Optimal B-spline bases for the numerical solution of fractional differential problems

Efficient numerical methods to solve fractional differential problems are particularly required in order to approximate accurately the nonlocal behavior of the fractional derivative. The aim of the paper is to show how optimal B-spline bases allow us to construct accurate numerical methods that have a low computational cost. First of all, we describe in detail how to construct optimal B-spline bases on bounded intervals and recall their main properties. Then, we give the analytical expression of their derivatives of fractional order and use these bases in the numerical solution of fractional differential problems. Some numerical tests showing the good performances of the bases in solving a time-fractional diffusion problem by a collocation-Galerkin method are also displayed

A fractional B-spline collocation method for the numerical solution of fractional predator-prey models

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. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost

Bell-shaped nonstationary refinable ripplets

We study the approximation properties of the class of nonstationary refinable ripplets introduced in \cite{GP08}. These functions are solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions and convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the $C^2$ nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented. Keywords: total positivity, variation-dimishing, refinable ripplet, bell-shaped function, nonstationary prewavelet, nonstationary biorthogonal basisComment: 30 pages, 10 figure

A fractional wavelet Galerkin method for the fractional diffusion problem

The aim of this paper is to solve some fractional differential problems hav- ing time fractional derivative by means of a wavelet Galerkin method that uses the fractional scaling functions introduced in a previpous paper as approximating functions. These refinable functions, which are a generalization of the fractional B-splines, have many interesting approximation properties. In particular, their fractional derivatives have a closed form that involves just the fractional difference operator. This allows us to construct accurate and efficient numerical methods to solve fractional differential problems. Some numerical tests on a fractional diffusion problem will be given

An inversion method based on random sampling for real-time MEG neuroimaging

The MagnetoEncephaloGraphy (MEG) is a non-invasive neuroimaging technique with a high temporal resolution which can be successfully used in real-time applications, such as brain-computer interface training or neurofeedback rehabilitation. The localization of the active area of the brain from MEG data results in a highly ill-posed and ill-conditioned inverse problem that requires fast and efficient inversion methods to be solved. In this paper we use an inversion method based on random spatial sampling to solve the MEG inverse problem. The method is fast, efficient and has a low computational load. The numerical tests show that the method can produce accurate map of the electric activity inside the brain even in case of deep neural sources

A fractional spline collocation-Galerkin method for the time-fractional diffusion equation

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. Several numerical tests showing the effectiveness of the proposed method are presented.Comment: 15 page

Ternary shape-preserving subdivision schemes

We analyze the shape-preserving properties of ternary subdivision schemes generated by bell-shaped masks. We prove that any bell-shaped mask, satisfying the basic sum rules, gives rise to a convergent monotonicity preserving subdivision scheme, but convexity preservation is not guaranteed. We show that to reach convexity preservation the first order divided difference scheme needs to be bell-shaped, too. Finally, we show that ternary subdivision schemes associated with certain refinable functions with dilation 3 have shape-preserving properties of higher order

Neuroelectric source localization by random spatial sampling

The magnetoencephalography (MEG) aims at reconstructing the unknown neuroelectric activity in the brain from the measurements of the neuromagnetic field in the outer space. The localization of neuroelectric sources from MEG data results in an ill-posed and ill-conditioned inverse problem that requires regularization techniques to be solved. In this paper we propose a new inversion method based on random spatial sampling that is suitable to localize focal neuroelectric sources. The method is fast, efficient and requires little memory storage. Moreover, the numerical tests show that the random sampling method has a high spatial resolution even in the case of deep source localization from noisy magnetic data

Totally positive refinable functions with general dilation M

We construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M=4and M=5are also given

On the numerical solution of fractional boundary value problems by a spline quasi-interpolant operator

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. The unknown coefficients of the approximating operator are determined by a collocation method whose collocation matrices can be constructed efficiently by explicit formulas. The numerical experiments we conducted show that the proposed method is efficient and accurate