Ricerc@Sapienza

In many applications, real phenomena are modeled by differential problems having a time fractional derivative that depends on the history of the unknown function. For the numerical solution of time fractional differential equations, we propose a new method that combines spline quasi-interpolatory operators and collocation methods. We show that the method is convergent and reproduces polynomials of suitable degree. The numerical tests demonstrate the validity and applicability of the proposed method when used to solve linear time fractional differential equations.

Fractional Calculus is widely used to model real-world phenomena. In fact, the fractional derivative allows one to easily introduce into the model memory effects in time or nonlocality in space. To solve fractional differential problems efficient numerical methods are required. In this paper we solve the fractional oscillation equation by a collocation method based on refinable bases on the semi-infinite interval. We carry out some numerical tests showing the good performance of the method.

The aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The main advantage is in that the derivatives of integer and fractional order of the fractional splines can be expressed in a closed form that involves just the generalized finite difference operator. This allows us to construct an accurate and efficient numerical method.

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator.

We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional
derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage
of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale

Boundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative in space. We approximate the solution by the Schoenberg-Bernstein operator, which is a spline positive operator having shape-preserving properties.

We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.