration matrix to approximate the highest order derivatives to the lower order derivatives. By this approximation the problem is reduced to a constrained optimization problem which can be solved by using the penalty quadratic interpolation method. Numerical examples are included to confirm the efficiency and accuracy of the proposed method.

A Hilbert Space Approach to Fractional Differential Equations

We study fractional differential equations of Riemann–Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $${\mathbb {R}}$$ R , we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existence and uniqueness of solutions and the causality of solution operators for non-linear fractional differential equations.

Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order α∈(0,1)∪(1,2). It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations.

Numerical Solutions Based on a Collocation Method Combined with Euler Polynomials for Linear Fractional Differential Equations with Delay

In this study, a method combined with both Euler polynomials and the collocation method is proposed for solving linear fractional differential equations with delay. The proposed method yields an approximate series solution expressed in the truncated series form in which terms are constituted of unknown coefficients that are to be determined according to Euler polynomials. The matrix method developed for the linear fractional differential equations is improved to the case of having delay terms. Furthermore, while putting the effect of conditions into the algebraic system written in the augmented form in which the coefficients of Euler polynomials are unknowns, the condition matrix scans the rows one by one. Thus, by using our program written in Mathematica there can be obtained more than one semi-analytic solutions that approach to exact solutions. Some numerical examples are given to demonstrate the efficiency of the proposed method.